error analysis in numerical solutions of various shock ... · error analysis in numerical solutions...

Error Analysis in Numerical Solutions of

Various Shock Physics Problems

A Dissertation Presented

by

Ming Zhao

to

The Graduate School in Partial Fulfillment of the Requirements for the

Degree of

Doctor of Philosophy

in

Applied Mathematics and Statistics

Stony Brook University

August 2005


The Graduate School

Ming Zhao

We, the dissertation committee for the above candidate for the Doctor ofPhilosophy degree, hereby recommend acceptance of this dissertation.

James GlimmAdviser

Department of Applied Mathematics and Statistics

Xiaolin LiChairman


Stephen FinchMember


David G. EbinOutside Member

Department of MathematicsStony Brook University

This dissertation is accepted by the Graduate School.

Graduate School

ii

Abstract of the Dissertation

Error Analysis in Numerical Solutions ofVarious Shock Physics Problems

by

Ming Zhao

Doctor of Philosophy

in

Applied Mathematics and Statistics


2005

Our purpose is to seek robust and understandable error models for shock

physics simulations. First, for 1D planar problems, we developed statistical

models for ensemble uncertainty and numerical errors. A composition law

was further formulated and validated to estimate errors in the solutions of

composite problems in terms of errors from simpler ones.

In a further study of spherically symmetric 1D shock interactions, the

error analysis is complicated by a nonconstant power law growth or decay of

error between interactions (the waves themselves are also non constant and

growing or decaying by power laws).

For shock interactions in spherical geometry, we conduct a detailed analy-

iii

sis of the errors. One of our goals is to understand the relative magnitude of

the input uncertainty vs. the errors created within the numerical solution. In

more detail, we wish to understand the contribution of each wave interaction

to the errors observed at the end of the simulation.

Key Words: uncertainty quantification, error model, composition law,

Riemann problem.

iv

To my loving parents Jixin Gu and Gai Zhao ...

Table of Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . xxi

1 Introduction and Background . . . . . . . . . . . . . . . . . . 1

1.1 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 1D Simulation Formulation . . . . . . . . . . . . . . . 6

1.2.2 Introduction to FronTier . . . . . . . . . . . . . . . . . 11

1.3 Organization of This Dissertation . . . . . . . . . . . . . . . . 12

2 Statistical Riemann Problems . . . . . . . . . . . . . . . . . . 15

2.1 A Multinomial Expansion for the SRP Output . . . . . . . . . 18

2.2 Evaluation of the Expansion Coefficients . . . . . . . . . . . . 21

2.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

vi

3 The Statistical Numerical Riemann Problems for Planar Flows

29

3.1 The Wave Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Isolated Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Interactions with Contact Waves . . . . . . . . . . . . . . . . 38

3.4 Shock Crossing Shock Interactions . . . . . . . . . . . . . . . . 43

3.5 The Contact Reshock Interactions . . . . . . . . . . . . . . . . 45

4 The Statistical Numerical Riemann Problems for Spherical

Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 The Single Propagating Wave . . . . . . . . . . . . . . . . . . 48

4.2 The Shock Contact Interaction . . . . . . . . . . . . . . . . . 52

4.3 Shock Reflection at the Origin . . . . . . . . . . . . . . . . . . 56

4.4 The Contact Reshock Interaction . . . . . . . . . . . . . . . . 57

5 Composite Shock Interaction Problems . . . . . . . . . . . . 61

5.1 Composite Shock Interaction Problems In Planar Flows . . . . 61

5.1.1 A Multipath Integral for a Nonlinear Multiscattering

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1.2 Evaluation of the Multipath Integral . . . . . . . . . . 63

5.1.3 Errors in Resolved Calculations . . . . . . . . . . . . . 69

5.1.4 Errors in Under Resolved Calculations . . . . . . . . . 69

5.2 Composite Shock Interaction Problems in Spherical Flows . . 71

5.2.1 The Multipath Error Analysis Fromula . . . . . . . . . 72

5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

vii

6 Conclusion And Future Work in 2D Chaotic Flows . . . . . 82

6.1 Conclusion from 1D Flow Study . . . . . . . . . . . . . . . . . 82

6.2 Future Work in 2D Chaotic Flows . . . . . . . . . . . . . . . . 85

6.3 The 2D Wave Filter . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 88

7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.1 Appendix 1: The Rest of Statistical Numerical Riemann Prob-

lems in Planar Flows . . . . . . . . . . . . . . . . . . . . . . . 93

7.1.1 Rarefaction Crossing Rarefaction Interaction (Case 4) . 94

7.1.2 The Contact Rarefaction Interaction (Case 5) . . . . . 95

7.1.3 Shock Overtaking Shock Interaction (Case 6) . . . . . 96

7.1.4 Compression Crossing Compression Interaction (Case 7) 99

7.1.5 The Contact Compression Interaction (Case 8) . . . . . 100

7.1.6 Rarefaction Crossing Rarefaction Interaction (Case 9) . 101

7.1.7 The Contact Rarefaction Interaction (Case 10) . . . . . 101

7.2 Appendix 2: Composite Shock Interaction Problems in Planar

Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.2.1 Errors in Resolved Calculations . . . . . . . . . . . . . 103

7.2.2 Errors in Under Resolved Calculations . . . . . . . . . 104

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

viii

List of Figures

1.1 Schematic plot of Riemann solution, which is constructed by 3

waves, 4 states. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Flow chart for the front tracking computation. With the ex-

ception of the i/o and the sweep and communication of interior

points, all solution steps indicated here are specific to the front

tracking algorithm itself. . . . . . . . . . . . . . . . . . . . . . 13

2.1 Problem 1: Shock-contact (step up). The mid state is held fixed,

and the two wave strengths are varied. . . . . . . . . . . . . . . 16

2.2 Problem 2: Shock-wall interaction. The right state is held fixed

and the left state is varied. . . . . . . . . . . . . . . . . . . . . . 16

2.3 Left. Space time density contour plots for the multiple wave inter-

action problem studied in this section. Right. Pressure contour

plots for the base case considered here. . . . . . . . . . . . . . . 17

2.4 Left: Type and location of waves as determined by our wave

filter analysis for the base case considered here. Right: Schematic

representation of the waves and the interactions, with labels for

the interactions, taken from the left frame. . . . . . . . . . . . 18

ix

3.1 Schematic diagram illustrating the operation of a wave filter. Left:

computational data (squares) are fit to an error function. The er-

ror function depends on four parameters, a position, a width, and

two asymptotic values. These determine the wave position, width

and height, with subgrid accuracy. Right: a piecewise linear con-

struction is fit to the rarefaction or compression wave data. . . 33

3.2 Ensemble mean shock width (the green dots on the right) and the

standard deviation (the red dots on the left) of the shock width

(left frame). The mean width, equal to about 2∆x, is much larger

than the standard deviation, indicating that the mean width is

essentially a deterministic feature of the solution. Convergence

properties of the traveling wave to the steady state values on

each side of the wave (right frame). The straight line in the right

frame is the asymptote to the exponential convergence rate, with

slope 0.01 in units of ∆x. . . . . . . . . . . . . . . . . . . . . . 35

3.3 Ensemble mean contact width for isolated noninteracting waves.

Because the width is entirely grid related, we record width in units

of ∆x and time in units of the number of time steps. The standard

deviations are also plotted as the red points to the extreme left in

each frame. Left (step down): we observe an increase from 2 cells

to 30 over 104 steps and an asymptotic growth rate cct1/3, where

cc ∼ 1 depends on the flow Mach number. The straight line in

the left frame is the asymptote to the contact width, with slope

3. Right (step up): We observe a bound on the contact width. 37

x

3.4 Ensemble mean shock and contact position errors as a function

of time, expressed in grid units. Step up case. In the label, B.S.:

Backward Shock, C: Contact, F.S.: Forward Shock. . . . . . . . 42

3.5 Ensemble mean shock and contact position errors as a function of

time, expressed in grid units. To emphasize the transient errors,

a smaller number of time steps are shown. Left: shock crossing

shock case. Right: shock crossing contact (step down) case. . . 43

3.6 Problem 3: Contact-shock (step down). . . . . . . . . . . . . . . 45

4.1 Left. Space time density contour plot for the multiple wave in-

teraction problem studied in this section, in spherical geometry.

Right. Type and location of waves determined by the wave filter

analysis. For both plots, R indicates the radius distance from the

origin. And B.S.: Backward Shock, C: Contact, and F.S.:Forward

Shock in the label. . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Left. Mach number vs radius for a single inward propagating

shock. Right. The same data plotted on log-log scale. . . . . . . 50

4.3 Left. Mach number vs radius for a single outgoing propagating

shock. Right. The same data plotted on log-log scale. . . . . . . 51

4.4 Ensemble mean contact width for single propagating contact. We

record width in units of ∆x. The standard deviations are also

plotted. and are the points to the extreme left in the frame. . . 52

xi

4.5 Left: ensemble mean inward/outward moving shock and contact

widths after a shock contact interaction. Right: ensemble mean

shock and contact position errors as a function of time, expressed

in grid units. The associated standard deviations are extremely

small, not shown in the plots. In the legend, C. denotes the

contact while I.S. and O.S. are the inward and outward moving

shocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 The solution and its errors at the point (x, t) can be obtained

by “adding up” the solution and errors for the waves within the

domain of dependence . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Schematic graphs illustrating all contributions to the errors or

uncertainty in the output from a single Riemann solution, namely

the reshock interaction (case 3) of the reflected shock from the

wall as it crosses the contact. The numbers labeling the black

circles refer to the Riemann interactions contributing to the error.

The letter I in the first two diagrams indicates input uncertainty. 66

5.3 Schematic graphs, showing all six wave diagram contributions to

the errors or uncertainty in the output from a single Riemann

solution, namely the reshock interaction (numbered 3 in the right

frame of Fig. 4.1) of the reflected shock from the origin as it

crosses the contact. . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Pie charts showing the contribution of each wave interaction dia-

gram to the error variance of the wave strength at the output of

interaction 3, for a solution using 500 mesh units. . . . . . . . . 80

xii

5.5 Pie charts showing the contribution of each wave interaction dia-

gram to the error variance of the wave strength at the output of

interaction 3, for a solution using 100 mesh units. . . . . . . . . 81

6.1 Left: Initial geometry of a circular shock imploding a perturbed

circular contact discontinuity. The two circles are offset relative

to each other. Right: Chaotic flow observed after reshock by the

outgoing shock reflected from the origin. Gray scale in both plots

indicates density. The grid is 800 × 1600. . . . . . . . . . . . . 86

6.2 Space time (r, t) contours of the primary waves, as detected by our

wave filter algorithm. These are the inward (direct) and outward

(reflected) shocks and the inner and outer edges of the mixing

zone, all detected within a single averaging window, in this case

θ ∈ [−45o, 0o]. Note the rarefaction wave near t = 80, r = 24,

due to the shock reflection off of the constant pressure boundary.

Contour plots of the mean density are also shown. The shock has

been offset relative to the contact, and the grid is 800 × 1600. . 91

6.3 Radial shock position as a function of angle for a series of times.

The grid level is 200 and there is no offset of the center. Left:

inward shock, right: outward shock. . . . . . . . . . . . . . . . 92

7.1 Problem 4: Rarefaction-wall. The right state velocity v = 0.0

is fixed and the left state densities and pressures are held fixed.

The input rarefaction wave width is an input parameter. Same

comments applied to later cases. . . . . . . . . . . . . . . . . . . 94

xiii

7.2 Problem 5: Contact-rarefaction. The right state velocity v = 0.0

is fixed and the left state densities and pressures are held fixed. 95

7.3 Problem 6: Shock-shock overtake (two waves of the same family).

The left state is held fixed and the two wave strengths are varied. 97

7.4 Because the width is entirely grid related, we record width in

units of ∆x and time in units of the number of time steps. . . . 98

7.5 Problem 7: Compression-wall. The right state is held fixed. . . . 99

7.6 Problem 8: Contact-compression. The right state velocity v = 0.0


7.7 Problem 9: Rarefaction-wall. The right state velocity v = 0.0 is

fixed and the left state densities and pressures are held fixed. . . 101

7.8 Problem 10: Contact-rarefaction. The right state velocity v = 0.0


xiv

List of Tables

2.1 The shock-contact (step up) interaction. Expansion coefficients

for output wave strengths for input variation ±10%. Compari-

son of linear, bilinear, and full quadratic models for the output

variables wok. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 The shock-contact (step up) interaction. Expansion coefficients

for output wave strengths. Comparison of input variation (±1%,

±10%, ±50%). The linear model is used for the first two cases

while the bilinear model is required for the largest variation. . . 23

2.3 The shock crossing equal shock (wall reflection) interaction. Ex-

pansion coefficients for output wave strengths (linear model) for

input variation ±10%. . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 The SNRP shock contact (step up) interaction. Expansion coeffi-

cients for output wave strengths (linear model) for input variation

±10%. Here the base case input contact wave width is zero. . . 40

3.2 The SNRP defined by the crossing of two shocks. Expansion

coefficients for output wave strengths, widths and position errors

(linear model) for input variation ±10%. . . . . . . . . . . . . . 44

xv

3.3 The SNRP shock contact (step down) interaction. Expansion

coefficients for output wave strengths, widths and position errors

(linear model) for input variation ±10%. . . . . . . . . . . . . . 46

4.1 Comparison of the exponents from the approximate and the exact

similarity solutions. . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 The SNRP shock contact interaction. Expansion coefficients for

output wave strengths, wave strength errors, wave width errors

and wave position errors (linear model) for the initial shock con-

tact interaction. Here the base case input contact wave width

is zero. The final columns refer to difference between the linear

model (2.1) and the exact quantity. The errors in rows 4-12 refer

to the difference between the numerical solution on 100 cells and

the exact solution using 2000 cells. . . . . . . . . . . . . . . . . 58

4.3 The SNRP defined by the shock reflection at the origin. Expan-

sion coefficients for output wave strengths, wave strength errors,

wave width errors and wave position errors (linear model) for in-

put variation ±10%. . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 The SNRP contact reshock interaction. Expansion coefficients for

output wave strengths, wave strength errors, wave width errors

and wave position errors (linear model). . . . . . . . . . . . . . 60

5.1 Predicted and simulated errors for output wave strengths, wave

widths and wave positions, Case 1. . . . . . . . . . . . . . . . . 70

xvi





5.4 Case 1. The contact-shock interaction (step up). Errors for out-

put wave strengths, wave widths and wave position. Comparison

of under resolved simulation and prediction. . . . . . . . . . . . 73

5.5 Case 2. The shock crossing equal shock (wave reflection) inter-

action. Errors for output wave strengths, wave width and wave

position. Comparison of under resolved simulation and prediction. 74

5.6 Case 3. The contact-shock interaction (step down). Errors for

output wave strengths, wave width and wave position. Compari-

son of under resolved simulation and prediction. . . . . . . . . . 75


widths and wave positions, output to interaction 3. The inward

rarefaction and contact strengths are expressed dimensionlessly as

Atwood numbers. The outward shock strengths are in the units of

Mach number. The width and position errors are in mesh units.

The wave strength errors are expressed as mean ± 2σ where σ is

the ensemble STD of the error/uncertainty. . . . . . . . . . . . 78

xvii

5.8 The contribution of each interaction to the mean value of the total

error in each of three output waves at the output to interaction 3,

for 100 and 500 mesh units. Units are dimensionless and represent

the error expressed as a fraction of the total wave strength. The

last two rows compare the total of the mean error as given by the

model to the directly observed mean error. The columns I.R., C.,

and O. S. are labeled as in Fig. 4.1, Right frame. . . . . . . . . . 79

7.1 Case 4. The SNRP defined by the crossing of two rarefactions.

Expansion coefficients for output wave strengths (linear model)

for input variation ±10%. . . . . . . . . . . . . . . . . . . . . . 95

7.2 Case 5. The SNRP defined by the contact rarefaction interaction.



7.3 Case 6. The SNRP defined by the shock shock overtake (two

waves of the same family). Expansion coefficients for output wave

strengths (linear model) for input variation ±10%. . . . . . . . . 99

7.4 Case 7. The SNRP defined by the crossing of two compressions.



7.5 Case 8. The SNRP defined by the contact compression inter-

action. Expansion coefficients for output wave strengths (linear

model) for input varation ±10%. . . . . . . . . . . . . . . . . . 102

xviii

7.6 Case 9. The SNRP defined by the crossing of two rarefactions.



7.7 Case 4. The crossing of two rarefactions. Predicted and simulated

errors for output wave strengths, wave widths and wave positions. 104

7.8 Case 5. The contact rarefaction interaction. Predicted and sim-

ulated errors for output wave strengths, wave widths and wave

positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.9 Case 6. The shock shock overtake. Predicted and simulated errors

for output wave strengths, wave widths and wave positions. . . . 106

7.10 Case 7. The crossing of two compressions. Predicted and sim-


positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.11 Case 8. The contact compression interaction. Predicted and sim-


positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107


errors for output wave strengths, wave widths and wave positions. 107


errors for output wave strengths, wave widths and wave positions.

Comparison of under resolved simulation and prediction. . . . . 108

7.14 Case 5. The contact rarefaction interaction. Predicted and sim-


positions. Comparison of under resolved simulation and prediction.108

xix

7.15 Case 6. The shock shock overtake. Predicted and simulated er-

rors for output wave strengths, wave widths and wave positions.


7.16 Case 7. The crossing of two compressions. Predicted and sim-



7.17 Case 8. The contact compression interaction. Predicted and sim-




errors for output wave strengths, wave widths and wave positions.


xx

Acknowledgements

My thanks go to everyone who has been involved in making me into this

picture.

I would like to extend my deepest gratitude and thanks to James Glimm,

my adviser, for his insightful suggestions and constant support throughout my

research. I often felt fortunate to be under his advisement, as a great math-

ematician and brilliant physicist, he not only gave guidance on my scientific

research, but also set up a lifetime role model to follow. I am also thankful to

Xiaolin Li for his authentic encouragement, which led me through the early

years of chaos and confusion.

