introduction to computational continnum dynamics: a personal perspective

5

INTRODUCTION TO COHPUTATIONAL CONTINUUH DYNAMICS: A PERSONAL PERSPECTIVE

DARRELL L. HICKS

Department of Mathematical and Computer Sciences,

Michigan Technological University, Houghton, MI 49931

ABSTRACT

This is a survey of some open questions in the compu-

tational aspects of continuum (solid, liquid, gas, plasma,

multiphase or multimaterial) dynamics.

INTRODUCTION

Continuum dynamics studies include solid, liquid, gas, plasma, multi-

phase and multimaterial dynamics in 1, 2 and 3 spatial dimensions. Compu-

tational continuum dynamics studies include the creation, design, and

analysis of discrete analogs of the conservation (or balance) laws of

continuum dynamics. This is a very broad field with a long history and

many applications (see the bibliography). I will restrict my scope to

problems that I have worked on and problems that are interesting to me. One of my main motivations in ,this tutorial is to introduce mire mathema-

ticians in academia to home of the fascinating, important and difficult

mathematical questions in computational continuum dynamics.

An important and very basic axiom of continuum dyxlamics that should

not be overlooked is that whatever the phase of the material is, the

balance (or conservation) laws of continuum dynamics apply. These laws are

named conservation of mass, conservation of volume, conservation of momen-

tum and conservation of energy; they are derived from the first principles

of physics. To these laws must be added a material law. Material laws are

also known as equations of state, constitutive relations, rate relations.

First let's consider the simple case of one-dimensional motion in material (or Lagrangean) coordinates.

The Lagrangean reference frame is fixed in the material. Let X be

the label of a point of the material. Let t be the label of a point in time. Our initial data will be given at t=o. Let the mass density at time t=o and material point X be denoted P,(X) . A convenient independent variable for the one-dimensional Lagrangean reference frame is

the Lagrangean mass variable, p . The relationship between X and p is

dv = po(X)dX .

The object of the game is to solve for x , the motion of the material. The motion, x , is a function of u and t . The evaluation of the motion, x(lJ, t) , is the position in the spatial reference frame of the material point labeled u at the time point labeled t . Assuming differ- entiability of x we have:

V = adalJ and " = ax/at

where V is called the specific volume (V = l/p) and " is called the specific momentum. Other dependent variables are: E is specific (total) energy, e is specific internal energy, "212 is specific kinetic energy,

P is the pressure, T is the absolute temperature, s is the specific entropy. The energy partition is E = e + u2/2 . Given a thermomechanical equation of state where e is a function of V and s we have the thermodynamic definitions

P(V, S) = -ae(v, s)/av

T(v, S) = ae(v, s)/as

S(v, S) = -ap(v, s)/av

where a is the acoustic impedence; it is related to the isentropic sound speed by a= cp. For example, the ideal gas law is

e = e. exp((s - s,)/cv)(v/vo)l-~

where e, and so are constants, cv > 0 is a constant called the specific heat at constant volume and y > 1 is a constant called the gas ex- ponent. It follows that

P = W(Y - 1)

and

T = e/cv

for an ideal gas. Another example of a material law category is the mechanical equation

of state which allows e and p to depend only on V . A specific example of this category is the isentropic gas law:

e = eo(V/Vo)l-y

and

P = P,WVJY

Another specific example is the linear pV law (a one-dimensional version of Ilooke's law):

e=e 0 - aZ(V - Vo)

If the material is not in thermodynamic equilibrium then a rate rela- tion may be specified. For example, consider the generalized Maxwell (1877) (see Kolsky (1963), Herrmann et al (1971), Asay et al (1975)) rela- tion

ae ,+a ZaV.,=O at

where 2 = aZ(p, V) and R = R(p, V) . A special case of this is the Malvem material law where a2 is constant and R = (p - peq)/~ where

peq is the equilibrium pressure and T is the relaxation time of the material.

Another way of introducing rate dependence is to specify the viscous stress or viscosity:

where A 2 0 is the coefficient of viscosity. For example, the Navier- Stokes viscosity is of the form A = A,p where A, > 0 is a constant.

Let 2 = (v, u, E)T then the one-dimensional conservation laws in Lagrangean coordinates for a non-heat-conducting material expressed in- differential form are:

where

KC@ = c-u, P + 4. U(P + q))T

is called the flux vector. Typical problems are mixed initial-boundary value problems: the

initial data, U,(P) is given on some finite interval PL (U 2 PR ; the boundary data tGpical;y specifies either u or p at uL and "R .

DISCRETIZATION METHODS

There are many ways to discretize partial differential equations:

finite difference, finite element, spectral, etc. In the final analysis they all boil down to a projection onto a finite dimensional solution

space. There are many good articles and books on discretizing PDEs. My

favorite is Richtmyer and Morton (1967), but there is also Roache (1972), Lapidus and Pinder (1982), Meis and Marcowitz (1981), Peyret and Taylor

(1983), Halt (1984), Herrmann et al (1973) and Hicks and Walsh (1976). Because of the limited scope of this brief report we shall only dis-

cuss a few discretization methods in detail here and that will be done in

the next section.

