an approximate riemann solver for the euler …oden/dr._oden_reprints/1993-013.an_h-r-adaptive.pdfis...

SIAM J. SCI. COMPUT.Vol. 14, No. I, pp. 1811-217.Janu&ry 1993

© 1993 Society for Industrial and Applied Mathematics011

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AN h-r-ADAPTIVE APPROXIMATE RIEMANN SOLVER FOR THEEULER EQUATIONS IN TWO DIMENSIONS·

MICHAEL G. EDWARDSU, J. TINSLEY ODEN', AND LESZEK DEMKOWICZ'

Abstract. A new adaptive st.rategy for solving the Euler equations of compressible flow ispresented. The method of Roe [J. Comput. Phys., 43 (1981). pp. 357-372] is extended into twodimensions for an arbitrary quadrilateral grid and is coupled with the h-adaptive quadrilateralrefinement-unrefinement algorithm of Demkowicz and Oden [T/COM Report 88-02]. Refinementof a quadrilateral grid retains a certain grid structure which is fully exploited by the extension ofthe higher-order version of the method into two dimensions. A total variation diminishing (TVD)analysis is presented for a nonuniform grid, together with an assessment of the solution error inducedby the nonuniformity in the grid. Grid movement is also considered and adaptive strategies are dis-cussed and tested. The adaptive scheme proves to be highly robust. Improved accuracy and largesavings in computer time are obtained.

Key words. adaptive. grid refinement, higher-order, TVD, Riemann solver, compressible Eulerequations

AMS(MOS) 8ubject classifications. 65~f06, 65M50

1. Introduction. In the early seventies, much of the research on numericalmethods for hyperbolic conservation laws focused on producing schemes that wereknown to produce physically meaningful solutions whenever they converged to theexact solution. At the same time, schemes were sought that did not oscillate in thevicinity of shocks. The first family of schemes that fulfilled these requirements werethe so-called monotone difference schemes in which the numerical fluxes are monotonefunctions of the cell-centered values of the discrete solutions.

Unfortunately, monotone schemes were found to suffer from two major deficien-cies: they are no more than first-order accurate, and they are usually overdissipative,smearing shocks over several grid spacings. These defects promoted an extensive se-ries of investigations for the "holy grail" in conservation law solvers: schemes that didnot oscillate but did yield higber-order (e.g., second-order) accuracy.

An advance in this direction for the case of one-dimensional scalar conservationlaws was made by Harten [3], who introduced the notion of a TVD (total variationdiminishing) scheme. The idea is that if the total variation of the solution can becontrolled so that it never increases over a timestep, then a nonoscillating solutionwith second-order accuracy can be obtained. This can be accomplished by limiting thevalues of the numerical flux ("flux-limiting methods"); several alternative flux-limitingstrategies were discussed by Sweby [41. Among tbese is the method of Roe [11, which,while usually very effective, may violate the entropy condition for the conservationlaw. An "entropy fix" was proposed by Harten and Hyman [5] for overcoming thisdefect in Roe's approach .

It should also be noted that most of the theoretical results were developed forone-dimensional scalar conservation laws. Goodman and LeVeque [61 argued that

•Received by the editors June 6, 1989;accepted for publication (in revised form) March 31. 1992.'Texas Institute for Computational Mechanics. The University of Texas at Austin, Austin, Texas

78712. The research of the first two authors was supported in part. by Army Research OfficegrantDAAL03-89-K-0120and by the Office of Naval Research. The research of the third author wassupported by NASA Langley Research Center.

tPrcsent address, B. P. Research Centre, Chertsey Road, Sunbury-on-Thames, MiddlesexTW167LN, United Kingdom.

185

186 EDWARDS. ODEN, AND DEMKOWICZ

any two-dimensional TVD scheme had to also be monotone, and hence, Illay be onlyfirst-order accurate. However, numerical experiments suggest that two-dimensionalgeneralizations of one-dimensional TVD schemes (which are not strictly TVD schemesin two dimensions) can be second-order accurate in many cases. Moreover, they canalso be robust and yield nonoscillating, high-resolution simulation of shocks (see, e.g.,Yee (71).

The simple but less reliable alternative to using a TVD scheme is to resort to somevariant of a Lax-Wendroff scheme and combine it with an ad hoc artificial viscositymethod, which can be tuned to eliminate spurious oscillations in some cases [8].