Professor Kenny Ye has been deeply involved in the early stage of this

project, he has led me through the door of the practical statistical data analy-

sis. Professor Wei Zhu expressed her interest in my work and shared with me

her knowledge of statistics and provided many useful references and friendly

encouragement. Professor Stephen Finch and David G. Ebin, thank you for

your valuable suggestion and sincere comments for the completion of this dis-

sertation.

I should also mention that my graduate studies in Stony Brook were

supported by NSF and the U. S. Department of Energy. Their support makes

research like this possible.

Of course, I am most grateful to my parents, who have been always

supportive to whatever decisions that came to me, and their unconditional

love. Without them this work would never have come into existence (literally).

Finally, I wish to thank the following the most: Yan Yu, Taewon Lee(for

our four years’ friendship and collaboration on this project to make this thesis

complete); Jean Nick Pestieau and Wurigen Bo (for their later contributions

into this project); my most special friend Fay Y. Zhao (for all your support and

care); Xiaofei Fan, Masha Prodanovic, Dasha Eremina, Hyunsun Lee, So-Youn

Shin, Xiangfeng Wu ... (for all the good and bad times we had together).

xxii

Chapter 1

Introduction and Background

1.1 Uncertainty Quantification

Uncertainty quantification is at its essence a study of errors, both their

description and their consequences. Addressed here are the errors arising from

the finite resolution numerical discretization used to obtain solutions. In this

sense, the problem, a bit simplistically, can be viewed as the determination of

error bars to be assigned to the numerical solution algorithms. For a Bayesian

decision framework which leads to this point of view see [19–21]. Unlike other

authors [3, 5, 8, 24, 26] who usually use observational errors or expert opinions

[8] to form the probability model for the likelihood, our approach is to use

solution error models for the likelihood. We provide a scientific basis for the

probabilities associated with numerical solution errors. The authors are not

aware of comparable error analysis studies, but of course numerical simulation

errors have been long studied from different points of view.

An early focus of numerical error modeling was round off errors. For the

hyperbolic systems we study, modern 64-bit processors with double precision

1

arithmetic appear, as a practical matter, not to be sensitive to this class of

errors, while they are difficult to analyze theoretically. A common approach to

error analysis in numerical analysis is the study of the asymptotic behavior of

errors under mesh refinement. The method of asymptotic analysis of numerical

solution errors is so old and well established that it is difficult to cite its origins

[32]. This is a useful approach, and one we refer to in the case of well resolved

simulations. However, since many simulations are under resolved, as used in

practice, asymptotic analysis of convergence does not address the issue of error

bars and uncertainty quantification. Moreover, the coefficients which multiply

powers of ∆x in the asymptotic expressions cannot be determined theoreti-

cally. A third main theme in the analysis of errors is the use of a posteriori

error estimators. The method of a a posteriori analysis aims to construct an

upper bound on the solution error, either theoretically validated or based on

numerical experiments [2, 7, 28, 31]. But for hyperbolically dominated flows,

with poor theoretical foundations, a posteriori methods are generally inap-

plicable. We seek to characterize the error, not just to bound it. Moreover,

a posteriori methods are most fully developed and justified theoretically for

elliptic problems, and have only a partial or preliminary development for the

shock interaction problems we consider here. For this reason, we consider er-

rors from a statistical point of view, and examine an ensemble of coarse and

fine grid pairs, and their differences, which are assumed to represent the coarse

grid solution error.

We have developed an approach to uncertainty quantification for numer-

ical simulations that is based on three ideas:

2

1. A combined approach for forward as well as inverse propagation of un-

certainty [19, 20]. This combined approach is important when the use of

disparate sources of data, including data pertaining to observations of

full system performance, is important.

2. Models for numerical solution errors [9, 16–18]. For multi-scale problems

and complex multi-physics problems, under-resolved simulations and ac-

companying simulation error are frequently unavoidable in practice.

3. Parametrization, comprehensibility, and validation of simulation error

models [15]. This step allows testing and validation to occur in somewhat

idealized situations, less complex than the full system simulations, but

still applicable to them.

Our principal concern is to develop a method for determining solution

errors in shock wave interaction problems having a significant degree of com-

plexity. Our strategy takes advantage of the fact that shock problems typically

consist of smooth regions separated by discontinuities (actually narrow regions

with strong solution gradients), and consists of three main steps.

Step 1: Determine solution errors for a comprehensive set of elementary wave

interactions. These are summarized as a set of input/output relations

for the errors in such interactions;

Step 2: Construct wave filters that decompose a complex flow into approximately

independent components consisting of elementary waves;

3

Step 3: Formulate a composition law that constructs the total solution error at

any space-time point in terms of errors from repeated elementary inter-

actions.

Step 1 is carried out in Chap. 2, 3 and 4. Step 2 is discussed in [25]. The

main idea is summarized and extended in Chap. 3 and 4. The composition

law is presented in Chap. 5.

We validate this method by predicting the errors in the composite (com-

plex) simulation with the errors calculated using the composition law [10]. The

results are derived by a study of errors in the solution of Riemann problems,

for details see Sec. 1.2.1.

The uncertainty of a numerical simulation comes from three different

sources, the errors in the physical model, the errors in the numerical model

(ı.e. simulation errors), and the uncertainty on the input parameters of the

model(ı.e. initial uncertainty). In our study, we mainly focused on the latter

two categories of errors. Simulation errors typically consist of:

• position errors in the location of the traveling wave discontinuities or

sharp solution gradients;

• wave width errors arising from the numerical vs. the physical width of

the traveling waves;

• solution state errors in the smooth regions bordering the regions with

discontinuities or sharp solution gradients.

Any or all or these errors may arise in the input data to the Riemann

problem. Output errors, however, have two sources. Those arising from inac-

4

curacies in the solution algorithm are called created errors, while those that

can be ascribed directly to input error or uncertainty are called transmitted

errors or transmitted uncertainty.

In contrast to the case of planar waves, the complications that addition-

ally result from spherical geometry are the following:

1. The solution waves are not of constant strength between wave interac-

tions, but evolve approximately according to a power law as a function

of the radius.

2. The solution is not spatially constant between waves.

The radially dependent strength of spherical waves is discussed in [36]. The

spatial variation of spherical waves is contained in the Guderley solution [22],

which gives simple power laws that predict this dependence of the solution on

the radius. We find that these power laws also predict the evolution of the

error, which, once formed, propagates according to the same laws as govern

the solution the waves themselves.

In summarizing and extending the results of [25], we also introduced

diagnostic windows, which measure the solution state in the constant regions

between the waves, as well as window filters, which identify individual wave

types (The filters look for regions containing one single wave passing through

the filter).

5

1.2 Background

1.2.1 1D Simulation Formulation

The 1D simulations that will be discussed in this dissertation, both planar

and spherical, are based on numerical solutions to the Euler equations that

describe the conservation of mass, momentum, and energy for a γ-law gas:

ρt + (ρu)x = 0,

(ρu)t + (ρu2 + p)x = 0,

et + ((e + p)u)x = 0, (1.1)

where e = ρu2/2 + pε, ε = p/ρ(γ − 1). In the above formulas, ρ is the mass

density, u the gas velocity, p the thermodynamics pressure, ε the internal

energy per unit mass and e the total energy per unit volume. The equation of

state for γ-law gas is given by [36]

p = κργeS,

where S is the entropy, κ, γ = constants, γ > 1. Specifically the simulations

in this dissertation use the value γ = 1.67.

The state of the gas is a vector

u =

⎡⎢⎢⎢⎢⎣

ρ

u

e

⎤⎥⎥⎥⎥⎦ .

6

The Riemann problem [6] is the initial value problem for equation 1.1 with

special initial data of the form

u(x, 0) =

⎧⎪⎨⎪⎩

ur, x ≥ 0,

ul, x ≤ 0,

where

ur =

⎡⎢⎢⎢⎢⎣

ρr

ur

er

⎤⎥⎥⎥⎥⎦ and ul =

⎡⎢⎢⎢⎢⎣

ρl

ul

el

⎤⎥⎥⎥⎥⎦

are two constant states of the gas. The solutions of Riemann problem will be

made up of three types of elementary waves (solutions) in the region t > 0.

We define c as speed with c2 = γp/ρ. Characteristics for a simple wave

are

C+ :dx

dt= u + c and C− :

dx

dt= u − c .

Riemann invariants are found to be

Γ± = u ±∫

c(ρ)

ρdρ.

Γ± are constant along the characteristics C±, respectively.

At a point where the solution has a jump discontinuity propagating with

7

the speed s = dx/dt, the Rankine-Hugoniot jump conditions

s[ρ] = [ρu],

s[ρu] = [ρu2 + p],

s[e] = [(e + p)u],

corresponding to Eq. 1.1 must hold; these jump conditions reflect conservation

of mass, momentum, and energy through the discontinuity. Here the jump [A]

in a quantity A is defined by [A] = A+ − A−, where A± is the limiting value

of A on the right(left) of the discontinuity.

Pick a coordinate system in which the discontinuity stays stationary. In

this coordination system, the Rankine-Hugoniot relations at the origin become

ρ1u1 = ρ0u0,

ρ1u21 + p1 = ρ0u

20 + p0,

(e1 + p1)u1 = (e0 + p0)u0. (1.2)

Let mass flux M = ρ1u1 = ρ0u0. If M = 0 we call the discontinuity a

contact-discontinuity. u1 = u0 = 0, which indicates the discontinuity moves

with the flow, p1 = p0, ρ1 �= ρ0.

If M �= 0, this discontinuity is called a shock. With subscript 0/1 indicate

the states in the front/back of shock, respectively. We can derive some simple

8

identities from Eqs. 1.1 and 1.2:

M2 = −p0 − p1

ν0 − ν1

,

u0u1 =p0 − p1

ρ0 − ρ1

,

ε1 − ε0 +p0 + p1

2(ν1 − ν0) = 0,

where ν = 1/ρ, and the third equation is called Hugoniot equation for the

shock.

Figure 1.1: Schematic plot of Riemann solution, which is constructed by 3waves, 4 states.

Fig. 1.1 shows the Riemann solution, which are are given as follows [35]:

9

1-family

pr/pl = e−x ,

ρr/ρl = f1(x) ≡

⎧⎪⎨⎪⎩

e−x/y, x ≥ 0,

β+ex

1+βex , x ≤ 0,

(ur − ul)/cl = h1(x) ≡

⎧⎪⎨⎪⎩

2(1 − e−τx)/(γ − 1), x ≥ 0,

2√

τ(1−e−x)

(γ−1)(1+βe−x)1/2 , x ≤ 0,

2-family

pr/pl = 1, ρr/ρl = ex, ur − ul = 0;

3-family

pr/pl = ex ,

ρr/ρl = f3(x) ≡ 1/f1(x),

(ur − ul)/cl = h3(x) ≡

⎧⎪⎨⎪⎩

2(eτx − 1)/(γ − 1), x ≥ 0,

2√

τ(ex−1)

(γ−1)(1+βex)1/2 , x ≤ 0,

where constants β = (γ + 1)/(γ − 1), τ = (γ − 1)/2γ.

For equations of state 1.1, there is a unique solution (in the class of shocks,

centered simple waves, and contact discontinuities separating constant states)

to the Riemann problem with these initial states, if and only if

ur − ul <2

γ − 1(cl + cr). (1.3)

If 1.3 is violated, then a vacuum is present in the solution [35]. We define

10

A = ρr/ρl, B = pr/pl, and C = (ur − ul)/cl, then the following hold given the

condition 1.3:

1. The 1-family of the solution is a simple wave if and only if

√2

γ − 1h1logB < C <

2

γ − 1

[1 +

√B

A

],

and is a shock otherwise

2. The 3-family of the solution is a simple wave if and only if

h1(−logB) < C <2

γ − 1

[1 +

√B

A

],

and is a shock otherwise.

Illustrations of the setup specifications for the base case of one dimen-

sional planar geometry simulation are given in Fig. 2.1 of Chap. 2. Reflecting

boundary conditions are applied at both x = 0 and x = xmax (r = 0 and

r = rmax in spherical geometry).

1.2.2 Introduction to FronTier

Since our work is based on numerical simulations using the FronT ier

code, we also present a brief introduction to this code.

The FronT ier code is based on front tracking, a numerical method which

regards the fluid interface as an evolving, lower dimensional mesh. The Front

Tracking method, initiated by Richtmyer and Morton [33], is designed to pro-

vide piecewise smooth solutions as well as distinguished discontinuities which

11

separate the smooth solutions in solving systems of hyperbolic partial differ-

ential equations. Fig. 1.2 is a flow chart taken from [13] describing the front

tracking computation in FronT ier. For more details, see [12–14].

1.3 Organization of This Dissertation

In the Chap. 1, the concept of uncertainty quantification was introduced.

We then discussed our general steps to address uncertainty quantification prob-

lems in numerical solutions, which, to be specific, are error analysis. A back-

ground review of FronT ier code and Front Tracking algorithm used for our

numerical simulations was also given.

In Chap. 2 we studied the statistical Riemann problem (SRP), which is

to describe the non-linear problem of the interaction of strong hydrodynamic

waves by our simple linear input-output models. Statistical Riemann problems

are illustrated to show how to map the distributions of errors from input to

output.

In Chap. 3 we study the statistical numerical Riemann problem (SNRP).

This differs from the statistical Riemann problem in that incoming shock

waves, specified numerically, have finite width determined from numerical as

well as physical considerations. We note that incoming rarefaction and com-

pression waves have physical time dependent widths. The SNRP also differs

in that the numerical algorithm that solves the Riemann problem creates (as

well as propagates) errors. We will show that, to fairly good accuracy, one

can model the errors in the outgoing waves as affine linear, i.e., constant plus

linear, (or perhaps bilinear) statistical expressions in the strength of the incom-

12

Figure 1.2: Flow chart for the front tracking computation. With the exceptionof the i/o and the sweep and communication of interior points, all solutionsteps indicated here are specific to the front tracking algorithm itself.

13

ing waves. The coefficients in these linear expressions depend parametrically

on the incoming waves, and here they are taken as defined by a character-

istic of base case. Diagnostic window and wave filters were introduced. A

wave filter is an automated pattern recognition algorithm which locates shock

wave, rarefaction wave and contact discontinuities in numerical solutions of the

Euler equations for compressible fluids. These filters locate the shock waves

and identify wave widths. Based on these wave filters, we can also study the

behavior of isolated waves.

Chap. 4 led our discussion of SNRP into 1D spherical flows. The spherical

geometry complicated our previous models by a nonconstant power law growth

or decay of error between interactions.

A composition law for complex 1D shock wave interaction problems was

elaborated in Chap. 5 in both planar and spherical geometry, based on our

detail studies of SNRP in Chap. 3 and Chap. 4.

Several conclusions for uncertainty quantification studies in 1D flows were

presented in Chap. 6. Also in this chapter, some open problems in 2D chaotic

flows were discussed, which are our future work.

The rest 7 (besides the 3 modeled in Chap. 3) of the 10 Riemann Problems

showed in Fig. 2.4 are modeled as SNRP and presented in the Appendix 7.1.

The continued error study of later interactions for both resolved and under

resolved calculation can also be found in the Appendix 7.2.

14

Chapter 2

Statistical Riemann Problems

The main result of this chapter is that a linear input-output model will

serve to describe the manifestly nonlinear problem of the interaction of strong

hydrodynamic waves.

In this chapter, the solution operator is considered to be exact, and acts

to propagate the input error (or uncertainty) to the output error. The out-

put errors can be represented conceptually as a multinomial in powers of the

strengths of the waves that are interacting. We characterize the statistics of

the errors as Gaussian, and thus the error distribution is determined by its

mean and covariance. For weak waves these quantities (mean and covariance)

as a description of the input, are transformed via the solution operator Ja-

cobean to the mean and covariance of the output. We are more interested in

strong waves, but find a description of the solution operator as linear pertur-

bation about base case still provides a useful description of the input to output

transformed error statistics.

We consider, in one spatial dimension, the interaction of a shock wave

15

Figure 2.1: Problem 1: Shock-contact (step up). The mid state is held fixed,and the two wave strengths are varied.

Figure 2.2: Problem 2: Shock-wall interaction. The right state is held fixedand the left state is varied.

with a contact located near a reflecting wall. The base case for the first wave

interaction, see Fig. 2.1 coincides with the base case for the shock contact

interaction studied in both this chapter and Chap. 3. We further specify the

wall location as 1.5 units to the right of the initial contact location. The

transmitted shock, after interaction with the contact, progresses to interact

with (i.e. reflect off) the wall. This interaction was also studied both in this

chapter and Chap. 3. Subsequently, there are a number of reverberations, of

reflected rarefaction and compression waves, between the contact and the wall

and between a new contact formed by a shock overtake interaction and the

16

original contact. The new contact is clearly visible in Fig. 2.3 (left frame),

as the vertical line near the left border, starting at a time about t = 5.2.

The interactions are illustrated by the space time contour plots of the density,

shown in Fig. 2.3, left and pressure contours, right. In Fig. 2.4 (left), we show

the type and location of the waves, as determined by our wave filter analysis

program. Both figures refer to the base case.

Figure 2.3: Left. Space time density contour plots for the multiple waveinteraction problem studied in this section. Right. Pressure contour plots forthe base case considered here.

Ten Riemann problems are extracted from the complex wave problem

interaction illustrated in Fig. 2.4, Left. A schematic representation of the

wave interactions, identifying these ten Riemann problems is given in Fig. 2.4,

17

Figure 2.4: Left: Type and location of waves as determined by our wave filteranalysis for the base case considered here. Right: Schematic representationof the waves and the interactions, with labels for the interactions, taken fromthe left frame.

Right.

2.1 A Multinomial Expansion for the SRP Output

The SRP considered here has two incoming waves, each of which may be

one of the followings: contact, shock, rarefaction, or compression. Rarefaction

and compression waves only interact when they belong to opposite families,

while two contacts never interact. For pairwise interactions (shock interactions

with other shocks, with rarefaction or with compression waves), we distinguish

18

whether the two interacting waves are of the same (left moving vs. right

moving) family or opposite families. For contact waves, we specify the direction

of the density change (step up or down) as seen from the side of the wave

approaching the contact. Thus there are a total of 14 elementary cases for the

incoming waves. Each case is associated with a wave amplitude of a specific

sign, so that there is no loss of generality in assuming that the signs are chosen

so that the amplitudes are non-negative. Fixing one case, we specify a starting

state for the incoming waves, and a base case for the (strong) wave strengths.