THE HYDROCODE CONVERGENCE PROBLEM

John van Neumann (1944) proposed the first hydrocode (a computer pro-

gram for numerical hydrodynamics), conjectured convergence "in the weak

sense", and stated that "a mathematical proof of this surmise would be most

important, but seems very difficult, even in the simplest special cases". This problem we shall refer to as the hydrocode convergence problem. To

describe his scheme we set up a grid. Let t" = n A t , n = 0, 1, 2, 3,...

and Uj = UL + jAp , j = 0, 1, 2,..., (UR - vL)/Au . Let crete approximation to f at the point (t", Uj) .

fy be the dis-

set

A " = n+1

"j+I/2 - $1,:

and

A u = ,“+% . - ,:+‘; j+l J

then van Neumann's discretization of the conservation of volume law is

A'V/At = A.u/Au .

set

a.u = u;+% _ “;-4

and

A.P = P;+~ - p: J-k

8

then his discretization of the conservation of momentum law is

A’ulAt = -A.p/Au .

In 1950, "on Nemann and Richtmyer proposed their artificial viscosity

method. Their artificial (or pseudo) viscosity has the form

q = -c~P~A.+J

where is a dimensionless constant near unity. Their discretization of the conzrvation of volume and momentum are the same as van Neumann's (1944) except p is replaced with p + q in the conservation of momentum law.

Set

11+1 11 a-e = e. 34 - 'j+_S

aud

n+1 I1 P + ‘I = (Pj+ti

114 + Pj+Jz)/2 + qj+k

then their discretization of the evolution equation for the specific inter-

nal energy

se/at = -(p + q)av/at

is

__- A'e/At = -(p + q)A'V/At .

The evolution equation for e is generated by subtracting the u multiple of the conservation of momentum from the conservation of energy and then

using the conservation of volume.

Landshoff (1955) proposed a modification of the van Neumann-Richtmyer

scheme which we shall refer to as the van Neumalm-Richtmyer-Landshoff (VNRL) scheme. In this scheme the q is modified to

q = -cQaA.u - cq p/A.u[A.u

where cI? is a dimensionless constant near one tenth of unity.

The van Neumann-Richtmyer-Landshoff scheme is the basis for several

one, two and three-dimensional hydrocodes in the national laboratories.

For example, see Kipp (1978), Swegle (1978), Thompson (1975), Wilkins

(1964), Brodie and Aubrey (1965), and Herrmann et al (1973). This hydrocode scheme or variations of it is probably used more than any other in the

national laboratories. The hydrocodes that are based on this scheme are used in making national decisions about energy and defense systems.

For these reasons we agree with van Neumann that a hydrocode convergence proof for the van Neumann-Richtmyer-Landshoff scheme "would be most

important". Unfortunately, we also have to agree with van Nanann that it "seems very difficult, even in the simplest special cases".

SOME RELEVANT RESULTS

If we assume the existence of smooth solutions then the convergence

results of Lax and Keller (1951) and Strang (1964) apply. However, the conservation laws allow discontinuities in U to evolve from flows involving compressions in materials with ada; > 0 even when the initial data is arbitrarily smooth. Therefore solutions are sought to the integral

9

form of the conservation laws and these are called weak or generalized

solutions. The integral form of the conservation laws can be derived by

integrating equation (1) in u and t . The hydrocode convergence problem is closely related LO constructive,

global existence, uniqueness and well-posedness proofs for weak solutions

of the conservation laws (see Lax (1973)) . For over twenty years the

prospects of proving "on Neumann's conjecture in the nonlinear case looked

gloomy but 1965 saw a glimmer of hope. Glimm (1965), guided by the illum-

inating results of Friedrichs (1948), G d o uno" (1961), Lax (1957, 1963) and

Oleinik (1957), cast light on the problem by giving the first convergence and global existence proof for the case of the isentropic gas law. Glimm's

scheme is a "Monte-Carlo-ization" of Godunov's (1959) method. That is,

Glimm introduced a random choice scheme and proved convergence with prob-

ability one via a weak law of large numbers argument. The method of Godunov (1959) splits the initial value problem into a

set of Riemann problems, In the Riemann problem the initial data is com-

posed of two states (a left state and a right state); the solution has

three wave types (shocks, rarefactions and contact discontinuities) issuing

from the original discontinuity between the two states. A shock wave is a

compression wave with a discontinuity in the fluid velocity and pressure. A rarefaction wave is an expansion wave with a ramp-shaped profile in the

fluid velocity and pressure. A contact discontinuity is a discontinuity in

the mass density but with continuity in the fluid velocity and pressure. The next paragraph gives a step-by-step description of Godunov's method.

Godunov's method:

i) Start with a step function (,U

val).

constant in each (uj, uj+l) inter-

ii) Solve the Riemann problems that arise between the states in adjacent

intervals. iii) Let the waves run from the interval boundaries into the intervals for

time At where wAt < Au with w being the maximum magnitude of

the wave speeds.

iv) Replace the solution at time t t At with a step function U such

that the measures of the components of ~ U within each inter&l are

conserved.