However, the resolution of discontinuities obtained by any discrete scheme willstill be limited by the local cell size. Adaptive grid techniques have been developedby several authors, (e.g., [9]. [10], [11], [12]. and [13]) with a common aim of con-ccntrating (either by inserting or moving) grid nodes into the areas of the flow fieldwhere they are most needed (where the flow gradients are large), thereby reducingthe discretization error as the element size reduces. For a numerical scheme of givenaccuracy. grid adaption can lead to increased accuracy and a reduction in computertime and computer storage requirements.

Although adaptive techniques can improve on the results obtained with artificialviscosity methods, any spurious oscillations that occur can be highlighted by thcadaptive grid and amplified [9]. This provides a strong motivation for combiningTVD schemes with adaptivity. In particular, the combination of a second-order flux-limited scheme on an adaptive quadrilateral grid has a two-fold advantage over itstriangular counterpart.

First, a dynamically refined quadrilateral grid is far less likely to produce badlydistorted elements than a dynamically refined triangular grid, and therefore, we canexpect convergence on a quadrilateral grid.

Second, the flux-limiting concept can be naturally generalized to an adaptivequadrilateral grid.

In this paper, we develop an adaptive, two-dimensional TVD scheme for systemsof hyperbolic conservation laws, particularly Euler's equations. This scheme functionson an arbitrary unstructured mesh of quadrilateral cells (unstructured in the sensethat the cells may be dynamically refined or unrefined). We employ an extendedversion of Roe's scheme in two dimensions with flux limiting and an entropy fix.

Error indicators arc computed at the end of a designated number of timesteps,and the mesh is automatically refined (an h-method) or unrefined using the techniqueof Demkowicz and Oden [2]. This adaptive technique is applicable to both time-dependent and steady-state problems.

In addition (currently only) for steady-state problems, grid node relocation (anr-method) is induced according to an equidistribution principle. The grid is post pro-cessed and automatically aligns grid lines along discontinuities.

The r-method is also combined with the h-method (currently) for steady-stateproblems. After the grid has been relocated and subsequently refined, results superiorto those of h-adaptive schemes without node relocation are obtained.

In all of the numerical experiments performed, the adaptive h-scheme (for time-dependent and steady-state problems) and the h-r scheme (for steady-state problems)perform surprisingly well, giving excellent resolution of flow features on rather coarsegrids. Large savings in computer time were also observed with the same code takingfour times longer to run on equivalent uniform fine grids.

AN h-r-ADAPTIVE RIEMANN SOLVER FOR EULER EQUATIONS 187

2. Roe scheme. We begin by reviewing the scheme proposed by Roe [14], [15]for the scalar conservation law

(2.1)ut+f(u)x=O t>O xEIR,

U(X,O) = Uo(X),

where u(x, t) has initial data u(x, 0) = uo(x). We also write (2.1) in the form

Ut + a(u)ux = 0, a(u) = dfduo

We next partition the domain lIt into finite difference cells Ii = [Xi_1, Xi+1] i E Z,2 2

with centroids Xi. A piecewise constant mesh function W takes on values Wi = W(Xi)'We use the following standard notation for differences:

for the grid ratio µ:

(2.2a)

for the local CFL number v:

µ= 6.t/6.Xj

(2.2b)

and a( Ui, Ui+ 1) is the discrete wave speed at i+ ~.The first-order Roe [14] scheme (in space and time) is the conservative upwind

scheme

(2.3)Vi_! ~ 0,

vi+! < 0,

which is stable and oscillation free for

(2.4)

We note that (2.3) can be written as

Ivl ::S 1.

(2.5a)

where

(2.5b)

u~+1 = ui - µ(hi+t - hi_!),

is a consistent numerical flux function with

h(u, u) = f(u).

It is well known that this scheme must be modified to ensure entropy satisfaction,which is achieved here by employing the Harten and Hyman entropy fix [51. Roe [14]extended his scheme to second-order accuracy and assured monotonicity preservation

188 EDWARDS, ODEN. AND DEMKOWICZ

(monotone data at time level n remains monotone at time level n + 1) by the use of aflux limiter, a device previously used by van Leer [16]. The monotonicity-preservingschemes of Chakravarthy and Osher [17]and Harten [3] employ similar devices, andthe relationships between the schemes is discussed by Sweby [4],who presents a unifiedapproach to designing such schemes.