Then we consider variation about this base case by ±1%, ±10% and ±50% in

the density ratio (for contacts) or pressure (for all other waves). Within this

formulation, we can describe the output wave strengths by an expression linear

in the two input wave strengths, with a possible additional bilinear term, i.e.

linear in the the product of the input wave strengths.

The range of validity of such a bilinearization will depend on the details

of how the problem is formulated; perhaps the most important factor is the

choice of variables to describe wave strengths. We choose two dimensionless

variables to measure these wave strengths; a modified Atwood number A =

(ρ2−ρ1)/(ρ02 +ρ0

1) to measure contact strengths, and a similar expression built

out of the pressures, P = (P2 −P1)/(P02 +P 0

1 ), for the other wave types. Here

the quantities ρ0i and P 0

i denote densities and pressures from the base case

associated with the ensemble. Of course the solution of a Riemann problem,

being fully nonlinear, will require an infinite Taylor’s series for its complete

description. Terms in the series expansion can be determined by statistical

study, which is fit to linear/bilinear regression models for a given finite sample

19

size.

However, this statistical test misses the point and the value of a simple

model. We want to capture the main effects, and regard the remainder not

as a statistical sampling error but as a modeling error. Thus we consider an

expansion with more terms than we ultimately want, and discard terms with

small coefficients even if the associated p-value is not near 1. For input wave

strengths wi1, wi

2 (i ≡ in) and output wave strengths wo1, wo

2 and wo3 (o ≡ out)

(ordered from left to right in real 1D space), the expansion is defined by its

coefficients αk,J , for J a multi index, J = (j1, j2). The expansion has the form

wok =

∑J

αk,Jwi,J , (2.1)

where wi,J = (wi1)

j1(wi2)

j2 . The coefficients αk,J depend parametrically on the

base case Riemann problem. Given a statistical ensemble of input and output

values wi and wo, we use a least squares algorithm to determine the best

fitting model parameters αk,J , for any given polynomial order of model. We

use (2.1) variationally, that is to map input variation (about the base case for

the ensemble) to output variability. In other words, (2.1), which is a formula

for wave strengths, implies a similar formula with different but computable

coefficients αk,J , in which all ω’s are defined as variations from the base case,

so that they represent uncertainty or error. For simplicity we consider the case

of a linear input-output relation, ωok = αk,0 +

∑J αk,Jωi,J . If we denote the

base case with an overbar, i.e. ωok and ωi,J , then the fluctuation δωo

k = ωok−ωo

k

satisfies δωok =

∑J αk,Jδωi,J .

20

2.2 Evaluation of the Expansion Coefficients

The main point of this section is to determine numerical values for the

expansion coefficients of the output wave strengths of the SRP. We also find

that, for many cases, the linear model is sufficient for scientific accuracy, while

the case of large variability will likely require additional expansion terms. We

show that bilinear terms are sufficient for large variability.

We consider, as a typical Riemann problem, the wave interaction of a

shock wave moving (to the right) into a contact (density increase, or step up

case). This wave interaction initiates the complex series of interactions studied

in Sec. 5.1.

The base case shock strength is given with a pressure ratio P2/P1 = 1337

(P = 0.999), and contact density ratio ρ2/ρ1 = 10 (A = 0.82). The equation

of state satisfies a gamma law gas with γ = 1.67.

Typical results on the variation of the polynomial order of the model are

presented in Table 2.1, while results for variation of the strength of the input

waves are shown in Table 2.2. To read these tables, we note that the first

row of Table 2.1 (labeled as wo1 (l.sonic)) lists coefficients α1,J for J = (0, 0),

J = (1, 0), etc. These coefficients are determined by a least squares algorithm,

that minimizes the mean error over the ensemble predictions to the exact

solution of the Riemann problem. The last two columns describe errors in the

model (2.1). The column labeled L∞ is the maximum of the absolute value,

over the ensemble, of the relative error, and it may be sensitive to the choice

of ensemble. Note that the sample size studied here and all cases after is 200.

The relative error is defined as (predicted - exact)/exact where exact is the

21

variable \ coef const wi1 wi

2 wi1w

i2 (wi

1)2 (wi

2)2 error

(r. sonic) (contact) L∞ STDlinear response model

wo1 (l. sonic) -0.206 0.452 0.252 0.55% 0.0008

wo2 (contact) -0.042 0.000 0.911 0.01% 0.00004

wo3 (r. sonic) -0.285 1.001 0.347 0.35% 0.001

bilinear response modelwo

1 (l. sonic) 0.011 0.234 -0.015 0.267 0.16% 0.0003wo

2 (contact) -0.043 0.001 0.912 -0.001 0.01% 0.00004wo

3 (r. sonic) 0.015 0.701 -0.020 0.368 0.10% 0.01quadratic response model

wo1 (l. sonic) -0.090 0.245 0.222 0.253 0.000 -0.136 0.01% 0.00002

wo2 (contact) -0.030 0.001 0.881 0.001 0.000 0.018 0.00% 1.1E-6

wo3 (r. sonic) -0.125 0.715 0.305 0.349 0.001 -0.187 0.02% 0.00002

Table 2.1: The shock-contact (step up) interaction. Expansion coefficientsfor output wave strengths for input variation ±10%. Comparison of linear,bilinear, and full quadratic models for the output variables wo

k.

result of the (exact) Riemann solution and predicted is the value given by

the finite polynomial model. The column STD is the standard deviation of

(predicted - exact). From the small values of these errors for the linear model,

as seen in Table 2.1, we conclude that the linear model is adequate for many

purposes. The two pure quadratic columns of the quadratic model are small,

indicating that the quadratic model is basically a correction to the bilinear

model. However, the 3× 3 linear sub block of the bilinear model is not a good

approximation to the corresponding values of the linear model. Moreover, in

this case the size of the pure bilinear terms in the model is large relative to the

decrease in error achieved by use of the bilinear model relative to the linear

one. Thus we conclude that the bilinear model contains a substantial amount

of cancelation between its terms, and in general the linear model may be more

satisfactory. All three models in Table 2.1 show an approximate diagonal,

22

identity matrix structure for the 2×2 subsystem formed by the right sonic and

contact waves, as would be expected from a linear (small amplitude) theory.

Observe that the approximate validity of a linear model has implications for

the statistics, as linear transformations map Gaussian statistics into Gaussian

statistics.

variable \ coef const wi1 wi

2 wi1w

i2 error

(r. sonic) (contact) L∞ STD1% input variation; linear model

wo1 (l. sonic) -0.207 0.453 0.252 0.01% 8.0E-6

wo2 (contact) -0.042 0.000 0.911 0.00% 3.6E-7

wo3 (r. sonic) -0.285 1.002 0.347 0.01% 0.0001

10% input variation; linear modelwo

1 (l. sonic) -0.206 0.452 0.252 0.55% 0.0008wo

2 (contact) -0.042 0.000 0.911 0.01% 0.00004wo

3 (r. sonic) -0.285 1.001 0.347 0.35% 0.00150% input variation; bilinear model

wo1 (l. sonic) 0.013 0.216 -0.016 0.280 4.15% 0.008

wo2 (contact) -0.041 0.002 0.911 -0.001 0.89% 0.001

wo3 (r. sonic) 0.017 0.676 -0.022 0.385 4.15% 0.116

Table 2.2: The shock-contact (step up) interaction. Expansion coefficients foroutput wave strengths. Comparison of input variation (±1%, ±10%, ±50%).The linear model is used for the first two cases while the bilinear model isrequired for the largest variation.

From Table 2.2 we see that the model coefficients show only a slight

sensitivity to the magnitude of fluctuation of the input variables. The small

(1% and 10%) fluctuation models have similar coefficients, but these coeffi-

cients do not define the best coefficients for the larger fluctuation models. The

linear model is badly fitted statistically so that it is not acceptable for the

larger (50%) variation, data not presented here. Surprisingly, the errors for

23

the quadratic model with input 50% variation are somewhat higher than for

the bilinear. We therefore use the bilinear model for 50% variation.

variable \ coef const wi1 error

(r. sonic) L∞ STDwo

1 (l. sonic) -0.0008 0.715 0.00% 4E-8

Table 2.3: The shock crossing equal shock (wall reflection) interaction. Expan-sion coefficients for output wave strengths (linear model) for input variation±10%.

In Table 2.3 we consider the shock crossing shock case. The two waves are

of opposite families and each has Mach number 53. This case is also the second

(in time) Riemann problem to occur within the complex problem studied in

Sec. 5.1, as shock reflection from a wall.

2.3 Variance

In essence our analysis will proceed by ignoring the correlation among

output waves. Having done so, we determine the variance of the output waves

as a function of the input distributions. The formulas are first derived with

general applicability, and then specialized to our situation, including the as-

sumption of a linear input-output model.

The Riemann solution wo = W o(wi) defines a mapping from the input to

the output statistics. The approximate mapping (2.1) defines an easily com-

putable approximation to this transformation of statistics. Thus for a given

probability distribution dνi(wi) defined on the input variables, and supported

in a domain Di of input variables, the output variables lie in a domain Do,

24

and on this domain the probability measure for the output variables is

dνo(wo) =dνo(W o)

dνi(wi)dνi(ωi) . (2.2)

From the above the formula, we see an immediate problem. The three

output variables, defined as a function of the two input variables, cannot be

statistically independent. They lie in a two dimensional sub-domain, Do, of

the three dimensional space of output wave parameters. However, these waves

separate in parameter space, and to a large extent carry out separate roles in

the composite wave interaction problem studied in Sec. 5.1. Thus we want to

generate independent statistical descriptions of each of the output waves. In

practice, for all Riemann problem simulations (except those with single input

wave), we generated 2 independent input waves (using Latin Hypercube ran-

dom generating algorithm) with parameter means equal to the corresponding

base case input wave parameters. This step, of ignoring the correlations in the

statistics of the outgoing waves, assumes that the successive interactions give

rise to a de-correlation of the solution from its dependence on the earlier inter-

actions. In a strict sense this assumption is not correct, but we believe it will

give a reasonably accurate description of the final composite statistics. The

validity of this assumption will be tested in the comparison given in Sec. 5.1,

where composition of errors from interacting Riemann problems are consid-

ered. For comparison, this assumption is analogous to the introduction of the

hypothesis of molecular chaos in the derivation of the Boltzmann equation.

We next deal with the output variance in a quantitative manner under

25

the assumptions of input independence. We write the domain Do,

Do ⊆ D1o ×D2

o ×D3o , (2.3)

as a subset of its projections Dko on the three output wave strength coordi-

nate axes. A marginal distribution is the projection of the three dimensional

measure dνo achieved through its integration over two neglected variables.

Assuming sufficient regularity, a marginal distribution can be written as

dνok(w

ok) =

dνok

dwok

dwok =

∫Do(wo

k,wok+dwo

k)

dνo(wo) , (2.4)

where

Do(wok, w

ok′) = Do ∩ R2 × [wo

k, wok′] . (2.5)

Let us suppose that the distribution of the incoming waves dνi is continu-

ous and that it is a product of the individual measures on the incoming waves.

(Thus the incoming waves are assumed to be statistically uncorrelated. Note

that they may also be output waves of other (distinct) Riemann problems.)

In this case,

dνi = f i(wi1, w

i2)dwi

1dwi2 = f i

1(wi1)f

i2(w

i2)dwi

1dwi2 (2.6)

for p.d.f.’s (probability density functions) f ik. It is convenient to introduce the

cumulative distribution function F i so that

dνi = dF i = f i1(w

i1)f

i2(w

i2)dwi

1dwi2 (2.7)

26

and the marginal cumulative distribution functions F oj for which

F oj (wo

j ) =

∫{W o

j (wi1,wi

2)<woj }

dF i =

∫{W o

j (wi1,wi

2)<woj }

f i1(w

i1)f

i2(w

i2)dwi

1dwi2 . (2.8)

We note that

f oj (wo

j ) =dF o

j (woj )

dwoj

. (2.9)

Next we assume a linear model

woj = αj +

2∑k=1

βijkw

ik (2.10)

for the SRP. Here we use α for the intercept, and β’s for coefficients of wave

strengths. We can now compute the marginal cumulative distribution func-

tions explicitly in terms of the coefficients for the linear model for the SRP,

F oj (wo

j ) =

∫ ∞

−∞

∫ (woj−αj−βi

j2wi2)/βj1

−∞f i

1(wi1)f

i(wi2)dwi

1dwi2 , (2.11)

f oj (wo

j ) =

∫ ∞

−∞

1

βij1

f i1((w

oj − αj − βi

j2wi2)/β

ij1)f

i2(w

i2)dwi

2 . (2.12)

The general form of variance transformation for linear model as 2.10 is

Var woj =

2∑k=1

(βijk)

2Var wik + 2βj1βj2Cov (wi

1, wi2) .

However, assuming statistical independence of the wik as above, the term

with Cov (wi1, w

i2) = 0. Thus, simple calculation shows that the variance,

27

Var woj , equals

Var woj =

2∑k=1

(βijk)

2Var wik . (2.13)

Using (2.13) and information such as that tabulated in Tables 2.1 and

2.2 to generate the β’s, we can predict the propagation of uncertainty through

a series of wave interactions. See Sec. 5.1.

28

Chapter 3

The Statistical Numerical Riemann Problems

for Planar Flows

The SNRP has a new feature beyond those of the SRP studied in Chap. 2.

It introduces errors (modeled as random) in addition to propagating errors or

uncertainty from input to output.

The waves in the SNRP have a finite width and the solution algorithm

in the SNRP has only finite accuracy. Because of the possible finite width to

the input waves, the problem and its solution are not strictly scale invariant,

and so we consider a generalization of the Riemann problem.

3.1 The Wave Filter

We introduce diagnostic windows, that measure the solution state in one

of the constant regions between the waves as well as wave filters, that diagnose

the wave type (only regions with a single wave pass through the filter). In doing

so, we first summarize and then extend the results of [25]. The moving window

29

in the wave filter has an initial width of 5 cells for shock and rarefaction waves

and 11 cells for contacts. The choice of these parameters appears to be suitable

for most higher order Godunov schemes. In this window, a Riemann problem is

solved using the extreme left and right states as input. The Riemann solution

has 3 outgoing waves, whose strength is assessed dimensionlessly as in Chap. 2

in terms of density and pressure differences and ratios. According to these

strengths, and a suitable cutoff for the strength, we identify from zero to three

of the waves as strong, and only the case of a single strong wave is analyzed

further. If adjacent or overlapping windows show a single identical wave, the

windows are merged, so that the full width of the wave will be brought into a

single window. This merging of adjacent or overlapping windows of increasing

size continues recursively until the same wave type fails to show up in the

adjacent windows. Wave profiles are reconstructed using fitting functions of

the form:

ρ(x, t) = ρ− +ρ− − ρ+

2

(f(

x − xc(t)√2σ

) + 1

)(3.1)

where ρ± refer to the asymptotic values for density ahead or behind the wave,

xc(t) is the moving center of the wave, and 2σ is the a measure of the wave

width. The fitting function f(x) is either the erf function

f(x) = erf(x) =2√π

∫ x

0

e−t2dt (3.2)

30

for contact or shock waves, or a linear ramp:

f(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−1

x

1

x < −1

−1 < x < 1

1 < x

(3.3)

for rarefactions and compressions. The four fitting parameters are determined

by solving the nonlinear least squares problem for the data in the interval

found by the Riemann problem filter described above. States identified as

within an active region for a single shock or contact wave by the filter are

fit to an error function, that allows determination of the location of the wave

(with subgrid accuracy, up to O(∆x2)), and its width. See Fig. 3.1, Left.

For single rarefaction or compression waves, the waves are fit to a straight

line segment, linear in the characteristic speed variable. Thus the region of

constant and variable states are fit to three straight line segments, the two

extreme ones being constants. See Fig. 3.1, Right. The width for a shock or

contact wave is defined in terms of the error function fit to the shock or contact

wave profile. Let σ be the standard deviation that enters into the definition

of the error function. Then the width (in units of ∆x) is the distance needed

for a 2σ transition (between 2.3% and 97.7%) of the jump in the density

(for a contact wave) or in the pressure (for a shock wave). The width of a

rarefaction or compression wave is defined as the distance between the edges

of the central linear piece for its piecewise linear description. The position is

defined, with subgrid accuracy, as the position of the mean value, at a point

half way through the jump. In this way we record the left and right states,

31

wave position (and hence speed) and wave width. These quantities are not

independent, as the speeds and jumps are connected by the Rankine-Hugoniot

jump relations. They are sufficient to fully characterize the numerical incoming

waves. The location where the linear fit attains the far field state is marked

as the edge of the wave, and its width is the distance between its two edges.

Finally, the filter regards any shock like wave that is ”too wide” in mesh units

to be a compression wave.

The wave filter is the fundamental diagnostic tool that identifies indi-

vidual waves, here within the solution of a numerical Riemann problem and

in Sec. 5.1 within the solution of a complex wave interaction problem. We

note immediately a limitation of the methodology, at least as presently de-

veloped. The definition of the wave filters assume that the individual waves

in the Riemann problem have separated. For sufficiently coarse grids in the

wave interaction problem of Sec. 5.1, the waves will enter into new interactions

before clearly separating as they leave an earlier interaction. A second, and re-

lated limitation concerns the relaxation of the left and right states at the edge

of a wave to their far field values, an issue studied in Sec. 3.2. If a subsequent

interaction occurs before this relaxation is complete, the associated errors will

be “frozen” into the input and output of this later interaction. We will assess

this issue in Sec. 5.1.

A statistical distribution of numerical incoming waves and starting states

determines the SNRP. Its solution gives the output waves, each of which gen-

erate the same type of data Thus we define the SNRP as a statistical (non-

deterministic) mapping from a statistical input wave description to a statistical

32

output wave description.