") Go back to (i) . Glimm's method: Modify step (iv) in Godunov's method as follows: In

each interval cast a random variable, evaluate U there at time

replace Q in the entire interval with this "al?&. t+At,

One of the main tools in Glimm's proof is a functional measuring the interaction between the waves arising from different points of discontinu-

ity. Showing the decrease in time of this functional leads to a bound on the variation of the solution and this leads to compactness via Helly's

Selection Theorem. At first glimpse Glimm's scheme did not appear to give any guidance to

hydrocoders looking for practical schemes. Numerical experiments by Moler and Smaller (1970), Chorin (1976) and Sod (1978) indicate that schemes

suggested by Glimm's method may be competitive on some shock wave problems in the one-dimensional flow of an ideal gas. HOWeVer, it is not as occur- ate as the "on Neumann-Richtmyer-Landshoff method on rarefaction waves (see Dahlgren and Thorne (1970)). There are also difficulties in generalizing the scheme to arbitrary geometries and arbitrary material laws and therefore we must recommend the "on Neumann-Richtmyrr-Landshoff scheme over the Glimm scheme for practical hydrocode methods.

In 1977, Tai-Ping Liu made several important extensions of Glimm's results. Unfortunately, there is not time in this brief report to discuss them all. The one which seems to be the most important to "on Neumann's

hydrocode convergence conjecture is the extension to a deterministic scheme. That is, Liu (1977) proved that the choice procedure in Glimm's method need not be random but that the scheme converges for any equidistributional

10

sequence. Liu accomplished this by introducing a wave tracing technique

yielding new and more precise information about the solution. Thus Liu

established convergence for a de*o.-.LLlnistic hydrocode scheme. This seems to move us closer to the posslt,;lity of proving the convergence of such

classic hydrocode schemes as "on Neumann-Richtmyer-Landshoff, Ilarlow (1957), and Lax-Wendroff (1960). Next we review progress on convergence proofs for these practical, much-used hydrocode schemes.

In Hicks (1969) a compactness result was proved for a conservative

scheme using conservative smoothing. Conservative smoothing is used in the Harlow (1957) PIC method and thus the results in Hicks (1969) are relevant to the much used PIC (Particle In Cell) schemes of Harlow and later modi-

fications by Johnson (1964). From the compactness one gets, of course,

the result that there exists a convergent subsequence. In Hicks (1969) compactness was achieved by using the conservative character of the dis-

cretization to prove bounds on the total volume and total energy. Then using extensions of Rellich's Lemma and the Helly Selection Theorem, com-

pactness was proved for the case of the ideal gas law. In Hicks (1978a, 1981) convergence results were established for the

"on Neumann-Richtmyer-Landshoff method for the case of the linear pV l.ZlWS with a q of the form

q = -cQaoA.u

where CL > 0 is an arbitrary constant. It was proved that the "on Neumann-Richtmyer-Landshoff method is stable under the modified CFL time-

step constraint

a'At < Au

where

a’ = ao(Ck

Convergence was also proved in Hicks

Glvern material law. Stability was straint

+vTKq.

(1978a, 1981) in the case of the proved in this case under the con-

C$L + h(1/2 - Ar) + Air < 1

where CFL = aoAt/Au , h = At/-r , r = At/Au and A = ciao . In both these cases (linear pV law and Malvern material law) the equations are linear and convergence follows from the Lax-Richtmyer Theorem (see Richtmyer and

Morton (1967)). Alternatively, one can subcycle the integration of the

material law with subcycle timestep, At, , while integrating the conser-

vation laws with a timestep, At, , that satisfies the modified CFL constraint above. In this case the stability constraint is

These two cases (linear pV and Malvern) are the only known material laws for which convergence has been proved for the "on Neumann-Richtmyer-Lands- hoff scheme.

In Hicks (1978b, 1979) compactness results were proved for a conservative modification of the "on Neumann-Richtmyer-Landshoff scheme in the case of the ideal gas law. It follows, of course, that a convergent subsequence exists. Certain motion compactness theorems were proved for conservative difference schemes. The compactness argument here is based on the embedding theorem in Sobolev spaces. That is, boundedness in a Sobolev space results in compactness in the relevant Banach space. These conver-

11

gence results are the only known results for the conservative modification of the van Neumann-Richtmyer-Landshoff scheme in the case of the ideal gas law.

See Smaller (1983) for some of the current research directions on the existence, uniqueness and stability of the conservation laws. As Smaller points out, there is a very interesting interplay here between analysis, topology and computational methods. It seems reasonable to add supercomputer technology to this list. That is, the architecture of today's supercomputers suggest computation methods. On the other hand, the need for greater computational capabilities for hydrocodes is a driving force for supercomputer development.

Applications of hydrocodes are many and varied. For example, they are used in simulations of loss-of-coolant accidents such as the infamous Three Mile Island Incident; they are also used in simulating the behavior of the fusion reactors of the future. Although we have only discussed the one- dimensional equations here, the two-and three-dimensional equations are the ones that make the strongest demands on supercomputers. See Richtmyer and Morton (1967) for the two and three dimensional equations and a discussion of the computational difficulties involved. For further entry points into the literature see the bibliography.

John van Neumann's (1946) prophecy continues to be fulfilled: 'I... we conclude by remarking that really efficient high-speed computing devices

may, in the field of nonlinear partial differential equations as well as in many other fields which are now difficult or entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress."