All of the above-mentioned schemes can be shown to be TVD, a notion firstintroduced by Harten [3]. The total variation, at time level n + 1, TV(unH), of thesolution is defined by

(2.6)

Harten [5Jdefined a TVD scheme for which

(2.7)

One immediate consequence of (2.7) is that the total variation remains bounded,which is one of the criteria that must be satisfied in order to establish convergence ofa difference scheme approximating (2.1) (see [18), [19), and [20]).

An important practical consequence of (2.7) is that the solutions obtained froma TVD scheme are free of spurious oscillations, which follows from Harten's [3]obser-vation that a TVD scheme is monotonicity preserving.

Harten [3] has also shown that sufficient conditions for a scheme of the form

(2.8a)

to be TVD are

(2.8b)

U~+l = u,n - C,' 1 ~U,n 1 + Di+1SU:+l,-, -, 2 2

From (2.3) and (2.4) , it follows that the first-order upwind scheme satisfies (2.8b)(and is therefore TVD) if for all i,

(2.9)

3. A TVD scheme on a nonuniform grid. While much attcntion has beenfocused on the application of the Roc scheme via the use of uniform grids, verylittle attention has been given in the case of nonuniform grids [21]. Since the usualdefinitions of accuracy 011 a uniform grid do not necessarily apply to nonuniform grids,we shall refer to the usual first-order scheme as the low-order scheme, and the usualsecond-order schemes as the high-order scheme on nonuniforIIl grids.

In this section we shall consider how to construct a high-order TVD scheme 011 anonuniform grid in one dimension. Our primary concern is that the scheme remainsconservative 011 a nonuniform grid. We shall consider the question of accuracy afterderiving a TVD scheme in conservation form on a nonuniform grid.

Consider the Lax-Wendroff scheme on a uniform grid:

(3.1) U~+l = ui - 4(Vi-12 ~U7_1 + vi+t~u7+!) + ~(IVi+1212 ~U~l -IVi __2112~U~_!).2 - 2 2 2

This schemc may be written in cOllservation form as

(3.2)

(3.3)


where

h (fi+fHl) I::1t I 12~ n

i+! = 2 - 21::1x aH! uH!·

The scheme (3.2) is said to be conservative since conservation may be demonstratedimmediately by summing (3.2) over all grid points i to obtain

(3.4) ~)u~+1 - u~)~x = -~t~)hH! - hi_!).i

Omitting boundary terms, the contribution on the right of (3.4) is zero, hence

(3.5)

and (3.5) indicates that the area (or "mass") J udx is conserved. So far we haveassumed that ~x is constant, but if we consider the nonuniform grid shown in Fig. 1,then we can replace ~x with the local element length ~Xi in (3.2)-(3.4) and retainconservation. The nonuniform grid approximation is

(3.6)

where

(3.7a)

n+1 n ~t (h h)ui = Ui - ~Xi i+~ - i-! '

(3.7b) ~Xi+! = (XH! - Xi) .

Note the introduction of ~XH;' in hH!, which ensures that the flux summation willcontinue to cancel for a nonul1lform grid.

•

(3.9)

FIG. 1. Nonuniform grid.

The low-order scheme of §2 can be similarly generalized to a nonuniform grid with(3.7a) replaced by

(3.8) hH! = ~(fi + fHJ) - ~Iai+! I~UH!'

while (3.6) remains unchanged. Expanding (3.6) and (3.8) and taking the case aH! 2:0, the scheme can be written in the upwind form

~ta· 1u~l+l = u~ - 1-2 ~u~ 1.

I 1 ~Xi I-lj

190 EDWARDS, ODEN, AND DEMKOWICZ

As in the case for uniform grids [4], the Lax-Wendroff scheme can be rearranged asthe first-order upwind scheme together with an antidiffusive term, i.e.,

(3.10)

Following the usual TVD analysis [4] an antidiffusive term can be added in the con-servation form

(3.11)

where ¢i+! is the flux limiter. The resulting scheme is

(3.12)

Rearranging (3.12) into the form

(3.13)

where

(3.15)

and comparing with (2.8), it follows that the scheme will be TVD for

(3.16)

Taking ¢ to be positive the left-hand inequality of (3.16) is satisfied for

(3.17)¢i+! 2----< )'rt - (1 - Vj_!