Figure 3.1: Schematic diagram illustrating the operation of a wave filter. Left:computational data (squares) are fit to an error function. The error functiondepends on four parameters, a position, a width, and two asymptotic values.These determine the wave position, width and height, with subgrid accuracy.Right: a piecewise linear construction is fit to the rarefaction or compressionwave data.

The statistics of the SNRP mapping function arise from grid errors, and

from the random placement of a traveling wave relative to the centers of the

finite difference lattice. Our first objective in this section is to compare and

contrast the SRP and the SNRP. Our second objective is to build up a library

of statistical input-output relations that will include all Riemann problems to

be encountered in Sec. 5.1. This library will be used to predict results for

the multiwave error and uncertainty analysis based on a multi-path scattering

formula.

33

3.2 Isolated Waves

We start with the analysis of the ensemble mean width of a single (non-

interacting) shock wave. Fig. 3.2 (left) shows the expected narrow and time

independent (∼ 2∆x) shock width. Among the several factors contributing to

wave strength and speed errors, we mention the finite accuracy of the Riemann

solution root solver, or some approximate Riemann solver and the numerical

(finite difference) nature of the solution. The latter arises in two ways, the

relaxation to a constant ambient state and the finite rate of convergence under

mesh refinement, both applicable on the post shock or up stream side of the

shock wave. The equivalence of these two effects can be seen from the self

similar nature of the Riemann solution, which implies that the fixed mesh,

large time limit, with evaluation of the ambient state at a large separation

from the traveling wave is equivalent to a fixed separation at a fixed time,

considered as a mesh refinement problem.

According to the theory of traveling waves for the viscous Riemann prob-

lem [36], considered as a model for numerically generated traveling waves, we

expect an exponential approach of the numerical shock profile to its limiting

values at x = ±∞. Depending on the numerical scheme this approach may be

oscillatory. For higher order flux or slope limited methods there is a trade-off

between the oscillations and the wave width, as the oscillations are reduced

through increase of the first order (diffusive) aspect of the algorithm, while the

wave width is reduced through decrease in this diffusiveness. The error occurs

on the upwind side of the shock, while the downwind states converge to their

far field values within a few mesh blocks. In Fig. 3.2 (right), we measure the

34

Figure 3.2: Ensemble mean shock width (the green dots on the right) andthe standard deviation (the red dots on the left) of the shock width (leftframe). The mean width, equal to about 2∆x, is much larger than the standarddeviation, indicating that the mean width is essentially a deterministic featureof the solution. Convergence properties of the traveling wave to the steadystate values on each side of the wave (right frame). The straight line in theright frame is the asymptote to the exponential convergence rate, with slope0.01 in units of ∆x.

local extremum in the error dimensionlessly as emax = |(ρ − ρ∞)/ρ∞| where

ρ∞ is the far field density. We model emax(n∆x) = c exp(−λn) where n is

the distance from the shock front in mesh units. We find λ = 1.0 × 10−2 and

c = 2.4 × 10−4 in the present case. The first extremum is a local maximum,

occurring about 4∆x from the center of the shock front. The details of the

shock error behavior will be sensitive to the numerical method, but the general

form of the error, should be somewhat universal.

35

Fig. 3.3 (left) shows the larger contact width wc ∼ cct1/3 growing from 2

to 30 cells with a rate asymptotically proportional to t1/3. Similar asymptotics

have been observed by Harten [23] for an ENO scheme. The rate t1/3 results

from the second order accuracy of the method used here. The coefficient cc of

the growth law is sensitive to the ambient Mach number M of the flow, and

more specifically to a transport CFL number θ = |v|/(|v| + c) = M/(M + 1),

assuming that the algorithm has a time step set by the CFL limit ∆t =

∆x/(|v|+ c). Here c = max{cl, cr} is the maximum of the left and right state

sound speeds cl and cr respectively, and v is the fluid velocity. The contact

advances a fraction θ of one mesh cell in a single time step. For θ = 0 or θ = 1,

the contact does not move in its position relative to the grid lines and cc = 0.0.

This value (cc = 0.0) is applicable for a very narrow range of theta. For most

of the range of θ, the numerical diffusion is sensitive to the direction of mixing.

For θ < 0.5 and the flow from high density to low, the numerical mixing is

that of heavy fluid into light. We call this the step down problem. For the step

down problem, the ensemble mean contact width is shown in Fig. 3.3 (left).

For most θ values, we find cc ∼ 1,

The reverse, called the step up problem, flows from light to heavy fluid.

It mixes small amounts of light fluid into heavy, an effect less noticeable in

terms of the diffusion width, especially for large density contrasts. For step up

flow, we find wc ∼ min{5, cct1/3}. See Fig. 3.3, in which the mean wave width

grows firstly following wc ∼ cct1/3, then stops around wc ∼ 5. As a partial

explanation of this difference between the step down and step up problems, we

note that the spreading is primarily associated with the up stream side of the

36

Figure 3.3: Ensemble mean contact width for isolated noninteracting waves.Because the width is entirely grid related, we record width in units of ∆x andtime in units of the number of time steps. The standard deviations are alsoplotted as the red points to the extreme left in each frame. Left (step down):we observe an increase from 2 cells to 30 over 104 steps and an asymptoticgrowth rate cct

1/3, where cc ∼ 1 depends on the flow Mach number. Thestraight line in the left frame is the asymptote to the contact width, withslope 3. Right (step up): We observe a bound on the contact width.

contact, and that continued spreading (the t1/3 asymptotic) depends on the

up stream flow being subsonic. The higher sound speed in the light fluid gives

a supersonic upstream state for the step up problem but not for the step down

problem, for the flow parameters considered here. These properties appear

to be sensitive to the details of the numerical algorithm, and specifically to

the form of the limiter employed. We have used a MUSCL algorithm. The

general form of the error model and even some of its specific features, such as

shock wave widths should be solver independent. The degree of recalibration

of the model presented here for other solvers, is an important question which

37

is beyond the scope of this study.

For some aspects of the solution error, the probabilistic error formalism

is more general than is required. When the standard deviation of the error

is much smaller than the mean error (when the coefficient of variation, their

ratio, is close to zero), then the error is essentially deterministic, and the

probabilistic formulation is unnecessary. The standard deviation of the width,

shown to the left in each frame of Fig. 3.3, is significantly smaller than the mean

width, indicating that the wave width is essentially a deterministic feature of

the numerical solution. The wave position errors have a statistical aspect,

especially in their transient behavior.

3.3 Interactions with Contact Waves

We study wave strength, speed and position errors after a wave interac-

tion. Note that the SNRP errors considered so far are not the same as errors

in the SRP.

We represent the wave properties as a quadruple

wak = (ωa

k , λak, s

ak, p

ak) , (3.4)

where ω, as in Chap. 2, is a wave strength, λ is a wave width(A or P ), s is a

wave speed error, and p is a position error. Also a = i for input and a = o for

output. The units for wave strengths are dimensionless. We measure the wave

width in units of mesh spacing with contacts having either a t1/3 or bounded

time dependence. Thus we interpret λi2 as the input contact wave width at the

38

time of interaction (no time dependence) and either λo2t

1/3 or λo2 as the output

contact wave width, depending on the up stream Mach number. Here t is the

elapsed time since the interaction, expressed in time step units. As the shock

width is deterministic and time independent, it is not needed as an input or an

output variable. The wave speed error s is specified dimensionlessly as a ratio

(relative to the exact speed value), and occurs only as an output variable. It

refers to a far field (large time) value, after the waves are well separated. This

large time asymptotic speed error is very small, and will not be considered

further. Near field (near interaction) state errors are given by an oscillatory

decaying exponential, approaching the far field value. The exponent, or decay

rate, characterizes this approach in a simple overall manner. See Sec. 3.2.

Wave position errors are specified in grid units, after an isolated transient

period.

The input multi-index J = (j1, j2, j3) now has three components while

the output multi index has six (to reflect the dependence on three position

errors pok). Using J , we define the power expressions

wi,J = (ωi1)

j1(ωi2)

j2(λi2)

j3 .

For the three interactions we will study, the input wave width (λi2)

j3 is not

used, i.e. modeled as being zero. With these conventions, Eq. (2.1) also de-

scribes the model for the SNRP, but now each coefficient αk,J is also a random

variable, reflecting randomness in the numerical solution algorithm, such as

the dependence of the solution on the sub-grid location of the interaction point

39

relative to the grid. As in Chap. 2, we can subtract the base case values from

the ωik and ωo

k and then regard ω (as with the other variables) as representing

error or uncertainty.

We begin with the analysis of the SNRP at the ensemble averaged level.

We present the mean model analysis in Table 3.1, for the same case as in

Table 2.1, with ±10% variation about the base case. We only examine the

linear model. The input contact width has been set to zero, as part of the

specification of this SNRP.

variable \ coef const ωi1 ωi

2 error(r. sonic) (contact) L∞ STD

ωo1 (l. sonic) -0.208 0.454 0.251 0.47% 0.001

ωo2 (contact) -0.042 0.000 0.912 0.03% 0.0001

ωo3 (r. sonic) -0.286 1.004 0.346 0.30% 0.001

λo1 (l. sonic) 2.184 -0.563 0.000 122% 0.240

λo2 (contact) 4.725 0.110 -1.466 0.67% 0.010

λo3 (r. sonic) 2.197 0.068 0.106 5.35% 0.057

po1 (l. sonic) 0.221 -0.014 0.023 27.1% 0.022

po2 (contact) 0.426 0.001 -0.092 1.78% 0.002

po3 (r. sonic) 0.332 -0.004 -0.099 3.47% 0.005

Table 3.1: The SNRP shock contact (step up) interaction. Expansion coef-ficients for output wave strengths (linear model) for input variation ±10%.Here the base case input contact wave width is zero.

The first three rows of ω terms of Table 3.1 can be compared to the

corresponding SRP values from Table 2.2. We note the near identity of the

SRP and SNRP values, indicating the numerical accuracy of the numerical

scheme applied to Riemann data.

The three wave width rows in Table 3.1 represent new errors relative to

40

the SRP. The large sup norm error for the λo1 results from a few outliers. The

standard deviation for this quantity is about 10% of the mean value, indicating

that the error model is (on the whole) satisfactory, and that the shock wave

widths are not fluctuating greatly. The outliers are mainly associated with

time steps and realizations for which the (narrow) reflected shock has at most

one internal mesh point. For these cases, our wave filter tool for assessing

the numerical shock width and position is not effective, so the outliers can be

viewed as a breakdown of the diagnosis methodology.

The shock speed errors are nearly zero (to 5 digit accuracy in relative

shock speed errors) after the interacting waves separate; these errors are not

presented in Table 3.1. Consistent with the convergence properties of the

traveling wave to its left and right states shown in Fig. 3.2, accurate determi-

nation of shock speed requires well separated waves, and thus a highly resolved

calculation.

We now consider the wave position errors. This would be equivalent to

studying timing errors (for example in wave arrival), since position errors are

the integral of the velocity errors. Fig. 3.4 shows the position errors as a

function of time. The fact that the position errors reach a steady nonzero

asymptotic value is equivalent to the fact that the speed errors tend to zero,

with a nonzero time averaged value. The initial transient error will be signif-

icant when we consider errors in under resolved simulations. Following this

transient, the position error is constant, reflecting convergence of the wave

speed to its exact value. The errors in the wave position rows of Table 3.1

present this constant error. All position errors are sub-grid. The standard

41

deviations are smaller than the means, indicating that the errors are basically

deterministic. The L∞ position error is the supremum of the relative error.

Occasional ensemble members with very small position error produce a small

denominator in the relative error, (model error)/(exact error). Thus the large

entries in this column (also in other tables) do not represent a deficiency in the

error model. Note that the standard deviation is comparable to mean position

error, so that occasional instances of nearly zero error are to be expected.

Figure 3.4: Ensemble mean shock and contact position errors as a functionof time, expressed in grid units. Step up case. In the label, B.S.: BackwardShock, C: Contact, F.S.: Forward Shock.

Continuing the study of position errors, we examine shock crossing shock

and the shock crossing contact (step down) cases, with an emphasis on the

42

transient error. Fig. 3.5 shows that the transient error is significant, on the

order of the mesh spacing, and the standard deviation is also significant, in

contrast to the steady or large time asymptotics emphasized in Fig. 3.4. These

transient errors are important for underresolved simulations.

Figure 3.5: Ensemble mean shock and contact position errors as a functionof time, expressed in grid units. To emphasize the transient errors, a smallernumber of time steps are shown. Left: shock crossing shock case. Right: shockcrossing contact (step down) case.

3.4 Shock Crossing Shock Interactions

Here we study the reflection of the shock off a wall, a special case of a

shock crossing shock interaction. We can ignore the shock wave width para-

meters, as these are narrow and deterministic. We only consider one output

wave position error, po1 as presented in Table 3.2. The shock wave strength

errors (the far field, large separation errors) are small.

43

variable \ coef const ωi1 error

(r. sonic) L∞ STDωo

1 (l. sonic) -0.002 0.716 0.014% 0.000032λo

1 (l. sonic) 2.291 -0.422 7.923% 0.062po

1 (l. sonic) 0.060 -0.039 5065% 0.009ωo

2 (contact) 0.057 0.0003 24.4% 0.005λo

2 (contact) 5.9 50% 0.7

Table 3.2: The SNRP defined by the crossing of two shocks. Expansion coef-ficients for output wave strengths, widths and position errors (linear model)for input variation ±10%.

The contact mode contributes an error, well known as shock wall heating.

The exact solution doesn’t have this error in this mode for a wall reflection.

The error is due to the entropy, temperature and density variables from entropy

errors made during the shock interaction process. Since entropy can only

increase these errors do not cancel. Because the solution algorithm conserves

mass locally, we expect the spatial integral of the density errors to cancel

approximately. Since the velocity of the fluid at the wall is zero, these errors

remain permanently attached to the wall. We do not have a theoretical model

for the form of these errors; therefore, our fitting of the errors will be less

precise than those discussed elsewhere in this paper. We define the ”wall error

width” to be the distance to the wall in mesh units of the furthest location

for which the density error is at least twice the background noise in the post

shock region, or about 1% of the base case density. This width is about 6∆x.

The wave strength error is defined dimensionlessly as the L1 error in density,

divided by ρ0∆x, where ρ0 is the base case post shock density after the wall

reflection.

44

3.5 The Contact Reshock Interactions

Figure 3.6: Problem 3: Contact-shock (step down).

After reflection from the wall, the transmitted lead shock wave re-crosses

the deflected contact, see Fig. 3.6. This is a step down interaction. We have

one input and one output wave width parameter, both are for the contact.

According to the analysis of Sec. 3.2, the contact width is modeled as cct1/3

and this formula is accurate after some 50 time steps. In Table 3.3, entry

λo2 = cc. The rarefaction width has the form constant + rate × time. We find

very small errors in this rate, not tabulated here. The entry λo3 refers to the

constant, which gives an offset for the centering of the rarefaction wave.

45


2 error(contact) (l. sonic) L∞ STD

ωo1 (l. sonic) 0.282 -0.314 0.645 0.57% 0.0008

ωo2 (contact) 0.013 0.819 0.118 0.20% 0.0003

ωo3 (r. sonic) -0.128 0.143 0.468 0.41% 0.0004

λo1 (l. sonic) 2.383 0.754 -1.307 5.47% 0.038

λo2 (contact) 0.909 0.011 0.216 1.00% 0.005

λo3 (r. sonic) 3.619 0.151 -0.974 14.8% 0.138

po1 (l. sonic) 0.242 0.043 0.042 10.4% 0.014

po2 (contact) -0.036 0.045 0.066 75.5% 0.008

po3 (r. sonic) -0.447 0.078 -0.036 16.7% 0.029

Table 3.3: The SNRP shock contact (step down) interaction. Expansion co-efficients for output wave strengths, widths and position errors (linear model)for input variation ±10%.

46

Chapter 4

The Statistical Numerical Riemann Problems

for Spherical Flows

We consider the interaction of a spherically symmetric shock wave with a

contact located near the origin. The interactions are illustrated by the space

time contour plots of the density, shown in Fig. 4.1 (left). In Fig. 4.1 (right),

we show the type and location of the waves, as determined by the wave filter

analysis program. Both figures show results for the base case. The build up

of complex wave patterns is evident.

We consider statistical numerical Riemann problems (SNRP) in spherical

geometry. Because of the possible width of the input waves, the problem and

its solution are not strictly scale invariant. Moreover, scale invariance is lost

through the length scale introduced by the radius at the time of interaction.

We are then considering a generalization of the spherically symmetric Riemann

problem.

The statistics of the SNRP mapping function arise from grid errors, and

from the random placement of a traveling wave relative to the centers of the

47

Figure 4.1: Left. Space time density contour plot for the multiple wave in-teraction problem studied in this section, in spherical geometry. Right. Typeand location of waves determined by the wave filter analysis. For both plots,R indicates the radius distance from the origin. And B.S.: Backward Shock,C: Contact, and F.S.:Forward Shock in the label.

finite difference lattice. Our objective in this section is to build up a library

of statistical input-output relations that will include all Riemann problems to

be encountered in Sec. 5.2. This library will be used to predict results for the

multi-wave error and uncertainty analysis based on a multi-path scattering

formula, as developed in the Appendix.

4.1 The Single Propagating Wave

We start with the analysis of a single propagating inward shock in spheri-

cal geometry. The radially dependent strength of spherical shocks is discussed

48

in [36]. The spatial variation of spherical shocks is contained in the Guderley

solution [22].

An inward moving shock does not maintain constant strength as in planar

geometry, but evolves approximately according to a power law as a function

of the radius. From Whitham’s approximation approach,

M ∝ r−1/n (4.1)

for cylindrical shocks, and

M ∝ r−2/n (4.2)

for spherical shock, where M is the Mach number of the shock, n = 1 +

2γ

+√

2γγ−1

, and γ is the adiabatic exponent (ratio of two specific heats). A

comparison with the exponents from Guderley’s exact similarity solution is

given in Table 4.1. A similar approximate power law also exists for the shock

velocity.

Cylindrical Sphericalγ Approximate Exact Approximate Exact

6/5 0.163112 0.161220 0.326223 0.3207527/5 0.197070 0.197294 0.394142 0.3943645/3 0.225425 0.226054 0.450850 0.452692

Table 4.1: Comparison of the exponents from the approximate and the exactsimilarity solutions.