REFERENCES

1.

2.

3.

4.

5.

6.

7.

Abarbanel, S. and G. Zwas (1969), "An Iterative Finite Difference Method for Hyperbolic Systems", Math. Comp. V23, pp. 549-565.

Abbot, M. B. (1966), An Introduction to the Method of Characteristics, American Elsevier Pub. Co.

Ablow, C. M. (1972), "A Characteristic Finite Difference Method for the Wave Equation in Two Dimensions", SIti J. Numer. Anal., V9, Nl pp. 152- 164.

Agmon, S., L. Nirenberg, and M. Protter (1953), "A Maximum Principle for a Class of Hyperbolic Equations and Applications to Equations of Mixed Elliptic-Hyperbolic Type", Comm. Pure Appl. Math., V6, pp. 455- 470.

Asay, J. R., D. L. Hicks, and D. B. Holdridge (1975), "Comparison of Experimental and Calculated Elastic-Plastic Wave Profiles in LiF", J. - of Applied Physics, V46, NlO, pp. 4316-4322.

Bagrinovski, K. A. and S. K. Godunov (1957), "Difference Schemes for Multidimensional Problems", Dok. Acad. Nauk., V115, p. 115.

Bakhvalov. N. S. (1970). "On the Existence in the Laree of Solutions of ,~ .~,, <, Quasilinear Hyperbolic Systems", J. Comp. Math. and Math. Phys., VlO, pp. 969-980.

12

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

Barker, L. M. (1969), "SWAP-9: An Improved Stress Wave Analyzing Program" , SC-RR-69-233.

Bellman, R. and R. H. Pennington (1954), "Taylor Instability", Quart. Appl. Math., V12, pp. 151-162.

Bertholf, L. D. and S. E. Benzley (1968), "TOODY II, A Computer Program for Two-Dimensional Wave Propagation", SC-RR-68-41.

Bethe, H. A., K. Fuchs, J. van Neumann, R. Peierls, and W. G. Penny (1944), "Shock Hydrodynamics and Blast Waves", LASL Report AECD-2860.

Birkhoff, G. (1950), Hydrodynamics, Dover Publications.

Birkhoff, 6. (1962), "Helmholtz and Taylor Instabilities", Proc. of Symp. in Appl. Math., V92, pp. 13-76.

Birkhoff, C,. and R. S. Varga (1959), "Implicit Alternating Direction Methods", Trans. A. M. S., V92, P. 13.

Blair, A., N. C. Metropolis, J. "on Neumann, A. H. Taub, and M. Tsingou (1959), "A Study of a Numerical Solution to a Two-Dimensional Hydrodynamical Problem", Math. Camp., V13, p. 145.

Brodie, R. N. and J. W. Aubrey (1965), "The PUFF and P-PUFF Computer Prog&ns", AFWL-TR-65-24.

Burst&n, S. Z., H. S. Schechter and E. L. Turkel (1975), "A Second Order Numerical Model for High Velocity Impact Phenomena", BRL-CR-239.

Butcher, B. M., D. L. Hicks, and D. B. Holdridge (1972), "Elastic Precursor Decay and Steady Wave Evolution in Porous Aluminum", Sandia National Labs., SC-RR-72-0627.

Cameron, I. G. (1966), "An Analysis of the Errors Caused by Using Artificial Viscosity Terms to Represent Steady-State Shock Waves", J- Comp. Phys., VI, pp. l-20.

Campbell, P. M. (1968), "An Annotated Bibliography on High Temperature Radiation Gas Dynamics", AFWL-TR-68-119.

Chorin, A. (1976), "Random Choice Solution of Hyperbolic Systems", J- Comp. Phys., V22, pp. 517-533.

Ciment, M. (1971), "Stable Difference Schemes with Uneven Mesh Spac- ings", Math. Camp., V25, N114, pp. 219-227.

Collatz, L. (1951), Numerische Behandlung van Differentialgleichungen, Springer-Verlag.

Conley, C. C. and J. A. Smaller (1972), "Shock Waves as Limits of Pro- gressive Wave Solutions of Higher Order Equations, II", Comm. Pure and Appl. Math., V25, pp. 133-146.

Conway, E. D. (1967), "Generalized Solutions of Linear Differential Equations with Discontinuous Coefficients and the Uniqueness Question for Multidimensional Conservation Laws", J. Math. Anal. and Appl., V18,

PP. 238-251.

26. courant, R. and R. 0. Friedrichs (1948), Supersonic Flow and Shock Waves, Interscience Publishers.

13

27. Courant, R., K. 0. Friedrichs, and H. Lewy (1928), "iiber die partiellcn Differenzen~leichungen der mathematische Physic", Math. Ann., VlOO,

PP. 32-74.

28. Courant, R. and D. Hilbert (1962), llethods of Mathematical Physics, V2,

Interscience Publishers.

29. Co"rant, R., E. Isaacson, and M. Rees (1952), "On the Solution of Non- linear Hyperbolic Differential Equations by Finite Differences", Comm.