The right-hand inequality is satisfied for

(3.18a)

where

(3.18b)

2 (1 )< - -1 ,¢i-~ - (1 -Vi_!) Vi

Gi_!AtVi = AXj

The bound on ¢ for a uniform grid

(3.19) ¢ $ min(2rt, 2)


is certainly obtained if

(3.20) 1max(vi) - 1~ I,

and 0 ~ vi_1 ~ 1, and (3.20) is satisfied if~

(3.21) ai_!!:it ~ !:ixi2 '

which corresponds to a maximum Courant-Friedrichs-Lewy (CFL) condition of ~,but with respect to the mean cell length. A similar analysis can be performed for anegative wave such that ¢ ~ min(2rH-!' 2), where ri+1 is defined below.

The general high-order TVD scheme for a nonuniform grid which is used here canbe written as

where hi+! is the first-order flux (3.8) and

(3.22b)f3~+1 = (1 + V~+1)a-:-+l/2.

1 2 1 2 1 2

Vi+! is defined in (3.15) together with rt. When the wave has a negative sign,

and ¢(r) can be any of the usual flux limiters ranging from minmod to superbee [41.4. Accuracy on a nonuniform grid. In Appendix A we show that while the

low- and high-order schemes of (3.6)-(3.8) have a local truncation error of 0(1) on anonuniform grid, the error in the solution converges with O(!:ix) and they are thereforesupraconvergent schemes [22].

While this result holds for an arbitrary fixed nonuniform grid, we stress thatthe grids used in this paper are obtained via an equidistribution principle. The gridcontinually adapts with the solution such that the finest grid zones overlay the regionsof the flowfield with steep flowgradients. Away from the interfaces between grids withdiffering levels of refinement the grid is uniform and, therefore, the local truncationerror of the scheme is restored to the uniform grid value, typically 0(!:ix2).

Therefore, although formally, only first-order cOIlvergence can be demonstrated,the adaptivity can be expected to provide a better convergence rate than that for afixed arbitrary nonuniform grid.


(5.1)

5. Roe scheme for systems of conservation laws. The extension of thescalar algorithms describcd above to systems of equations is performed by applyinga scalar algorithm to cach characteristic equation obtained by decomposition. Themain difficulty with this approach is ensuring that conservation is obtained uponrecomposition to the conservative variables.

In Roc's scheme [1] for solving the system of conservation laws

au + af(u) = 0at ax

with initial data

u(x,O) = uo(x),

a lincarized equation is used to approximate the solution to the Riemann problem.A constant mean-value Jacobian A(UL' UR) is cOIlStruCted such that the followingthree properties are satisfied:

(i) A(UL,UR) is consistent. as UL -+ UR A(UL, UR) -+ A(u), where A(u) = fu;(ii) For any UL, UR, A(UL. UR) (UR - UL) = f(UR) - f(uL);

(iii) The eigenvectors of A are linearly indcpendent.Condition (ii) ensures that the resulting scheme is conservative; (ii) and (iii)

cnsure that in the case of a single shock wave, the solution to the Riemann problemwill be exact [1]. In order to apply the scalar scheme, the system is dccomposed byexpanding ~u as

(5.2)

(5.3)

(5.4b)

and the flux difference ~f as

~f = A~u = L >'jejO'j,j

where ~u and OJ are the changes in the conservative and characteristic variables,respcctively, and )..i and ej are the cigenV'cUuesand eigenvectors of A, respectively.

Condition (ii) enables the first-order scheme to bc written as

(5.4a) u?+1 = ui - µ(hi+! - hi_!),whcre now

(fi+fi+l) 1~hi+! = 2 - 2 L.,ejl>'jIO'j·1

This scheme is immediately applicable to nonuniform grids provided that the gridratio µ dcfined in (3.22) is used in (5.4). The high-order version of this scheme canbe derived as in the scalar case (3.22), by the addition of an antidiffusive flux to thefirst-order flux hi+!' resulting in

wherc a superfix j rcpresents the jth charactcristic component; each component hasits OWIl wave speed )..j and gradient ai, and these replace the scalar wave spced a andthc gradient ~u, respectively. in (3.15) and (3.22).

AN h-T-ADAPTIVE RIEMANN SOLVER FOR EULER EQUATIONS 193

6. Two-dimensional scheme. Consider a system of conservation laws in twodimensions written in integral conservation form

(6.1)

with initial data

u(x, y, 0) = tto(X, y).