Fig. 4.2 shows the exponential divergence of the shock strength (here

characterized by the Mach number) at r → 0. The accuracy is amazing

49

Figure 4.2: Left. Mach number vs radius for a single inward propagatingshock. Right. The same data plotted on log-log scale.

in view of the simplicity of the approximate theory. The figure shows that

converging shocks are reacting primarily with the geometry, as assumed in the

approximate theory, and are affected very little by further disturbances from

the source of the motion; the strength of the initial shock enters only through

the constants of proportionality in (4.1) and (4.2). This is not true for outward

moving shocks. They slow down due to both the expanding geometry and to

the continuing interaction with the flow behind. From Fig. ??, however, we

find that the strength of an outward moving shock also follows a power law

which is similar to (4.2) but with a modified exponent, after the radius of the

outward moving shock is three times the initial radius. To develop a model for

shock wave propagation which has a smaller pre-asymptotic regime, we allow

50

two distinct exponents,

M ∝

⎧⎪⎨⎪⎩

ra1

ra2

r0 ≤ r ≤ 3r0

r ≥ 3r0.(4.3)

Here we choose a1 = −0.4, a2 = −1.0 for γ = 1.67, and r0 is the initial shock

radius.

Figure 4.3: Left. Mach number vs radius for a single outgoing propagatingshock. Right. The same data plotted on log-log scale.

We also study a single propagating contact. Fig. 4.4 shows that the

contact width grows from 2 to 5 cells with an asymptotically rate wc ∼ cct1/2.

For some aspects of the solution error, the probabilistic error formalism is more

general than is required. The standard deviation of the width, as shown to

the left in the frame of Fig. 4.4, is significantly smaller than the mean width,

indicating that the wave width is essentially a deterministic feature of the

numerical solution.

51

Figure 4.4: Ensemble mean contact width for single propagating contact. Werecord width in units of ∆x. The standard deviations are also plotted. andare the points to the extreme left in the frame.

4.2 The Shock Contact Interaction

We study the wave strength, speed, width and position errors after a

wave interaction. Similar to Eq. 3.4, we represent the wave properties as a

five tuple

wak = (ωa

k , δak , λ

ak, s

ak, p

ak) , (4.4)

where ω is a wave strength, δ is an error in the wave strength, λ is a wave

width, s is a wave speed error, and p is a position error. Also a = i denotes

input and a = o signifies output. We choose dimensionless variables to measure

wave strengths; the Atwood number A = (ρ2 − ρ1)/(ρ2 + ρ1) to measure the

contact strength, and the Mach number M defined as the ratio of the shock

52

speed to the ahead state sound speed, in the frame of a stationary ahead

state, for the shocks. We measure the wave widths in units of mesh spacing.

The wave position errors are specified in grid units, after an initial transient

period. Within this formulation, we can describe the output wave errors by an

expression multi-linear (and eventually linear) in the two input wave strengths,

i.e. linear in each of the two input wave strengths.

We begin with the analysis of the initial shock contact SNRP at the

ensemble averaged level. We present the linear model coefficients in Table 4.2,

with ±10% variation for the initial contact strength and ±5% variation for the

initial shock strength (consistent with ±10% variation in pressure ratio as used

in the planar study) about the base case. According to the analysis of Sec. 4.1,

the strength of this initial inward shock is not constant, and is increasing as it

moves toward the origin. We use the power law M = Cr−2/n to estimate the

initial shock strength at the interaction time and use this quantity represented

by the variable C as the input shock strength in the modeling. The input

contact width has been set to zero, as part of the specification of this SNRP.

To read Table 4.2, we note that the first (ωo1) row (labeled in the table

as wo1 (l. sonic)) lists coefficients α1,J for J = (0, 0), J = (1, 0), etc. These

coefficients are determined by a least squares algorithm that minimizes the

expected, or mean error over the ensemble, in comparing the linear predic-

tions to the exact solution of the Riemann problem. The last two columns

describe errors in the model (2.1). The presence of outliers was monitored and

the ensemble L∞ norm determined (results not tabulated); occasional outliers

indicate non Gaussian statistics. The model error is defined as (predicted - ex-

53

act) where exact is the result of the simulation and predicted is the value given

by the finite polynomial (linear) model (2.1). The column STD is the standard

deviation of (predicted - exact). Note that the STD errors, as defined, are di-

mensionful. To aid in interpreting the error magnitudes, we present in a final

column (labeled STD/ωo) the standard deviation of the error in the model

divided by the mean value of the variable predicted. This column represents

a fractional (dimensionless) error in the model.

According to the analysis of Sec. 4.1, the strength of the output inward

moving shock is modeled as Cr−2/n. This formula is accurate after some time,

and the Table 4.2 entry is ωo1 = C in this formula. We form a linear model for

this constant in this expression in Table 4.2. We find very small errors in the

exponent, not tabulated here. We developed a model (4.3) for the strength of

the output outward moving shock in Sec. 4.1. Here in our study, we are only

concerned with the first formula in (4.3). The entry ωo3 in the table represents

the coefficient multiplying the power term.

The three variable (λ) rows in Table 4.2 represent wave width errors.

The standard deviation for this quantity is about 10% of the mean value,

indicating that the error model is (on the whole) satisfactory, and that the

shock wave widths are not (mostly) fluctuating greatly. The inward moving

shock width decreased about 10% relative to the wave width at the interaction

time, while the outward moving shock width increased about 10%. See Fig. 4.5,

left frame. The contact width is modeled as cct1/5 where both the width and t

are expressed in mesh units. The Table 4.2 entry λo2 = cc in this formula. We

form a linear model for this constant in this expression in Table 4.2.

54

Figure 4.5: Left: ensemble mean inward/outward moving shock and contactwidths after a shock contact interaction. Right: ensemble mean shock andcontact position errors as a function of time, expressed in grid units. Theassociated standard deviations are extremely small, not shown in the plots.In the legend, C. denotes the contact while I.S. and O.S. are the inward andoutward moving shocks.

We also study the wave position errors. Fig. 4.5, right frame, shows the

position errors as a function of time. The entries in the wave position rows

of Table 4.2 present those errors, given in mesh units. All position errors are

subgrid. The standard deviations are smaller than the means, indicating that

the errors are basically deterministic.

All solution errors are sensitive to the grid spacing, taken to be 100

computational cells in Table 4.2-4.4. This sensitivity is not extreme. For

example, if the 100 cell model is used to analyze the 500 cell data, the model

errors (STD) approximately double, but remain small.

55

4.3 Shock Reflection at the Origin

Here we study the reflection of the shock off the origin. According to the

analysis of Sec. 4.1, the input inward moving shock has infinite strength at the

origin. We used the strength at the radius r = 1 as the initial state and the

input shock strength in the modeling process. We study the wave strength,

wave strength errors, wave width and wave position errors. See Table 4.3.

We found that the Mach number of the outward moving shock (reflected

shock) was essentially independent of the input variation in Mach number. To

explain this phenomena, we recall that the ambient state ahead of the outward

moving reflected shock is an incoming continuously variable flow. The sound

speed ahead of this flow is affine linearly dependent on the strength of the

incoming shock wave, as is the shock speed of the reflected outward moving

shock wave. Thus the outward moving Mach number, as a ratio of two quan-

tities varying affine linearly with the incoming shock strength, has a fractional

linear form in the incoming wave strength. A simple calculation shows that

the variation in the outward moving shock Mach number Mo contains the fac-

tor (1−Mo) and since Mo ≈ 1.2, this small factor suppressed variation in Mo

as a function of Mi, the Mach number of the incoming shock. Thus the Mach

number is not a good measure for the outward moving shock strength. We

choose the pressure behind the reflected shock instead as ωo1 in Table 4.3. The

pressure also follows the power law. The large entries in this row result from

the fact that the (dimensional) pressure (ωo1) is much larger in pressure units

than the Mach number (ωi1).

56

4.4 The Contact Reshock Interaction

After reflection from the origin, the transmitted lead shock wave re-

crosses the deflected contact. The outgoing waves from this interaction consist

of a rarefaction wave propagating toward the origin, a contact and a shock

propagating outward. The region inside of the outward propagating shock, on

both sides of the contact is not piecewise constant, but contains an inward

propagating compression, which eventually breaks to form an inward moving

shock, reaching the origin at interaction 4. This inward moving compression

is generated from the geometrically caused weakening of the outward moving

shock, and is a well recognized aspect of spherical shock wave dynamics. The

shock and the rarefaction interact, and eventually the rarefaction disappears

in this interaction. Here we only follow the waves through the output of in-

teraction 3, and thus avoid much of this interaction. Specifically, we focus on

the inward moving rarefaction and not the inward moving shock. We study

the wave strength, wave strength errors, wave width and wave position errors

resulting from interaction 3. See Table 4.4.

According to the analysis of Sec. 4.1, this is a step down interaction and

the contact width is modeled as cct1/5 where both the width and t are expressed

in mesh units. We form a linear model for the coefficient cc in this expression

in Table 4.4. The rarefaction width has the form constant + rate × time. The

entry λo1 refers to the constant, which gives an offset for the centering of the

rarefaction wave. This entry is expressed in mesh units.

57


2 model error

(contact) (l. sonic) STD STD/ωo

wave strengths (100 cells)

ωo1 (l. sonic) -33.353 19.521 2.501 0.860 0.954%

ωo2 (contact) 0.374 0.200 0.0003 0.042 7.650%

ωo3 (r. sonic) 3.568 0.402 -0.045 0.009 0.463%

wave strength errors (100 cells)

δo1 (l. sonic) 2.039 -3.200 -0.01 0.157 0.174%

δo2 (contact) 0.236 0.016 -0.002 0.021 3.825%

δo3 (r. sonic) 0.053 0.003 -0.001 0.0008 0.041%

wave width errors (100 cells)

λo1 (l. sonic) 1.675 0.305 0.017 0.085

λo2 (contact) 7.093 0.482 -0.146 0.239

λo3 (r. sonic) 2.829 0.302 -0.024 0.107

wave position errors (100 cells)

po1 (l. sonic) -0.247 0.242 0.005 0.009

po2 (contact) 0.643 0.065 -0.011 0.192

po3 (r. sonic) -0.042 0.062 0.004 0.009

Table 4.2: The SNRP shock contact interaction. Expansion coefficients foroutput wave strengths, wave strength errors, wave width errors and wave po-sition errors (linear model) for the initial shock contact interaction. Here thebase case input contact wave width is zero. The final columns refer to dif-ference between the linear model (2.1) and the exact quantity. The errors inrows 4-12 refer to the difference between the numerical solution on 100 cellsand the exact solution using 2000 cells.

58

variable \ coef const ωi1 model error

(l. sonic) STD STD/ωo

ωo1 (r. sonic) -242.394 5.606 1.137 0.468%

δo1 (r. sonic) -3.27 0.031 0.112 0.045%

λo1 (r. sonic) 1.221 0.018 0.099

po1 (r. sonic) 0.474 0.001 0.012

Table 4.3: The SNRP defined by the shock reflection at the origin. Expansioncoefficients for output wave strengths, wave strength errors, wave width errorsand wave position errors (linear model) for input variation ±10%.

59


2 model error

(r. sonic) (contact) STD STD/ωo

wave strengths (100 cells)

ωo1 (l. sonic) 0.097 -0.108 0.436 0.031 13.305%

ωo2 (contact) 0.103 -0.192 1.168 0.007 1.116%

ωo3 (r. sonic) 0.988 0.195 -0.225 0.003 0.262%

wave strength errors (100 cells)

δo1 (l. sonic) -0.291 0.161 -0.468 0.017 7.296%

δo2 (contact) -0.067 0.142 -0.125 0.006 0.957%

δo3 (r. sonic) -0.030 0.107 -0.0004 0.001 0.087%

wave width errors (100 cells)

λo1 (l. sonic) 9.776 -6.372 5.091 0.484

λo2 (contact) 1.903 0.156 -0.677 0.534

λo3 (r. sonic) 4.088 -1.401 1.549 0.168

wave position errors (100 cells)

po1 (l. sonic) 4.782 -3.602 2.372 0.379

po2 (contact) -0.453 0.409 -0.054 0.177

po3 (r. sonic) -0.199 -0.685 3.213 0.052

Table 4.4: The SNRP contact reshock interaction. Expansion coefficientsfor output wave strengths, wave strength errors, wave width errors and waveposition errors (linear model).

60

Chapter 5

Composite Shock Interaction Problems

5.1 Composite Shock Interaction Problems In Planar

Flows

Here we introduce a formula for combining the wave interaction errors

defined in Chap. 2 and 3 for isolated Riemann problems, to yield the error

for arbitrary points in the colution of the complex wave interaction problem.

The formula is validated for fully resolved simulations and it is shown to be

partially correct and partially incorrect for under resolved simulations.

5.1.1 A Multipath Integral for a Nonlinear Multiscat-

tering Problem

We begin with a formula expressing the error in a given Riemann problem

R0 as multinomial expansion associated with initial waves and errors located

inside its domain of dependence. For a 1D shock wave interaction problem,

think of the solution as being primarily composed of localized waves, interact-

61

ing through Riemann problems and generating outgoing waves, that further

interact in the same manner. Each wave w is described by a vector νw that

records its strength, location in state space, speed and starting location and

time, and the errors or uncertainty associated with these quantities. The in-

teraction of waves generates a planar (1D space and time) graph, the vertices

of which are the Riemann problems and the bonds are the traveling waves,

between Riemann problem interactions. Call this graph G. Starting from a

given Riemann problem (vertex) or wave (bond), we can trace backward and

determine its domain of dependence.

For each Riemann problem, we consider three types of vertices, corre-

sponding to the constant, linear and bilinear terms in the parameterized ap-

proximate solution and error terms developed in Chap. 3. We treat the linear

terms separately from the others, as they allow a simple propagation law,

SL =

∫w(t = 0)dω (5.1)

where w(t = 0) is a vector representing the strength of the time zero wave and

its error or uncertainty, evaluated at the beginning of the path ω, and SL is the

purely linear propagation contribution to a final time error. The path space

integral dω is taken over all paths progressing in time order through G from

the initial time to the final vertex, with each term weighted by the appropriate

linear factors from the formula for the approximate solution of the Riemann

problems transversed. This path space representation makes evident the point

that the solution SL is that of a multiple (linear) scattering problem.

62

The amplitude S at the final time (vertex of G) can similarly be thought

of as a solution of a nonlinear multiple scattering problem, leading to a rep-

resentation in terms of multipath integrals. To allow nonlinear (constant and

bilinear) interactions, we re-introduce the vertices from these other terms. Let

V = V(G) be the set of vertices of G, and let B ⊂ V be a subset of V where

constant or bilinear terms occur. The total amplitude S will then be a sum

over terms SB indexed by B. For each v ∈ B, let Iv be the the interaction

coefficient, taken from a table of Chap. 3. We write

S =∑

B⊂V(G)

SB =∑

B⊂V(G)

∫ ∏v∈B

IvdωB . (5.2)

Here dωB is a multipath integral over all multipaths (directed subgraphs of

G starting at t = 0 or at constant vertices, coalescing at bilinear vertices

v ∈ B and ending at the final vertex in G. The multipath propagator dωB

is a product of the individual propagators ω for each single path, as in (5.1).

The summation in (5.2) can be understood schematically as the sum over all

events within the domain of dependence of the evaluation point (x, t) at the

vertex of G. See Fig. 5.1.

5.1.2 Evaluation of the Multipath Integral

In Fig. 5.2 we illustrate the distinct terms contributing to (5.2). Each

graph is a single term, for the error associated with the output to interaction

3, in which the shock wave reflected from the wall recrosses (re-shocks) the

contact. The first two graphs indicate the uncertainty originating with the

63

Figure 5.1: The solution and its errors at the point (x, t) can be obtainedby “adding up” the solution and errors for the waves within the domain ofdependence

initial conditions, i.e. with the choice of the ensemble. This uncertainty

propagates through two distinct paths, illustrated by the first two graphs of

Fig. 5.2, to reach the interaction site 3. The first graph follows the shocks,

the transmitted shock from the interaction 1 to the wall reflected shock and

back to the contact. The second graph follows the the contact from the lead

interaction 1, along the contact until it is reshocked at interaction 3. Next

we find two graphs that represent the errors originating during the interaction

1, and propagating to the output of 3 through the same two routes. Finally,

we find two graphs giving the errors that arise at the shock wall reflection

(interaction 2) and propagate to 3 and in the final graph, those arising during

interaction 3 directly.

From prior work [9], we know that the dominant errors in the composite

solution are located within the leading shock and contact waves of the problem.

64

A portion of these errors are simple resolution errors. This means that they

are errors due to the difference between the numerical and the exact wave

forms as traveling waves. This portion of the error is independent of the wave

interactions, and in the variables we use to describe the traveling waves, and

shows up in the wave width only. Any other errors, e.g. in strength or position,

or errors in width beyond these resolution errors can be attributed to the wave

interactions.

Now we explain in detail the definition of each of the terms associated

with these graphs. The simplest is the last. It is the error created during

the solution of the interaction at Riemann problem case 3. For this term,

we accept the numerical input to case 3 (as defined by a coarse grid solution,

including the numerical error). This input data is solved using both a fine grid

and a coarse grid, and the difference is the error associated with this graph.

The same formula applied to the initial point of the other graphs, defines the

beginning of each of the graphs 3 - 5. From this input error, we proceed as

follows: The error is transmitted without change along the edges of the graph.

For graph 3, the error is transmitted from output of the first interaction of

the graph, that is the pure transmitted shock wave output of interaction case

1 to the input of interaction case 2 in the same graph. The case 3 Riemann

problem then sees this error as an initial uncertainty and transforms it via a

linear transformation, to an output error or uncertainty located at the output

of interaction 3. The definition of the other graphs is similar, containing a

sequence of linear transformations, one for each Riemann problem the error

signal passes through.

65

Figure 5.2: Schematic graphs illustrating all contributions to the errors or un-certainty in the output from a single Riemann solution, namely the reshockinteraction (case 3) of the reflected shock from the wall as it crosses the con-tact. The numbers labeling the black circles refer to the Riemann interactionscontributing to the error. The letter I in the first two diagrams indicates inputuncertainty.