Pure and Appl. Math., V5, pp. 243-255. I__

30. Dafermos, C. M. (1973), "The Entropy Rate Admissibility Criterion for

Solutions of Hyperbolic Conservation Laws", J. Diff. Eqns., V14, N2,

pp. 202-212.

31. Douglis, A. (1961), "The Continuous Dependence of Generalized Solutions of Nonlinear Partial Differential Equations Upon Initial Data", Comm.

Pure and Appl. Math., V14, pp. 267-284.

32. Elliot, L. A. (1962), "Shock Fitting in Two-Dimensional Flow", Proc.

Roy. Sot., VA 267, pp. 558-565.

33. Epstein, B. (1962), Partial Differential Equations, McGraw-Hill Book

co.

34. Epstein, B. (197O), Linear Functional Analysis, W. B. Saunders Pub-

lishers.

35. Flowerdew, S. .J. (1969), "An Analysis of Some Eulerian !lethods for One-Dimensional, Inviscid, Compressible Hydrodynamics", Master's The-

sis, Univ. of Illinois.

36. Fey, R. L. (1964), "Steady-State Solutions of Hyperbolic Systems of Conservation Laws with Viscosity Terms", Comm. Pure and Appl. Math.,

V17, pp. 177-188.

37. Frank, R. M. and R. B. Lazarus (1964), "Mixed Culerian-Lagrangian Method", Meth. in Comp. Phys., V3, p. 47, Academic Press.

3x. Friedman, A. (1962), Partial Differential Equations, Halt, Rinehart, and Wins ton.

39. Friedrichs, K. 0. (1948), "Formation and Decay of Shock Waves", Comm. Lure and Appl. Ma&, Vl, pp. 211-245.

40. Friedrichs, K. 0. (1954), "Symmetric Hyperbolic Differential Equa- tions 'I, +nm. Pure and Appl. Math., V7, pp. 345-392.

41. Friedrichs, K. 0. and H. Lewy (1928), " "Llber die Eindeutigkeit und das

Abhangigkeitsgebiet der Losungen beim Anfangswert problem linearer hyperbolischen l)ifferentialgleichullgell", Math. Ann., V98, pp. 192-204.

42. Garabedian, P. (1964), Partial Differential Equations, J. Wiley Pub- lishing Co.

43. Gary, J. (1966), "A Generalization of the Lax-Richtmyer Theorem on Finite Difference Schemes", J. SIAM Numer. Anal., V3, N3, pp. 467-473.

44. Gelfand, I. M. (1959), "Some Problems in the Theory of Quasilinear Equations", _Llspekhi Mat. Nauk, V14, pp. 87-158.

14

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

Glimm, J. (1965), "Solutions in the Large for Nonlinear Hyperbolic

Systems of Equations", Comm. Pure and Appl. Ilath., V18, N4, pp. 697-

715.

Glimm, J. and P. D. Lax (1969), "Decay of Solutions of Systems of Non- linear Hyperbolic Conservation Laws", AEC Report NYO-1480-115.

Goad, W. B. (1960), "WAT: A Numerical Method for Two-Dimensional Un-

steady Fluid Flow", LASL Report LAMS 2365.

Godunov, S. K. (1959), "Finite Difference Method for Numerical Compu-

tation of Discontinuous Solutions of the Equations of Fluid Dynamics",

Mat. Sbornik, V47 (89), N3, pp. 271-306.

Godunov, S. 1:. (1961), "Error Estimates for the Approximate Solutions

of the Simplest Equations in Gas Dynamics", Zhurnal Vych. Nat. i Mat.

a, Vl, N4, pp. 622-637.

Godunov, S. K. and G. P. Prokopov (1972), "The Use of Moving Meshes in Gas-Dynamical Computations", USSR Camp. Math. and Math. Phys., VU, N2,

pp. 182-195.

Gottlieb, D. (1972), "Strang-Type Difference Schemes for Multidimen-

sional Problems", SIAM J. Numer. Analy., V9, N4, pp. 650-661.

Gourlay, A. R. and J. L. Morris (1968), "Finite Difference Methods for Nonlinear Hyperbolic Systems, II", Math. Comp. V22, N103, pp. 549-556.

Grad, H. (1961), "The Many Faces of Entropy", Comm. Pure and Appl.

Math., V24, pp. 323-354.

Grandey, R. A. (1963), "The ROC-VTS Computer Program", AFSWC-TDR-62-

114.

Harlow, F. H. (1957), "Hydrodynamic Problems Involving Large Fluid Distortions", J. Assoc. Comp. Mach., V4, pp. 137-142.

Uarlow, F. H. (1963), "The Particle-In-Cell (PIG) Method for Numerical Solutions of Problems of Fluid Mechanics", Proc. Symp. in Appl. Math.,

V15, AMS Providence, RI, p. 269.

Harlow, F. H. (1969), "Numerical Methods for Fluid Dynamics, An Anno-

tated Bibliography", LASL Report LA-4281.

Harten, A. and G. Zwas (1972), "Self-Adjusting Hybrid Schemes for Shock Computations", J. Comp. Phys., V9, pp. 568-583.

Hedstrom, G. W. (1965), "The Near-Stability of the Lax-Wendroff Method",

Numer. Math,, V7, pp. 73-77.