In order to solve (6.1), the scheme of §5 is extended into two dimensions for an arbi-trary, quadrilateral grid. Assuming that the conservation variables u have a piecewiseconstant variation over each element, an application of the Gauss flux theorem to (6.1)over a given element results in

(6.2)

where

e~ = dx~ + dy~,

(Ck, Sk) are tbe direction cosines of the kth outward normal, and Te is the element (e)area. By further assuming a one-dimensional variation in the variables u across eachelement face k, the resolved flux Fk can be regarded as being locally one-dimensional.Thus the scheme of §5 can be applied with the Roe flux (corresponding to F) substi-tuted in the above summation (6.2) for each face of the element. The local Jacobianmatrix is given by

(6.3a)aF at ag

A(u) = au = auc+ aus,

and the corresponding eigenvectors and eigenvalues of this matrix (which appear inthe Roe flux) must satisfy

(6.3b) t:.F = I:>'jejQj.j

Finally this scheme can be written in the classical finite-volume form as

where

(6.4b)


and the summation index i varies over the sides of the element e, (AXi, AYi) beingthe vector element of the length tangential to side i and ei being the ith neighboringelement number.

The high-order version of this scheme can also be written in the form (6.4a).The antidiffusive flux of (5.5) is generalized into two dimensions by associating anantidiffusive flux contribution with each face of the element. The local eigenvaluesand eigenvectors of (6.3) are used in the construction of the antidiffusive flux at eachface. The flux limiters corresponding to the even- and odd-numbered faces of theelement (Fig. 2) are evaluated with respect to the underlying x and Y directions ofthe grid, respectively. Each antidiffusive term is then multiplied by its correspondingelement face length and the sign of the local outward normal vector before beingassembled on the right-hand side of (6.4).

4

3

I

L

2

FIG. 2. Local coordinate system.

For a regular grid, the resulting scheme is second-order accurate with respectto the x - Y coordinate directions. Formal second-order accuracy, including cross-derivative terms, can be achieved via the time-splitting technique of Strang [23].

The Roe matrix A, together with the eigenvectors, eigenvalues, and characteristicincrements OJ, are presented in Appendix B for the Euler equations of compressibleflow in two dimensions.

7. Adaptive refinement and unrefinement, In this section we shall brieflydescribe some of the principle features pertaining to h-refinements which are con-tained within the general data structure developed by Demkowicz and Oden [2] forquadrilateral grids. A description of the logic and account of the structure is givenby Oemkowicz in [24].

Following the nomenclature of Demkowicz and Oden [2], when a refinement of aquadrilateral element takes place the element is divided into four subelcments, whicharc called "sons." The group of four sons is called It "family" and the original elementis called the "father" element (Fig. 3).

AN h-r-ADAPTIVE RIEMANN SOLVER FOR EULER EQUATIONS

4 I 3

1 I 2

Father Four sons

FIG. 3. Initial refinement.

195

A typical refinement, shown in Fig. 4, gives rise to nodes such as p which arecalled hanging nodes or irregular nodes. Whereas each regular node is located atthe corner in common with all of its ncighboring clcments, the irregular node is notlocated at a corner for all of its neighboring elements. In this data structure only oneirregular node per element side is permitted.

d cq

a b

p

FIG. 4. Refinement with ilTegular node.~p, q.

Due to this constraint (one irregular node per side) any element can have a maxi-mum of eight sides and eight neighbors. Following Devloo [25]the sides of the elementare labeled one to eight in the sequence shown in Fig. 5.

7 3

4

f-8

+ 5

FIG. 5. Element side numbering.


This numbering sequence gives rise to some attractive logical relations which areexploited within the data structure [24], [25] and within the extension of the abovetwo-dimensional scheme to an unstructured grid.

Returning to Fig. 4, if one of the sons (b) of the family is refined, the constraint(one irregular node per element side) causes the neighboring element to be refined sothat some spreading of the refinement process can occur. This is illustrated in Fig. 6.

die

a b ~

FIG. 6. Spreading of refinement.

Adaptive strategy. As in [9], a crude computational grid is generated by dividingthe flow domain into a suitably small number of quadrilaterals. The initial crudegrid is then globally refined to create a reasonable coarse grid (e.g., Fig. 12), which isused for the first part of the computation. After a specified numbcr of timcsteps (oriterations) the adaptive strategy (described bclow) is invoked.