We develop these formulas explicitly for the first two graphs of Fig. 5.2,

representing one component of the propagation of the initial uncertainty (ini-

tial ensemble) to the output uncertainty of interaction 3. The initial uncer-

tainty is reflected in the wave strength variables, according to the definition

of the ensemble of initial conditions. The transmission of the mean values

through a linear model is standard and is not detailed here. The transforma-

tion of the variance follows formulas from Chap. 2. Let B(l) with matrix entries

β(l)jk be the matrix that gives the linear transformation of these variables due

to interaction l, as in (2.10). We note that the matrix entries β(l)jk are defined

by the ωi columns and ωo rows of Table 3.1 (linear; l = 1), Table 3.2 (l = 2),

66

and Table 3.3 (l = 3). The output to interaction 3 has three components, and

we compute the variance of each, labeled j = 1, 2, 3. By formula (2.13), we

have

Var ωo(3)j = (β

(3)j2 )2Var ω

i(3)2 = (β

(3)j2 )2Var ω

o(2)1

= (β(3)j2 )2(β

(2)11 )2Var ω

i(2)1 = (β

(3)j2 )2(β

(2)11 )2Var ω

o(1)3

= (β(3)j2 )2(β

(2)11 )2

2∑k=1

Var (β(1)3k )2ω

i(1)k (5.3)

This formula can be evaluated explicitly. We denote the LHS as SB1 in accor-

dance with (5.2).

SB1 =

⎛⎜⎜⎜⎜⎝

0.3122 0.6452

0.8092 0.1242

0.1422 0.4682

⎞⎟⎟⎟⎟⎠⎛⎝ 0

Var (wi32 )

⎞⎠

=

⎛⎜⎜⎜⎜⎝

0.3122 0.6452

0.8092 0.1242

0.1422 0.4682

⎞⎟⎟⎟⎟⎠ 0.7152

⎛⎝ 0 0

1.0012 0.3472

⎞⎠⎛⎝ Var (wi

1(t = 0))

Var (wi2(t = 0))

⎞⎠(5.4)

Similarly, the variance associated with the second graph can be evaluated

as

SB2 =

⎛⎜⎜⎜⎜⎝

0.3122 0.6452

0.8092 0.1242

0.1422 0.4682

⎞⎟⎟⎟⎟⎠⎛⎝ Var (wi3

1 )

0

⎞⎠

=

⎛⎜⎜⎜⎜⎝

0.3122 0.6452

0.8092 0.1242

0.1422 0.4682

⎞⎟⎟⎟⎟⎠⎛⎝ 0.0002 0.9112

0 0

⎞⎠⎛⎝ Var (wi

1(t = 0))

Var (wi2(t = 0))

⎞⎠ (5.5)

67

We also need to calculate formulas giving the transmission of position errors

through the various Riemann problems. For interaction with a wall (e.g. case

2), the formula is elementary. Assuming no error in the wall position, let pi and

po denote input and output position errors for a reflection off of a stationary

wall, where the output is due transmission of error, i.e. due only to the input,

as opposed to Chap. 3, where there is no input position error and the output

position error is created during the interaction. Then we have

po = pi vo

vi(5.6)

where vi and vo are the incoming and outgoing wave speeds for the waves

involved in the wall reflection. For the interaction of two incoming waves, the

result is slightly more complicated. Each of the two terms in the formulas

below is due to the input error in one of the input waves. That error can be

computed by (5.6) if we perform the analysis in the frame in which the other

wave is stationary. The result is

po1 =

pi1(v

o1 − vi

2) + pi2(v

i1 − vo

1)

vi1 − vi

2

(5.7)

po2 =

pi1(v

o2 − vi

2) + pi2(v

i1 − vo

2)

vi1 − vi

2

(5.8)

po3 =

pi1(v

o3 − vi

2) + pi2(v

i1 − vo

3)

vi1 − vi

2

(5.9)

where pij is the position error of the incoming contact wave (j = 1) or left facing

shock (j = 2). The complete position error model is obtained by adding the

68

results of (5.7) - (5.9) to those of Table 3.3 for the position errors created at

the interactions. For interactions 1 - 3, we model the wave width (error) as a

created error only. Thus only the final graph of Fig. 5.2 contributes to this.

5.1.3 Errors in Resolved Calculations

We regard a calculation as resolved if all (the principal) waves have sep-

arated, with converged left and right asymptotic states, before they interact

with another wave. For this type of simulation, we choose 500 mesh cells in

our basic simulation study, and assess errors in comparison to a 5000 cell sim-

ulation. We examine errors in wave strength, wave position and wave width,

based on the graphical expansion given in Sec. 5.1.1, 5.1.2. The wave strength

errors are dominated by the transmission of error (or uncertainty) from the

initial conditions, and thus are given by the first two diagrams of Fig. 5.2. In

Tables 5.1, 5.2 and 5.3 we compare the predicted error with the error computed

directly, taken from a full solution of the multiple wave interaction problem.

The model for the prediction of the error is satisfactory for all cases: the wave

strength and its errors, the wave width errors, and the wave position errors.

5.1.4 Errors in Under Resolved Calculations

Here we allow 100 cells for the coarse grid simulation. This resolution

allows 10 cells between the contact and the reflection wall at the time of

interaction 3 and beyond. Since the contact has a width of 5 cells, since the

right facing rarefaction is about this size and since the wall has inaccurate

states in a region of several mesh blocks neighboring it, we are clearly at the

69

variable \ error Simulation Predictionmean wave strengths

ωo1 (l. sonic) 0.451 0.452

ωo2 (contact) 0.704 0.703

ωo3 (r. sonic) 0.999 0.998

wave strength errorsVar ωo

1 (l. sonic) 0.0008 0.0008Var ωo

2 (contact) 0.0019 0.0018Var ωo

3 (r. sonic) 0.0035 0.0036wave width errors

λo1 (l. sonic) 1.630 1.622

λo2 (contact) 3.636 3.635

λo3 (r. sonic) 2.346 2.352

wave position errorspo

1 (l. sonic) 0.220 0.226po

2 (contact) 0.313 0.312po

3 (r. sonic) 0.200 0.202

Table 5.1: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, Case 1.

limit of the present diagnostic methods based upon the wave filter. For the

same reasons, the calculation is clearly under resolved. For this reason we are

not able to analyze data for the case of a coarse grid simulation with 50 cells

using the present version of our wave filter. Again we present the first three

interactions in detail, at 100 cell resolution, comparing the predicted to the

directly simulated errors. See Tables 5.4, 5.5, 5.6. We see good results for the

wave strengths and their errors and for the wave width errors, and poor results

for the comparison of position errors. This can be understood in terms of the

decay time for convergence to asymptotic large time values for the position

errors, an explanation that also accounts for the difference with the resolved

70


ωo1 (l. sonic) 0.713 0.714


1 (l. sonic) 0.0018 0.0018wave width errors

λo1 (l. sonic) 1.868 1.869


1 (l. sonic) -0.118 -0.092


case, for which the simulated and predicted position errors agree. The position

errors have a relatively slower decay time. The other three quantities show a

high level of agreement between the resolved and under resolved cases. The

wave width error is expressed in grid units, and so should be the same in the

two cases differing in grid resolution only. For the wave strength entries, the

lack of dependence on grid resolution is due to the fact that these quantities are

dominated by the uncertainty expressed in the ensemble of initial conditions,

which is independent of grid resolution.

5.2 Composite Shock Interaction Problems in Spherical

Flows

The main point of this section is to formulate and validate the multipath

scattering formula of Sec. 5.1 for analysis of spherical flow errors.

71


ωo1 (l. sonic) 0.520 0.519

ωo2 (contact) 0.674 0.674

ωo3 (r. sonic) 0.306 0.305


1 (l. sonic) 0.0009 0.0010Var ωo

2 (contact) 0.0012 0.0013Var ωo


λo1 (l. sonic) 2.097 1.982

λo2 (contact) 5.027 4.918

λo3 (r. sonic) 2.875 3.033


1 (l. sonic) -0.097 -0.105po

2 (contact) -0.003 0.013po

3 (r. sonic) -0.151 -0.134


5.2.1 The Multipath Error Analysis Fromula

We consider the repeated interactions of a spherically symmetric shock

wave with a spherical contact located near the origin. The base case for each

wave interaction coincides with the base case assumed for the interactions

studied in Chap. 4. The transmitted shock, after interaction with the contact,

progresses to interact with (i.e. reflect off) the origin. This interaction was

also studied in Chap. 4. Subsequently, there are a number of reverberations,

of reflected rarefaction and compression waves, between the contact and the

origin. The interactions are illustrated by the space time contour plots of

the density, shown in Fig. 4.1 (left), and Fig. 4.1 (right) shows the type and

72


ωo1 (l. sonic) 0.451 0.452

ωo2 (contact) 0.741 0.703

ωo3 (r. sonic) 0.996 0.998


1 (l. sonic) 0.0008 0.0008Var ωo

2 (contact) 0.0022 0.0018Var ωo


λo1 (l. sonic) 1.381 1.621

λo2 (contact) 3.498 3.635

λo3 (r. sonic) 2.347 2.352


1 (l. sonic) 0.972 0.226po

2 (contact) 1.539 0.312po

3 (r. sonic) 0.785 0.202

Table 5.4: Case 1. The contact-shock interaction (step up). Errors for outputwave strengths, wave widths and wave position. Comparison of under resolvedsimulation and prediction.

location of the waves determined by the wave filter analysis program.

We first appeal to the fine grid simulation data (as a stand in for the exact

solution) to develop or parameterize an affine linear model for the output

wave strengths at interaction j in terms of the input wave strengths. This

formula, considered variationally, also yields a model for the transmission of

error/uncertainty through interaction j. We additionally initialize comparable

input on the fine and coarse grids at interaction j, so that the difference

between the fine and coarse grid solutions is the created error at interaction j.

The created error, C(j)s , in the wave strength w at interaction j is thus defined

73


ωo1 (l. sonic) 0.721 0.712



λo1 (l. sonic) 1.718 1.871


1 (l. sonic) -0.401 -0.092

Table 5.5: Case 2. The shock crossing equal shock (wave reflection) interaction.Errors for output wave strengths, wave width and wave position. Comparisonof under resolved simulation and prediction.

as

C(j)s = coarse − fine = w

o(j)s(c) − w

o(j)s(f) .

Here the three output waves are indexed form left to right by s, 1 ≤ s ≤ 3.

The superscripts o and i represent output and input strengths respectively,

and the subscripts f and c represent fine and coarse grid solutions.

The linear regression, or linear approximate formula for the output wave

strength at interaction j is obtained from the fine grid simulation data,

wo(j)s(f) = α(j)

s +2∑

k=1

β(j)sk w

i(j)k(f) . (5.10)

We use this formula variationally to obtain a formula for the transmission of

input error δwi(j)k . The input error itself is a combination of transmitted input

uncertainties Ic, Is and created errors C(i)s also transmitted from interactions

74


ωo1 (l. sonic) 0.523 0.514

ωo2 (contact) 0.669 0.705

ωo3 (r. sonic) 0.318 0.315


1 (l. sonic) 0.0009 0.0010Var ωo

2 (contact) 0.0013 0.0013Var ωo


λo1 (l. sonic) 1.964 2.000

λo2 (contact) 4.606 4.928

λo3 (r. sonic) 2.169 3.029


1 (l. sonic) -0.551 -0.103po

2 (contact) 0.301 0.015po

3 (r. sonic) 0.432 -0.131

Table 5.6: Case 3. The contact-shock interaction (step down). Errors foroutput wave strengths, wave width and wave position. Comparison of underresolved simulation and prediction.

i < j,

δwo(j)s(f) =

2∑k=1

β(j)sk δw

i(j)k . (5.11)

A source S(i)r of errors contributes to the output error through wave interaction

diagrams g ∈ G(j)s(i)r , which connect the rth source at interaction i to the sth

output at the interaction j. Such diagrams were introduced in Sec. 5.1. A

shock wave as input to an intermediate interaction m with radius r(m) in a

diagram g ∈ G is propagated from the previous interaction m′ occurring at

the radius r(m′) according to a power law. The power law gives the definition

of a propagator with a proper power a(m,g) for the m′ m bond of the diagram

75

Figure 5.3: Schematic graphs, showing all six wave diagram contributionsto the errors or uncertainty in the output from a single Riemann solution,namely the reshock interaction (numbered 3 in the right frame of Fig. 4.1) ofthe reflected shock from the origin as it crosses the contact.

g,

P(m)(m′) =

(r(m)

r(m′)

)a(m,g)

. (5.12)

We define γ(m)sk for the sth output and the kth input at interaction m as the

multiple of a coefficient and a propagator, which are defined by a bond in the

diagram g (i.e.β is related to g and P to g)

γ(m)sk = β

(m)sk P(m)

(m′) . (5.13)

The coarse grid output error is the sum of the transmitted errors from (5.11)

and the error created at interaction j,

δwo(j)s(c) =

∑1≤r≤30≤i≤j

∑g∈G(j)s

(i)r

CgS(i)r , (5.14)

76

where Cg =∏

B(g)γ

(m)sk and the product runs over all bonds B(g) of the diagram

g. Observe that the created error at interaction j is included in the i = j term

in (5.14).

At the output to the interaction 3, we have the following closed form

expression for the mean,

〈δwo(3)s(c) 〉 = γ

(3)s1 γ

(2)11 〈C(1)

1 〉 + γ(3)s2 〈C(1)

2 〉 + γ(3)s1 〈C(2)

1 〉 + 〈C(3)s 〉 . (5.15)

Observe that 〈Is〉 = 〈Ic〉 = 0, so that with a linear propagation model and

assumed mean zero initial uncertainty, the two initial uncertainties do not con-

tribute to the mean error. We assume statistical independence of the sources,

to obtain a closed form expression for the variance of the error,

Var δwo(3)s(c) =

{(γ

(3)s1 γ

(2)11 γ

(1)11

)2

+(γ

(3)s2 γ

(1)21

)2}

Var Ic

+

{(γ

(3)s1 γ

(2)11 γ

(1)12

)2

+(γ

(3)s2 γ

(1)22

)2}

Var Is

+(γ

(3)s1 γ

(2)11

)2

Var C(1)1 +

(γ

(3)s2

)2

Var C(1)2

+(γ

(3)s1

)2

Var C(2)1 + Var C(3)

s . (5.16)

5.2.2 Results

We analyze errors at the output to interaction 3 directly, comparing the

100 mesh and 500 mesh simulation to a 2000 mesh, fine grid simulation, here

taken as a substitute for the exact solution. These errors are compared to those

generated by adding up and propagating errors from the input data and from

77

variable \ error Simulation Prediction Simulation Prediction100 vs. 2000 mesh 500 vs. 2000 meshwave strength errors and propagated initial uncertainties

δo1 (l. sonic) 0.04±2(0.03) 0.03±2(0.02) 0.01±2(0.02) 0.009±2(0.01)

δo2 (contact) 0.14±2(0.05) 0.12±2(0.02) 0.03±2(0.01) 0.03±2(0.008)

δo3 (r. sonic) -0.02±2(0.02) -0.02±2(0.01) -0.006±2(0.005) -0.007±2(0.004)

mean wave width errors mean wave width errorsλo

1 (l. sonic) 3.04 2.83 2.63 2.72λo

2 (contact) 5.36 6.11 5.56 6.08λo

3 (r. sonic) 2.71 3.04 2.92 2.98mean wave position errors mean wave position errors

po1 (l. sonic) 1.25 0.23 0.12 0.18

po2 (contact) 0.43 0.06 0.05 0.04

po3 (r. sonic) -0.73 -0.15 -0.08 -0.11

Table 5.7: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, output to interaction 3. The inward rarefactionand contact strengths are expressed dimensionlessly as Atwood numbers. Theoutward shock strengths are in the units of Mach number. The width andposition errors are in mesh units. The wave strength errors are expressed asmean ± 2σ where σ is the ensemble STD of the error/uncertainty.

each of the interactions 1 to 3, using the multipath scattering formula. Thus,

for example, a position error as input to interaction 1 is translated geometri-

cally to a position error for the output to interaction 1 via simple geometric

considerations as in Sec.5.1. This error is propagated to an input error for

interaction 2 through solutions of radial differential equations. Propagation

continues, and yields an error at the output to interaction 3. See Table 5.7.

The wave strength rows present the result of initial uncertainty propagated to

the output of interaction 3 as well as the accumulation of solution errors. The

multipath scattering formula gives reasonable prediction of error magnitudes

in all cases except the wave position errors for the under resolved (100 mesh)

78

simulation. We see that the created numerical solution errors are important.

We also find that a major portion of the created numerical solution errors

come from the second interaction, the shock reflection interaction.

In Figs. 5.4, 5.5, we present three pie charts representing fractional con-

tribution from each of the six interactions to the error variance for the inward

rarefaction, contact and outward shock, respectively, as output to interaction

3. From these charts, we can infer the relative importance between the in-

put uncertainty and the solution error and determine the contribution of each

interaction to the total error variance.

We also show the contributions of each interaction to the mean value of

the final total error. See Table. 5.8. We only show the values correspond-

ing to diagrams 3 to 6, as the contribution of the first two diagrams (input

uncertainties) is observed to be zero.

Wave Diagram I.R. C. O.S.Number 100 500 100 500 100 500

3 0.10 -0.01 -0.01 0.001 0.09 -0.0094 0.05 0.009 0.1 0.02 -0.02 -0.0045 -0.05 -0.005 0.006 0.0006 -0.04 -0.0046 -0.07 0.015 0.03 0.01 -0.05 0.01

Total Prediction 0.03 0.009 0.12 0.03 -0.02 -0.007Total Simulation 0.04 0.01 0.14 0.03 -0.02 -0.006

Table 5.8: The contribution of each interaction to the mean value of the totalerror in each of three output waves at the output to interaction 3, for 100 and500 mesh units. Units are dimensionless and represent the error expressed asa fraction of the total wave strength. The last two rows compare the totalof the mean error as given by the model to the directly observed mean error.The columns I.R., C., and O. S. are labeled as in Fig. 4.1, Right frame.