Herrmann, W. and D. L. Hicks (1973), "Numerical Analysis Methods", pp. 57-91 of Metallurgical Effects at Hi&h Strain Rates, Ed. Rohde, Butcher, Holland and Karnes, Plenum-New York-London.

Herrmann, W., D. L. Hicks, and E. G. Young (1971), "Attenuation of Elastic-Plastic Stress Waves", pp. 23-63 in Shockwaves and the Mech- anical Properties of Solids, ed. J. J. Burke and V. Weiss; published by Syracuse Univ. Press.

Hersh, R. (1963), "Mixed Problems in Several Variables", J. Math. x, v12, p. 317.

62.

15

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

79.

Hicks, D. L. (1978a), "The Hydrocode Convergence Problem, Part I_", Camp. Meth. in Appl. Mech. and Eng., V13, Nl, pp. 79-88.

Hicks, D. L. (1978b), "Stability Analysis of WONDY (a Hydrocode based on the Artificial Viscosity Method of "on Neumann and Richtmyer) for a Special Case of Maxwell's Material Law", Math. Comp., V32, N144, pp. 1123-1130.

Hicks, D. L. (1979), "The Hydrocode Convergence Problem, Part 2", s Meth. in Appl. Mech. and Eng., VZO, N3, pp. 303-316.

Hicks, D. L. (1980, "Hydrocode Subcycling Stability", Math. Camp., V37, N155, pp. 69-78.

Hicks, D. L. (1985), "Hydrocodes on the HEP", a chapter in the book MIMD Computation: HEP Supercomputer and its Applications, ed. by J. Kowalik, MIT Press.

Hicks, D. L. and R. T. Walsh (1976), "Numerical Analysis of the Partial Differential Equations in Hydrocodes and Wavecodes", Sandia National Laboratories, SAND-75-0448.

Holt, M. (1984), Numerical Methods in Fluid Dynamics, Springer-Verlag.

Hopf, E. (1969), "On the Right Weak Solution of the Cauchy Problem for a Quasilinear Equation of First Order", J. Math. Mech., V19, pp. 483- 487.

Hoskin, N. E. (1964), "Solution by Characteristics of the Equations of One-Dimensional Unsteady Flow", Meth. in Comp. Physics, V3, p. 265.

Hugoniot, II. (1889), "Sur la Propagation du Mouvement daus les Corps et Specialment daus les gas Parfaits", J. de 1'Ecole Polytechnique, V58, p. 80.

llurd, W. (1969), "A Uniqueness Theorem for Weak Solutions", Pat J. A Math., V28, pp. 556-559.

Johnson, J. L. and J. A. Smaller (1969), "Global Solutions for an Ex- tended Class of Hyperbolic Systems of Conservation Laws", Arch. Rat. Mech. Anal., V32, pp. 169-189.

Johnson, W. E. (1964), "OIL-A Continuous Two-Dimensional Hydrodynamic Code", CAMD-5580.

Kasahara, A., E. Isaacson, and J. Stoker (1964), "Numerical Studies of Frontal Motion in the Atmosphere", Courant Inst. Report NY0 1480-6.

Kidder, R., D. Steinberg, and A. Cecil (1966), "A One-Dimensional Magnetohydrodynamics Code (MAGPIE)", Lawrence Livermore Labs Report UCRL-14931.

Keller, J. B. and I. Kolodner (1954), "Taylor Instability of a Liquid Layer", J. Appl. Physics, V25, pp. 918-921.

Key, S. W. (1974), "HONDO-A Finite Element Computer Program for the Large Deformation Dynamic Response of Axisymmetric Solids", SLA-74- 0039.

80. Kipp, M. E. and R. J. Lawrence (1982), "WONDY V - A One-Dimensional Finite-Difference Wave Propagation Code, Sandia National Laboratories,

16

SAND-81-0930.

81. Kolsky, II. (1955), "A Method for the Numerical Solution of Transient

Hydrodynamic Shock Problems in Two Space Dimensions", LASL Report LA-

1867.

82. Kolsky, H. (1963), Stress Waves in Solids, Dover Publishers.

83. Kreiss, H. 0. (1966), "Difference Approximations for Hyperbolic

Differential Equations", pp. 125-152 in Numericai Solution of Partial

Differential Equations, Ed. by Bramble, Academic Press.

84. Landshoff, R. (1955), "A Numerical Method for Treating Fluid Flow in

the Presence of Shocks", LASL Keport LA-1930.

85. Lapidus, L. and G. F. Pinder (1982), N umrrical Solutions of Partial Differential Equations in Science and Engineering, Wiley-Interscience.

86. Lawrence, R. J. and D. S. Mason (1971), "WONDY IV - A Computer Program for One-Dimensional Wave Propagation with Rezoning", SC-RR-71-0284.

87. Lax, P. D. (1957), "Hyperbolic Systems of Conservation Laws, II",

Comm. Pure and Appl. Math., VlO, pp. 537-566.

88. Lax, P. D. (1963), "Nonlinear Hyperbolic Systems of Conservation Laws",

pp. 3-12 of Nonlinear Problems, ed. by Lager, Univ. of Wisconsin

Press.

89. Lax, P. D. (1973), "Hyperbolic Systems of Conservation Laws and the

Mathematical Theory of Shock Naves", Vll of SIAM CBMS Reg. Conf. Ser. in Appl. Math.

90. Lax, P. D. and J. B. Keller (1951), "The Initial and Mixed Initial

and Boundary Value Problem for Hyperbolic Systems", LASL Report, LAMS-

1205.

91. Lax, P. D. and B. Wendroff (1960), "Systems of Conservation Laws",

Comm. Pure and Appl. Math., V13, pp. 217-237.

92. Lax, P. D. and B. Wendroff (1962), "On the Stability of Difference

Schemes: A Sharp Form of Garding's Inequality", Comm. Pure and Appl.

u, V15, pp. 363-371.

93. Liu, T. P. (1977a), "The Deterministic Version of the Glimm Scheme",

Comm. Math. Phys., V57, N2, pp. 135-148.

94. Liu, T. P. (1977b), "Solutions in the Large for the Equations of

Nonesentropic Gas Dynamics", Indiana Univ. Math. .I., V26, Nl, pp. 147- 177.

95. Liu, T. P. (1977c), "Decay to N-waves of Solutions of General Systems of Nonlinear Hyperbolic Conservation Laws", Comm. Pure Appl. Math.,

V30, N5, pp. 586-611.

96. Liu, T. P. (1977d), "Large Time Behavior of Solutions of Initial and Initial-Boundary Value Problems of a General System of Hyperbolic Conservation Laws", Comm. Math. Phys., V55, N2, pp. 163-177.

97. Liu, T. P. (1977e), "Initial-Boundary Value Problems for Gas Dynamics", Arch. Rat. Mech. Anal., V64, N2, pp. 137-168.

17

98.

99.

100.

101.

102.

103.

104.

105.

106.

107.

108.

109.

110.

111.

112.

113.

114.

Malvern, L. E. (1969), Introduction to the Mechanics of a Continuous

Medium Prentice-Hall Pub. Co. -9

Maxwell, J. Clerk (1877), Matter and Motion, Dover Pub. Co.

McGrath, J. F., D. L. Hicks, and L. M. Liebrock (1985), "Parallel

Processing for Computational Continuum Dynamics", pp. 77-96, Workshop

on Parallel Processing Using the Heterogeneous Element Processor, Ed.

by S. Lakshmivarahan, Univ. of Oklahoma.

Meis, T. and U. Marcowitz (1981), Numerical Solution of Partial

Differential Equations, Springer-Verlag.

Moler, C. and J. Smaller (1970), "Elementary Interactions in Quasi-

linear Hyperbolic Systems", Arch. Rat. Mech. Anal., V37, pp. 309-322.

Moretti, G. (1974), "A Pragmatical Analysis of Discretization Proce-

dures for Initial and Boundary Value Problems in Gas Dynamics and Their Influence on Accuracy or "Look Ma, No Wiggles"," Polytech. Inst.

of New York Report AD/A-O02 293.

Morrey, C. B. (1955), "On the Derivation of the Equations of Hydro-

dynamics from Statistical Mechanics", Corn. Pure and Appl. Math., V13,

pp. 279-326.

Morrey, C. B. (1966), Multiple Integrals in the Calculus of Varia-

tions, Springer-Verlag.

Morton, K. W. (1964), "011 a Matrix Theorem Due to H. 0. Kreiss", Comm.

Pure and Appl. Math., V17, pp. 375-379.

Morton, K. W. and S. Schechter (1965), "On the Stability of Finite

Difference Matrices", J. Sot. Ind. Appl. Math. Ser. B. Num. Anal.,

V2, pp. 119-128.

van Neumann, J. (1944), "Proposal and Analysis of a New Numerical

Method for the Treatment of Hydrodynamical Shock Problems", Nat. Def.

and Res. Corn. Rep. AM-551. (Available in Collected Works, Vol. 6,

pp. 361-379).

van Neumann, J. (1945), "Memorandum to 0. Veblen on Variational Meth-

ods in Hydrodynamics", Collected Works, V7, pp. 357-359.

van Neumann, J. (1949), "Discussion on the Existence and Uniqueness on Multiplicity of Solutions of the Aerodynamical Equations", in

Collected Works, Vol. 6, pp. 348-356.

van Neumann, J. (1963), The Collected Works of John van Neumann, ed. by Taub, Pergamon Press.

van Neumann, J. and R. D. Richtmyer (1950), "A Method for the Numrri- cal Calculation of Hydrodynamic Shocks“, J. Appl. Physics, V21, N3,

PP- 232-237.

Oleinik, 0. A. (1957a), "Discontinuous Solutions of Nonlinear Differ- ential Equations", Usp. Mat. Nauk, v12, pp. 3-73.

Oleinik, 0. A. (1957b), "On the Uniqueness of Generalized Solutions of the Cauchy Problem for a Nonlinear System of Equations Occurring in

Mechanics", Usp. Mat. Nauk, V12, N6 (78), pp. 169-176.

18

115.

116.

117.

118.

119.

120.

121.

122.

123.

124.

125.

126.

127.

128.

129.

130.

131.

132.

133.