79

(a) Inward Rarefaction

(b) Contact

(c) Outward Shock

Figure 5.4: Pie charts showing the contribution of each wave interaction dia-gram to the error variance of the wave strength at the output of interaction 3,for a solution using 500 mesh units.

80

(a) Inward Rarefaction

(b) Contact

(c) Outward Shock

Figure 5.5: Pie charts showing the contribution of each wave interaction dia-gram to the error variance of the wave strength at the output of interaction 3,for a solution using 100 mesh units.

81

Chapter 6

Conclusion And Future Work in 2D Chaotic

Flows

6.1 Conclusion from 1D Flow Study

We have several main conclusions from the error study of numerical so-

lutions of 1D flow.

• From Chap. 2, 3 and 4, we showed that a very simple model of solution

error is sufficient for the study of (at least the present instance of) a

highly nonlinear problem. The error is linear in the input wave strengths.

• In Chap. 5, a composition law for combining errors and predicting er-

rors for composite interactions on the basis of an error model of the

simple constituent interactions has been formulated and validated. For

spherically symmetric shock physics problems, the main new difficulties

encountered were the non-constancy of the solution between interaction

events and the non-constancy of waves and errors between interactions.

82

For a planar geometry, the errors are constant between interactions, while

for a spherical geometry, the errors grow (if the wave which carries them

is moving inward) by a power law in the radius. Similarly outward mov-

ing waves and their errors weaken by a power law.

• In 1D flows of planar geometry, we find that although our formalism

allows for statistical errors in the ensemble that in fact, the dominant

part of all errors (excluding position errors) studied were deterministic,

in the sense that the ensemble mean error dominated the ensemble stan-

dard deviation. For spherical geometry, the composition model applied

to construct the variance of the error in the wave strength, generally

understates the STD by a factor generally between 1.5 and 2, for causes

not presently identified. Using the model, the total error is a sum of six

terms, each corresponding to a pattern of wave interactions and trans-

missions. Of these diagrams, two correspond to initial error, following

different transmission patterns, and four correspond to errors created

within the solution and transmitted to the output of the reshock inter-

action, where the errors are analyzed. We see that for a 500 cell grid,

the dominant error comes from the initial uncertainty, while for the 100

grid over 75% of the error arises within the numerical simulation. We

could conclude that for coarse grid simulations, there exists increased

importance of created errors.

• For planar case, the wave strength uncertainty which is dominated by

input uncertainty (i.e. the definition of the ensemble), is virtually un-

83

changed between the highly resolved and the under resolved simulations.

While for spherical case, this is not true. We concluded that for under

resolved spherical simulations, there exists increased importance of cre-

ated errors. The wave width errors are both expressed in grid units, and

are comparable between the two levels of resolution. The wave width

errors evidentially have a rapid relaxation to their asymptotic value.

• The primary solution errors created by the simulations are the wave

position errors in the under resolved simulations, on the order of a mesh

spacing. These errors are a transient phenomena, but become frozen

into the calculation as new interactions occur before the transient errors

have diminished. The wave position errors have a slow relaxation to

asymptotic values.

• We find that the wave filter performs well as a diagnostic tool, but that its

limitation (in its present version) lies in assuming well separated waves.

Thus we are limited in the degree of under resolution that we can analyze,

in that all waves must be at least partially separated from one another

before entering into a new interaction.

To the extent that a more detailed modeling of these errors is important,

a more accurate model that includes transient effects will be important. Even

with these limitations, the methods and results appear to be promising, and

should be extended to less idealized problems.

These conclusions are established only under several simplifying assump-

tions, namely restriction to one spatial dimension, use of a simplified (gamma

84

law gas) equation of state, and consideration of only one numerical method.

Further studies are needed to determine the extent that these conclusions have

a general validity.

6.2 Future Work in 2D Chaotic Flows

We are to apply the statistical approach to error analysis for the prob-

lems considered here, namely chaotic CFD. By definition, chaotic flows are

ones which exhibit sensitive dependence on initial conditions. Generally, they

also exhibit sensitive dependence on other parameters, not only the physical

parameters which define the flow, but also the numerical parameters which

define the solution algorithm.

There is no shortage of important chaotic flow problems. We mention

climatology [1], turbulence [4, 29], fluid mixing [34], flow in porous media [27],

and turbulent combustion [30]. Here we consider a representative but rela-

tively simple chaotic flow, the Richtmyer-Meshkov instability resulting from

a shock passing through a density discontinuity. We consider this problem in

a perturbed circular (2D) geometry. Namely, a circular shock wave implodes

a perturbed circular interface, and the two circles may or may not be offset

relative to one another. The shock proceeds to the origin, where it is reflected,

and the reflected shock reshocks the now highly perturbed interface, giving

rise to a highly chaotic flow. See Fig. 6.1.

85

Figure 6.1: Left: Initial geometry of a circular shock imploding a perturbedcircular contact discontinuity. The two circles are offset relative to each other.Right: Chaotic flow observed after reshock by the outgoing shock reflectedfrom the origin. Gray scale in both plots indicates density. The grid is800 × 1600.

6.3 The 2D Wave Filter

I Sec. 3.1, the one dimensional method examines states along a sequence

of adjacent points, fits the jump at the end points of the sequence to a Riemann

solver, and if a single wave type is detected in the Riemann solution, tries to

fit the solution through all the mesh points between the two mesh points to

an erfc or piecewise linear wave form (depending on the solution type). In

86

[11], this algorithm was extended to two dimensional flows. Starting from

an arbitrary point and in an arbitrary direction, a 1D wave filter looks for a

significant indication of a single wave type. The central location and direction

in which this single wave type occurs most clearly is the predictor for the wave

front position and normal. The predictor for the tangent to the wave front is

the normal to the predictor normal, passing through the predictor position.

Finding points at unit mesh spacing along this predictor for the tangent, we

repeat the 1D construction to find the best fitting position for the wave front.

From this construction, we have three points on the wave front, one to the left,

one to the right and one at the original location where the wave was detected.

We fit a circle to these three points, giving a corrected wave front position and

normal. For the present problem, with its approximate circular symmetry, and

our average of state variables over a 45o angular sector, we average the shock

position data in these sectors also to yield sector-averaged shock positions.

This algorithm is here applied in 2D for the first time, and we find some

modifications are needed for efficiency. First, as we are only interested in

the detection of shock waves, all the 1D passes only look for shock waves.

Secondly, we can prune the initial search locations and directions, omitting

points and directions in which an initial analysis indicates no activity. The

wave front data analysis is performed at a fixed time interval equal to six time

steps on the coarse grid, and occupies less than 10% of the total solution time.

A new wave type, the edge (inner or outer) of the mixing zone is also needed,

and for this purpose a new wave filter is constructed. We follow previously

accepted ideas in the analysis of Rayleigh-Taylor mixing data, and look for

87

the 5% and 95% volume fraction contours. For the present problem, with

approximate circular symmetry, we look for the 5% and 95% volume fraction

contours within a 45o sector. The location of these 5% and 95% contours can

be a noisy diagnostic for the edge of the mixing zone, which occasionally moves

abruptly as a function of time. While we have not solved this problem, we

have avoided it for the data presented in this paper.

In Fig. 6.2, we plot the space time contours in r, t space for the inward and

reflected shocks and for the inner and outer edges of the mixing zone. We have

adjusted the tolerances in the automated filter so that it picks up exclusively

the strong waves we are trying to locate (and thus misses numerous weak

waves). The thresholds are set in the variable (P2 − P1)× (pressure ratio +

Atwood number). The shock wave filter does not depend on angular averaging.

The angular dependence of the shock wave for a sequence of times is shown

in Fig. 6.3. This figure illustrates the robustness of the wave filter tool in its

2D application. The ripples in the wave front in the left frame near the center

(vertically) of the figure are at the time of passage of the shock front through

the perturbed interface, and reflect defraction events at the shock front to

advance or retard local portions of the shock front. Similar ripples occur in

the outgoing shock angular dependence during its transit through the mixing

zone. See Fig. 6.3, right frame, near the bottom.

6.4 Problem Formulation

We consider a computational domain x, y ∈ [0, 25] × [−25, 25] (units of

mm). With r denoting the radial coordinate in the x, y plane, the initial

88

contact is perturbed from a circle at r = 12.5. Outside this contact is a stiff-

ened gamma law gas, representing tin, with parameters given approximately

by γ = 3.72, p∞ = 0.15, e∞ = 0.0. Inside the contact is also a stiffened

gamma law gas, representing lucite, with approximate parameters γ = 1.85,

p∞ = 0.03, e∞ = 0.0. A constant pressure boundary is located at r = 24. The

initial ambient pressure is p = 10−6, and the imposed pressure at the boundary

is p = 0.687, giving rise to an inward propagating Mach M = 2 shock at t = 0.

The densities are approximately ρtin = 7.3 and ρlucite = 1.2 at t = 0, giving

an Atwood number A = (ρ2 − ρ1)/(ρ2 + ρ1) = 0.72. We consider a series of

grids, with sizes 100 × 200, 200 × 400, 400 × 800, and 800 × 1600. Errors will

be assessed in comparison of adjacent grid sizes. The runs continue until after

the passage of the reflected shock through the contact interface (reshock), and

terminate before the rarefaction wave which results from the reflected shock

interaction with the constant pressure boundary contaminates (very much of)

the rest of the flow. For the present problem dimensions, this time is t = 80.

With this convention, we observe errors at three levels of mesh refinement in

this study.

The contact interface has been perturbed by sine waves to have the initial

configuration

r(θ) = r0

(1 +

∑n

an sin(nθ)

), (6.1)

θ ∈ [−π/2, π/2]. Here r0 = 12.5 and sine modes are selected so that the

imposed reflection symmetry at θ = ±π/2 leads to a smooth curve. The

sum over n ranges from nmin = 8 to nmax = 16, so that the average number

of observed modes in the initial perturbation is about 12. The coefficients

89

an are chosen as Gaussian random variables, with mean zero and STD 0.2,

based on the C random number generator erand48(), mapped into a Gaussian

distribution. The observed mean peak to peak amplitude, determined by this

STD, is 0.25. Successive calls to the random number generator generate the

ensemble of initial conditions used in this study.

For the non-offset simulations, the circles defining the pressure boundary

conditions (and the initial shock) and the contact are both centered at the ori-

gin. For the offset simulations, the circle defining the (pre-perturbed) contact

has a center at x = 0, y = 5.

90

Radius

Tim

e

0 5 10 15 20

020

4060

80

Figure 6.2: Space time (r, t) contours of the primary waves, as detected by ourwave filter algorithm. These are the inward (direct) and outward (reflected)shocks and the inner and outer edges of the mixing zone, all detected within asingle averaging window, in this case θ ∈ [−45o, 0o]. Note the rarefaction wavenear t = 80, r = 24, due to the shock reflection off of the constant pressureboundary. Contour plots of the mean density are also shown. The shock hasbeen offset relative to the contact, and the grid is 800 × 1600.

91

Angle

Rad

ius

-90 -60 -30 0 30 60 900

5

10

15

20

25

Angle

Rad

ius

-90 -60 -30 0 30 60 900

5

10

15

20

25

Figure 6.3: Radial shock position as a function of angle for a series of times.The grid level is 200 and there is no offset of the center. Left: inward shock,right: outward shock.

92

Chapter 7

Appendix

7.1 Appendix 1: The Rest of Statistical Numerical Rie-

mann Problems in Planar Flows

For each of the 10 Riemann problems of Fig. 2.4, Right, we vary the

wave strength and for contacts only, we vary the wave width. Three variables

defining one reference state are not varied in this study; presumably similar

conclusions would be reached if they were also varied. We have two goals

in selecting the reference variables to hold fixed. If one of the states has a

reference ambient velocity, for example a velocity v = 0 for a state near a wall,

we want to preserve this property and freeze this velocity. For the pressure

and density values, we generally freeze those on the smaller side of the waves,

as this gives a more meaningful variation of the state, uniformly specified as

10% of the wave strength, as defined in Chap. 2.

93

7.1.1 Rarefaction Crossing Rarefaction Interaction (Case

4)

Here we study the reflection of the rarefaction off the wall, a special case

of the rarefaction crossing rarefaction interaction. We have one input and one

output wave width parameter, both for the rarefaction. The Rimann problem

was set up as Fig. 7.1. And we assume that at interaction location the input

rarefaction and the output rarefaction have the same width. The rarefaction

width has the form constant + initial width + rate × time. The entry λo1 in

Table 7.1 refers to the constant, which gives an offset for the centering of the

rarefaction wave. We find very small errors in the rate, not tabulated here.

This entry is expressed in mesh units. We form a linear model for this constant

in Table 7.1.

Figure 7.1: Problem 4: Rarefaction-wall. The right state velocity v = 0.0is fixed and the left state densities and pressures are held fixed. The inputrarefaction wave width is an input parameter. Same comments applied to latercases.

94



1 (l. sonic) 0.146 0.657 0.181% 0.0003λo

1 (l. sonic) 3.832 2.091 5.056% 0.096po

1 (l. sonic) 0.071 -0.237 8785% 0.006

Table 7.1: Case 4. The SNRP defined by the crossing of two rarefactions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

Figure 7.2: Problem 5: Contact-rarefaction. The right state velocity v = 0.0is fixed and the left state densities and pressures are held fixed.

7.1.2 The Contact Rarefaction Interaction (Case 5)

After reflection from the wall, the transmitted rarefaction wave re-crosses

the deflected contact. We have two input wave width parameters, see Fig. 7.2,

one each for the contact and the rarefaction, and three output wave width

parameters, one each for the left rarefaction, the contact and the right com-

pression wave. According to the analysis of single isolated waves, the output

contact width is bounded after some 100 time steps. The Table 7.2 entry λo2

refers to this bounded width. We assume that at interaction the input rarefac-

tion and the output rarefaction have the same width. The rarefaction width

95

has the form constant + initial width + rate × time. We find very small errors

in the rate, not tabulated here. The entry λo1 refers to the constant, which gives

an offset for the centering of the rarefaction wave. This entry is expressed in

mesh units. We form a linear model for this constant in Table 7.2. Similarly,

the compression wave width also has the form constant + initial width + rate

× time. Here the rate has the negative sign. We find very small errors in the

rate, not tabulated here. The entry λo3 refers to the constant, which gives an

offset for the centering of the compression wave. This entry is expressed in

mesh units. We form a linear model for this constant in Table 7.2.



ωo1 (l. sonic) 0.020 -0.073 0.697 0.33% 0.0002

ωo2 (contact) 0.198 1.076 -0.716 0.30% 0.0006

ωo3 (r. sonic) -0.029 0.107 0.295 0.76% 0.0003

λo1 (l. sonic) 4.613 0.018 7.378 10.5% 0.206

λo2 (contact) 4.933 -1.691 17.063 0.79% 0.024

λo3 (r. sonic) 6.260 0.777 3.812 3.09% 0.111

po1 (l. sonic) 6.671 -5.799 -20.655 90.8% 0.620

po2 (contact) -5.046 0.521 17.723 10.7% 0.050

po3 (r. sonic) -13.010 0.284 53.225 4.40% 0.089

Table 7.2: Case 5. The SNRP defined by the contact rarefaction interaction.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

7.1.3 Shock Overtaking Shock Interaction (Case 6)

In this Rimann problem, as see in Fig. 7.3, the overtake of reflected shocks

from interactions 1 and 3 is studied. The two input shocks are both left moving

96

forward shocks, which are produced by interactions 1 and 3, respectively. And

theoretically, one left moving forward shock, one contact and one right moving

backward shock are produced. We ignore the output backward shock because

it is too weak(ωo3 = 0.017) to be recognized by the filter program. Therefore,

the position error po3 and the width error λo

3 are not presented here.

Figure 7.3: Problem 6: Shock-shock overtake (two waves of the same family).The left state is held fixed and the two wave strengths are varied.

However, with an improved post-processing program, we can still get all

input and output gas states, so all input and output strengths are studied

here.

This case is a step down problem. From the analysis of the Sec. 3.2,

the contact width will be modeled as ωc cc ∼ t1/3, where the coefficient cc is

affected to Mach number, and the t denotes the time steps counted from the

interaction time. This also is showed by the Fig. 7.4, in which we take a log

for the time step axis to show the linear relationship. The time steps starts

from 51, because the contact widths near interaction time are unstable, which

are deleted for better result. And the we observe an increase in width from 4

cells to 9 over 1000 time steps.

97

In this plot there is a noticeable small disturbance after 1000 time steps,

beyond an otherwise good linear relation. That disturbance is due to an

additional wave interaction with the contact at that time, therefore affects

the contact width after 1000 time steps. This interaction is not presented here

since it is not one of the major interactions, but still it can be seen in Fig. 2.4.

Figure 7.4: Because the width is entirely grid related, we record width in unitsof ∆x and time in units of the number of time steps.

Thus, in the Table 7.3 we interpret λo2t

1/3 or λo2 as the output contact wave

width, also in grid units, depending on the Mach number. For λo1, it is the

shock wave width and independent of time steps. And pak and ωa

k (k = 1, 2, 3

and a = i, o) represent position errors and wave strengths as usual.

The width of the left most input forward shock, which is from interaction

1, is 1 grid unit. And The width of the other input forward shock, which is

from interaction 3, is 2 grid units.

98

The output position errors are subtracted by the input position error to

see their relationship with the input wave strengths, according to the formulae

introduced by Sec. 5.1.2.


2 error(l. sonic) (l. sonic) L∞ STD

ωo1 (l. sonic) 0.033 0.375 1.110 0.02% 0.0001

ωo2 (contact) -0.028 0.075 0.149 0.91% 0.0002

ωo3 (r. sonic) -0.018 0.014 0.056 0.57% 0.0000

λo1 (l. sonic) 2.050 -0.248 -0.224 9.57% 0.078

λo2 (contact) 0.092 0.278 1.418 1.71% 0.007

po1 (l. sonic) 0.132 -0.030 0.007 12.92 % 0.009

po2 (contact) -0.026 0.010 0.233 32.06% 0.009

Table 7.3: Case 6. The SNRP defined by the shock shock overtake (two wavesof the same family). Expansion coefficients for output wave strengths (linearmodel) for input variation ±10%.