Oleinik, 0. A. (1961), "The Convergence of Certain Difference Schemes", Dok. Akad. Nauk, V137, pp. 523-526.

Osher, S. and R. Sanders (1983), "Numerical Approximations to Non- linear Conservation Laws with Locally Varying Time and Space Grids", Math. Camp., V41, N164, pp. 321-336.

Parlett, B. (1966), "Accuracy and Dissipation in Difference Schemes", Comm. Pure and Appl. Math., V19, pp. 111-123.

Parlett, B. (1978), "Progress in Numerical Analysis", SIAM Rev., V20, pp. 443-457.

Peyret, R. and T. D. Taylor (1983), Computational Methods for Fluid Flow, Springer-Verlag.

Quarles, D. A. (1964), "A Moving Coordinate Method for Shockwave Calculations", Ph.D. Thesis at NYU.

Ransom, V. H. and D, L. Hicks (1984), "Hyperbolic Two-Pressure Models for Two-Phase Flow", J, Comp. Physics, V53, Nl, pp. 124-151.

Richtmyer, R. D. and K. W. Morton (1967), Difference Methods for Ini- tial Value Problems, Interscience Pub.

Roache, P. (1972), Computational, Hermosa Press.

Smaller, J. A., Ed. (1983), Nonlinear Partial Differential Equations", Contemp. Math. Series, V17, Amer. Math. Sot.

Smaller, J. A. (1983), Shock Waves and Reaction-Diffusion Equations, Springer-Verlag.

Smaller, J. and J. L. Johnson (1969), "Global Solutions for an Ex- tended Class of Hyperbolic Systems of Conservation Laws", Arch. for Rat.Mech. and Anal., V32, pp. 169-189.

Sod, G. A., "A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws", J. Comp. Phys., V27, pp. l-31.

Stetter, H. J. (1961), "On the Convergence of Characteristic Finite- Difference Methods of High Accuracy for Quasi-Linear Hyperbolic Equa- tions", Numer. Math., V3, pp. 321-344.

Stoker, J. J. (1957), water waves, Interscience Publishers.

Strang, W. 6. (1964), "Accurate Partial Difference Methods II: Non- linear Problems", Numer. Math., V6, pp. 37-46.

Swegle, J. W. (1978), "TOODY IV - A Computer Program for Two-Dimen- sional Wave Propagation", Sandia National Laboratories, SAND-78-0552.

Swegle, J. W. (1981), "ART00-An Automatically Rezoning Rro-Dimension- al Lagrangian Finite Difference Wavecode", Sandia National Labora- tories, SAND-81-2235.

Swegle, J. W. (1982), "Verification Problems for ARTOO, an Automati- cally Rezoning Two-Dimensional Finite-Difference Wavecode", Sandia National Laboratories, SANDBl-2236.

19

134.

135.

136.

137.

138.

139.

140.

141.

142.

143.

144.

145.

146.

Thompson, S. L. and H. S. Lauson (1972), "Improvements in the CHART D

Radiation Hydrodynamic Code II: A Revised Program", SC-RR-71-0713.

Thorne, B. .I. and D. B. Holdridge (1974), The TOOREZ Lagrangian Re- zoning Code", Sandia National Laboratories SLA-73-1057.

Truesdell, C. and W. No11 (1965), "The Nonlinear Field Theories of

Mechanics", VIII/3 of Handbuch der Physik, Springer-Verlag Pub.

Trulio, J. 6. (1973), "On the Eulerian-Afton Equations for Axisym-

metric Fluid Flow", Applied Theory, Inc. Report ATR-73-33-1, AD-758

010.

Ulam, S. (1962), "Stability of Many-Body Computations", Proc. of Symp.

in Appl. Math., V13, pp. 247-258.

Vichnevetsky, R. (1981), "Energy and Group Velocity in Semidiscreti-

zations of Hyperbolic Equations", Math. Comp. Sim., V23, pp. 333-343.

Vvendenskaya, N. D. (1956), "The Difference Method Solution of Cauchy's Problem for a Nonlinear Equation with Discontinuous Initial

Values", Dok. Akad. Nauk, Vlll, pp. 517-521.

Walsh, R. T. (1971), "Transformation of Conservation Laws", SC-TM-71-

0373.

Walsh, R. T. (1973), "Optimization and Comparison of Partial Differ-

ence Methods", SIAM J. Numer. Anal., VlO, N4, pp. 785-797.

Wendroff, B. (1972), "The Riemann Problem for Materials with Non-

convex Equation of State", J. of Math., V38, pp. 454-466.

Wendroff, B. (1976), "Review of Marchuk's Methods of Numerical Mathe- matics", Bull. Amer. Math. Sot., V82, p. 864.

Xiang, L. W., C. K. Cbu and Y. Baransky (19831, "Solitary Waves In- duced by Boundary Motion", Comm. Pure Appl. Math., V36, N4, pp. 495-

504.

Yanenko, N. N. (1971), The Method of Fractional Steps, Springer-Ver-

lag.

147. Zeldovich, Ya. and Yu. Raizer (1966), Physics of Shock Waves and High-

Temperature Hydrodynamic Phenomena, Academic Press.

introduction to computational continnum dynamics: a personal perspective

Documents