7.1.4 Compression Crossing Compression Interaction (Case

7)

Figure 7.5: Problem 7: Compression-wall. The right state is held fixed.

Here we study the reflection of the compression off the wall, see Fig. 7.5,

99

a special case of the compression crossing compression interaction. Similar to

the previous cases, the compression wave width has the form constant + initial

width + rate × time. Here the rate has a negative sign. We find very small

errors in the rate, not tabulated here. The entry λo1 refers to the constant,

which gives an offset for the centering of the compression wave. This entry is

expressed in mesh units. We form a linear model for this constant in Table 7.4.

We find less than 0.001% density oscillation near wall. So, there is no wall

errror in the compression wall interaction.



1 (l. sonic) -0.026 1.127 1.450% 0.00061λo

1 (l. sonic) -0.893 -2.323 31.41% 0.107po

1 (l. sonic) -0.049 0.516 847.5% 0.012

Table 7.4: Case 7. The SNRP defined by the crossing of two compressions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

7.1.5 The Contact Compression Interaction (Case 8)

After reflection from the wall, the transmitted compression wave re-

crosses the deflected contact. See Fig. 7.6. We have two input wave width

parameters, one each for the contact and the Compression, and three output

wave width parameters, each for the left compression, the contact and the

right rarefaction wave. The entry λo1 in Table 7.5 refers to the constant offset,

which is given by the form compression width constant + initial width + rate

× time.

100

Figure 7.6: Problem 8: Contact-compression. The right state velocity v = 0.0is fixed and the left state densities and pressures are held fixed.

7.1.6 Rarefaction Crossing Rarefaction Interaction (Case

9)

Figure 7.7: Problem 9: Rarefaction-wall. The right state velocity v = 0.0 isfixed and the left state densities and pressures are held fixed.

Case 9, as show in Fig. 7.7, is the reflection of the rarefaction off the wall

again. We used the same error model as in case 4.

7.1.7 The Contact Rarefaction Interaction (Case 10)

The Riemann Problem of case 10 is set up as Fig. 7.8. The output wave

strengths in case 10 are very weak (ωo1 = 0.033, ωo

2 = 0.674, ωo3 = 0.021). The

101



ωo1 (l. sonic) 0.030 -0.039 0.601 3.46% 0.0008

ωo2 (contact) -0.043 0.973 0.442 0.71% 0.0011

ωo3 (r. sonic) -0.026 0.034 0.402 4.29% 0.0007

λo1 (l. sonic) 3.298 0.116 -1.510 13.2% 0.080

λo2 (contact) -0.240 -0.081 -4.584 3480% 0.432

λo3 (r. sonic) 3.758 -1.265 9.379 19.7% 0.273

po1 (l. sonic) -0.086 1.295 -1.609 95.0% 0.186

po2 (contact) 0.218 -0.273 0.068 195.3% 0.147

po3 (r. sonic) -0.519 -0.772 2.298 44.8% 0.153

Table 7.5: Case 8. The SNRP defined by the contact compression interaction.Expansion coefficients for output wave strengths (linear model) for input va-ration ±10%.

wave filter lacks sufficient precision to allow analysis of the errors in this case,

because their small size are not important in any case.



1 (l. sonic) 0.004 0.938 0.55% 0.0001λo

1 (l. sonic) -0.060 5.290 206% 0.067po

1 (l. sonic) 0.119 -1.948 1938% 0.013

Table 7.6: Case 9. The SNRP defined by the crossing of two rarefactions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

102

Figure 7.8: Problem 10: Contact-rarefaction. The right state velocity v = 0.0is fixed and the left state densities and pressures are held fixed.

7.2 Appendix 2: Composite Shock Interaction Prob-

lems in Planar Flows

7.2.1 Errors in Resolved Calculations

We examine errors in wave strength, wave position and wave width, based

on the graphical expansion given in the fig. 5.2. The wave strength errors

are dominated by the transmission of error (or uncertainty) from the initial

conditions. The wave width errors and the wave position errors are dominated

by the created error in the current interaction and the transmission of errors

from previous interactions. In Tables 7.7, 7.8, 7.9, 7.10, 7.11 and 7.12, we

compare the predicted error with the error computed directly, taken from a

full solution of the multiple wave interaction problem. The model for the

prediction of the error is satisfactory for those cases: the wave strength and

its errors, the wave width errors.

103


ωo1 (l. sonic) 0.346 0.348



λo1 (l. sonic) 4.568 4.474


1 (l. sonic) 0.287 0.127

Table 7.7: Case 4. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.

7.2.2 Errors in Under Resolved Calculations

Here we allow 100 cells for the coarse grid simulation. This resolution

allows 10 cells between the contact and the reflection wall at the time of

interaction 3 and beyond. Since the contact has a width of 5 cells, since the

right facing rarefaction is about this size and since the wall has inaccurate

states in a region of several mesh blocks neighboring it, we are clearly at the

limit of the present diagnostic methods based upon the wave filter. For the

same reasons, the calculation is clearly under resolved. For this reason we

are not able to analyze data for the case of a coarse grid simulation with 50

cells using the present version of our wave filter. Again we present the last

six interactions in detail, at 100 cell resolution, comparing the predicted to

the directly simulated errors. See Tables 7.13, 7.14, 7.15, 7.16, 7.17 and 7.18.

We see good results for the wave strengths and their errors and for the wave

width errors, and poor results for the comparison of position errors. This

can be understood in terms of the decay time for convergence to asymptotic

104


ωo1 (l. sonic) 0.212 0.212

ωo2 (contact) 0.675 0.672

ωo3 (r. sonic) 0.147 0.145


1 (l. sonic) 0.0002 0.0002Var ωo

2 (contact) 0.0011 0.0014Var ωo


λo1 (l. sonic) 7.212 7.176

λo2 (contact) 9.183 9.693

λo3 (r. sonic) 7.895 8.101


1 (l. sonic) -4.932 -4.074po

2 (contact) 1.579 1.365po

3 (r. sonic) 6.989 5.389

Table 7.8: Case 5. The contact rarefaction interaction. Predicted and simu-lated errors for output wave strengths, wave widths and wave positions.

large time values for the position errors, an explanation that also accounts for

the difference with the resolved case, for which the simulated and predicted

position errors agree. The position errors have a relatively slower decay time.

The other three quantities show a high level of agreement between the resolved

and under resolved cases. The wave width error is expressed in grid units, and

so should be the same in the two cases differing in grid resolution only. For

the wave strength entries, the lack of dependence on grid resolution is due to

the fact that these quantities are dominated by the uncertainty expressed in

the ensemble of initial conditions, which is independent of grid resolution.

105


ωo1 (l. sonic) 0.783 0.781

ωo2 (contact) 0.079 0.083

ωo3 (r. sonic) 0.015 0.017


1 (l. sonic) 0.0016 0.0019Var ωo

2 (contact) 0.000001 0.000005Var ωo


λo1 (l. sonic) 1.808 1.821

λo2 (contact) 3.947 4.248


1 (l. sonic) 0.678 0.427po

2 (contact) 0.797 0.589

Table 7.9: Case 6. The shock shock overtake. Predicted and simulated errorsfor output wave strengths, wave widths and wave positions.


ωo1 (l. sonic) 0.138 0.139



λo1 (l. sonic) 8.183 6.662


1 (l. sonic) -5.957 -4.785

Table 7.10: Case 7. The crossing of two compressions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.

106


ωo1 (l. sonic) 0.086 0.086

ωo2 (contact) 0.675 0.674

ωo3 (r. sonic) 0.053 0.053


1 (l. sonic) 0.00003 0.00003Var ωo

2 (contact) 0.0013 0.0013Var ωo


λo1 (l. sonic) 2.843 3.168

λo2 (contact) -1.356 -0.927

λo3 (r. sonic) 12.794 11.381


1 (l. sonic) -7.258 -6.446po

2 (contact) 0.314 0.102po

3 (r. sonic) 5.316 5.260

Table 7.11: Case 8. The contact compression interaction. Predicted andsimulated errors for output wave strengths, wave widths and wave positions.


ωo1 (l. sonic) 0.054 0.053



λo1 (l. sonic) 11.293 10.013


1 (l. sonic) -7.610 -5.474

Table 7.12: Case 9. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.

107


ωo1 (l. sonic) 0.330 0.356



λo1 (l. sonic) 4.776 4.498


1 (l. sonic) 0.599 0.167

Table 7.13: Case 4. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions. Comparisonof under resolved simulation and prediction.


ωo1 (l. sonic) 0.218 0.244

ωo2 (contact) 0.692 0.676

ωo3 (r. sonic) 0.142 0.157


1 (l. sonic) 0.0002 0.0004Var ωo

2 (contact) 0.0009 0.0012Var ωo


λo1 (l. sonic) 8.163 7.046

λo2 (contact) 9.775 10.408

λo3 (r. sonic) 7.042 7.584


1 (l. sonic) -8.997 -4.235po

2 (contact) 4.789 1.738po

3 (r. sonic) 7.198 4.284

Table 7.14: Case 5. The contact rarefaction interaction. Predicted and simu-lated errors for output wave strengths, wave widths and wave positions. Com-parison of under resolved simulation and prediction.

108


ωo1 (l. sonic) 0.785 0.783

ωo2 (contact) 0.084 0.083

ωo3 (r. sonic) 0.019 0.017


1 (l. sonic) 0.0023 0.0019Var ωo

2 (contact) 0.000007 0.000004Var ωo


λo1 (l. sonic) 1.674 1.821

λo2 (contact) 3.194 4.264


1 (l. sonic) 1.606 0.472po

2 (contact) 2.524 0.598

Table 7.15: Case 6. The shock shock overtake. Predicted and simulated errorsfor output wave strengths, wave widths and wave positions. Comparison ofunder resolved simulation and prediction.


ωo1 (l. sonic) 0.126 0.134



λo1 (l. sonic) 7.156 6.215


1 (l. sonic) -8.217 -3.801

Table 7.16: Case 7. The crossing of two compressions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions. Comparisonof under resolved simulation and prediction.

109


ωo1 (l. sonic) 0.093 0.098

ωo2 (contact) 0.689 0.665

ωo3 (r. sonic) 0.041 0.031


1 (l. sonic) 0.00003 0.00004Var ωo

2 (contact) 0.0014 0.0012Var ωo


λo1 (l. sonic) 3.662 3.126

λo2 (contact) -1.739 -0.830

λo3 (r. sonic) 11.640 11.553


1 (l. sonic) -14.141 -9.202po

2 (contact) 1.391 0.139po

3 (r. sonic) 9.655 4.801

Table 7.17: Case 8. The contact compression interaction. Predicted andsimulated errors for output wave strengths, wave widths and wave positions.Comparison of under resolved simulation and prediction.


ωo1 (l. sonic) 0.050 0.043



λo1 (l. sonic) 10.122 11.800


1 (l. sonic) -9.299 -3.928

Table 7.18: Case 9. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions. Comparisonof under resolved simulation and prediction.

110

Bibliography

[1] Natural Climate Variability on Decade-to-Century Time Scales. NationalAcademy Press, 1995.

[2] I. Babuska and W. Rheinboldt. A posteriori error estimates for the fi-nite element method. Internat. J. Numer. Methods Engrg., 12:1597–1615,1978.

[3] J. W. Barker, M. Cuypers, and L. Holden. Quantifying uncertainty inproduction forecasts: Another look at the PUNQ-S3 problem. TechnicalReport SPE 62925, 2000.

[4] G. Batchelor. The Theory of Homogeneous Turbulence. The UniversityPress, Cambridge, 1955.

[5] L. Bonet-Cunha, D. S. Oliver, R. A. Redner, and A. C. Reynolds. A hy-brid markov chair monte carlo method for generating permeability fieldsconditioned to multiwell pressure data and prior information. SPE Jour-nal, 3:261–271, 1998.

[6] A. Chorin and J. Marsden. A Mathematical Introduction to Fluid Mech-nics. Springer Verlag, New York–Heidelberg–Berlin, 2000.

[7] B. Cockburn and H. Gau. A posteriori error estimates of general numeri-cal methods for scalar conservation laws. Mat. Aplic. Comp., 14(1):37–47,1995.

[8] P. S. Craig, M. Goldstein, J. C. Rougier, and A. H. Seheult. Bayesianforecasting for complex systems using computer simulators. Journal OfThe American Statistical Association, 96:717–729, 2001.

[9] B. DeVolder, J. Glimm, J. W. Grove, Y. Kang, Y. Lee, K. Pao, D. H.Sharp, and K. Ye. Uncertainty quantification for multiscale simulations.Journal of Fluids Engineering, 124:29–41, 2002. LANL report No. LA-UR-01-4022.

111

[10] J. Glimm, J. W. Grove, Y. Kang, T. Lee, X. Li, D. H. Sharp, Y. Yu,K. Ye, and M. Zhao. Statistical riemann problems and a composition lawfor errors in numerical solutions of shock physics problems. SISC, 26:666–697, 2004. University at Stony Brook Preprint Number SB-AMS-03-11,Los Alamos National Laboratory number LA-UR-03-2921.

[11] J. Glimm, J. W. Grove, Y. Kang, T. Lee, X. Li, D. H. Sharp, Y. Yu, K. Ye,and M. Zhao. Errors in numerical solutions of spherically symmetricshock physics problems. Contemporary Mathematics, 371:163–179, 2005.University at Stony Brook Preprint Number SB-AMS-04-03, Los AlamosNational Laboratory number LA-UR-04-0713.

[12] J. Glimm, J. W. Grove, X. L. Li, W. Oh, and D. H. Sharp. A criticalanalysis of Rayleigh-Taylor growth rates. J. Comp. Phys., 169:652–677,2001.

[13] J. Glimm, J. W. Grove, X.-L. Li, K.-M. Shyue, Q. Zhang, and Y. Zeng.Three dimensional front tracking. SIAM J. Sci. Comp., 19:703–727, 1998.

[14] J. Glimm, J. W. Grove, X.-L. Li, and D. C. Tan. Robust computationalalgorithms for dynamic interface tracking in three dimensions. SIAM J.Sci. Comp., 21:2240–2256, 2000.

[15] J. Glimm, Yoon ha Lee, and Kenny Ye. A simple model for scale up error.Contemporary Mathematics, 295:241–251, 2002.

[16] J. Glimm, S. Hou, H. Kim, Y. Lee, D. Sharp, K. Ye, and Q. Zou.Risk management for petroleum reservoir production: A simulation-basedstudy of prediction. J. Comp. Geosciences, 5:173–197, 2001.

[17] J. Glimm, S. Hou, H. Kim, D. Sharp, and K. Ye. A probability model forerrors in the numerical solutions of a partial differential equation. CFDJournal, 9, 2000.

[18] J. Glimm, S. Hou, Y. Lee, D. Sharp, and K. Ye. Prediction of oil pro-duction with confidence intervals. SPE 66350, Society of Petroelum En-gineers, 2001. SPE Reservoir Simulation Symposium held in Houston,Texas, 11-14 Feb.

[19] J. Glimm and D. H. Sharp. Stochastic methods for the prediction ofcomplex multiscale phenomena. Quarterly J. Appl. Math., 56:741–765,1998.

112

[20] J. Glimm and D. H. Sharp. Prediction and the quantification of uncer-tainty. Physica D, 133:152–170, 1999.

[21] James Glimm, Yunha Lee, David H Sharp, and Kenny Q Ye. Predic-tion using numerical simulations, A Bayesian framework for uncertaintyquantification and its statistical challenge. In Bilal M. Ayyub and Nii O.Attoh-Okine, editors, Proceedings of the Fourth International Symposiumon Uncertainty Modeling and Analysis (ISUMA 2003). IEEE, ComputerSociety, 2003.

[22] G. Guderley. Starke kugelige und zylindrische verdichtungsstosse in derNahe des kugelmittelpuktes bzw der zylinderachse. Luftfahrtforschung,19:302–312, 1942.

[23] A. Harten. ENO schemes with subcell resolution. J. Comp. Phys.,83:148–184, 1989.

[24] B. K. Hegstad and H. Omre. Uncertainty assessment in history matchingand forecasting. In E. Y. Baafi and N. A. Schofield, editors, GeostatisticsWollongong ’96, pages 585–596. Kluwer Academic Publishers, 1997.

[25] Yunghee Kang. Estimation of Computational Simulation Errors in GasDynamics. PhD thesis, State Univ. of New York at Stony Brook, 2003.LANL Report LA-UR-03-2700.

[26] M. C. Kennedy and A. O’Hagan. Bayesian calibration of computer mod-els. Journal Of The Royal Statistical Society Series B-Statistical Method-ology, 63(3):425–450, 2001.

[27] L. Lake. Enhanced Oil Recovery. Prentice-Hall, Englewood Cliffs, 1989.

[28] L. Machiels, A. Patera, J. Peraire, and Y. Maday. A general frameworkfor finite element a posteriori error control: Application to linear and non-linear convection-dominated problems. In Proceedings ICFD Conferenceon Numerical Methods for Fluid Dynamics, Oxford, England, March 31 -April 3. 1998.

[29] W. D. McComb. The Physics of Fluid Turbulence. Oxford UniversityPress, Oxford, 1990.

[30] J. C. Niemeyer and W. Hillebrandt. Turbulent nuclear flames in Type Iasupernovae. ApJ, 452:769–778, 1995.

113

[31] J. T. Oden and M. Ainsworth. A Posteriori Error Estimation in FiniteElement Analysis. John Wiley & Sons, New York, 2000.

[32] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery. NumericalRecipes in Fortran 77. Cambridge University Press, 1992.

[33] R. Richtmyer and K. Morton. Difference Methods for Initial Value Prob-lems. Interscience, New York, second edition, 1967.

[34] D. H. Sharp. An overview of Rayleigh-Taylor instability. Physica D,12:3–18, 1984.

[35] J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York, 1983.

[36] G. B. Whitham. Linear and Nonlinear Waves. John Wiley & Sons, NewYork, 1974.

114

error analysis in numerical solutions of various shock ... · error analysis in numerical solutions...

Documents