The adaptivc criteria for deciding when to refine an elemcnt and ullfcfillC a familyis based 011 monitoring the density gradient as in [9] and [10]. The density is chosenas the key variable because it is discontinuous at shocks and contact discontinuitics.

The actual adaptive strategy is listed below. The condition on the gradicnt ofthe sons of the family in (iii)(a) makes the procedure much more robust.

(i) Define the refinement and unrefiuement tolerances (user specificd constants)Er, Eu, rcspectively.

(ii) Find the maximum Loo density gradient throughout the field,

maxI<NEL<NRELEM

(iii) Looping over the families (NF = I,NRFArvl), find the density gradient mod-ulus 1\7pie, of each son ej(i = L 4) of the family (number NF)' Definc themean family density gradient by

4

1\7plNF = t L 1\7Pie •.i=l

If for each son (ei) of the family (NF),

(a) l\7ple; <Er lI\7plloo and

(b) l\7pINF <Eu lI\7pllocll

then unrefine the family and define the new element vector of conservationvariables by


4 / 4(7.1) UNF = t;AeiUe• t;Ae"

whcre uei are the conservative variables of the ith son of the family N F andA"i is the area of the ith son (element).

(iv) Looping over the elements (NEL = 1,NRELEM), find the density gradientmodulus l'Vp!NEL of each element NEL. If

refine the element NEL to produce four sons ei(i = 1,4). The conservationvariables for each of the four sons is defined by the value of the father, Le.,

(7.2)

The use of (7.1) and (7.2) ensure that conservation is maintained after asequence of ullfefinements and refinements.

Thc use of (7.2) is the most diffuse option. A bilinear intcrpolation would bemore accurate, but thcn conscrvation would be morc difficult to enforcc.

8. Extension of the Roe scheme to an unstructured grid. The low-orderscheme describcd in §6 is relatively easy to extend to an unstructured grid of thetype discussed above. For an element-wise implementation of the scheme, instead oflooping over sides one to four, as in the case of a structured grid, it is convenient toloop over sides one to eight, and test for the existence of a neighbor for any side N,say, between fivc and eight (5 S N S 8). If there is no neighbor, go to the end of theloop, otherwise the lengths ilx, ily, f. defined in §6 are halved before performing theflux calculations of §6 for sides Nand N - 4.

For side N - 4 the flux calculation will involve quantities at i and i2 while forside N the flux calculation will involve quantities at i and i6 (see Fig. 7).

N • i&

.i

N-4 •i2

FIG. 7. Flux calculation on unstructured grids.

The extcnsion of the higher-order scheme is far more complicated. The support ofthe scheme in the case of a structured grid is sketched in Fig. 8. Unlike the low-orderscheme, which involves five elements, the higher-order scheme involves nine elements,the additional four elements being "neighbors of the neighbors" of the element inquestion.

Thc higher-order upwinding can be conveniently obtained 011 a structured gridwhere global curvilinear ~ and 1] "directions" may be identified and the flux limitingperformed along these Iidirections."


x

x

FIG. 8. Support of two-dimensional first-order scheme 0 and second-order scheme x.

(8.1)

Turning now to the case of an "unstructured grid" of the type discussed above,it is easy to see that while thc grid is unstructured in the sense that certain gridlines tcrminate in the field, the grid still rctains a certain structure in thc sense thatglobal ~ and 1J "directions" can still be identified throughout the grid. This propertyis completely exploited by constructing directional flux limiters corresponding to theeven-numbered sidcs (~ direction) and odd-numbered sides (1J direction) of the ele-ment, respectively. (This extension of the flux-limited scheme would not be applicableto an unstructured triangular grid such as that generated in [10].)

For illustration we shall consider the construction of the flux limiters in the ~direction (corresponding to a positive wave) which contribute to the solution at thecentral element (i). Various grid configurations that may occur in the "~ direction"which affect the positive wave limiters are skctched in Fig. 9.

We introduce a notation (consistent with Fig. 6) that relates the standard fluxlimiters (§3) in the ~ direction to the even-numbered sides of the element, and thatdenotes the flux limiters corresponding to sides 2, 4, 6, and 8 by, ¢2, tP4., tP6, and tPs,respectively. For a regular (structured) grid (Fig. 9(a)), the solution at element i willinvolve the usual flux limiters tP2 = tP(rt) and tP4 = tP(rt_l)' where rt is defined in§4 as

+ Vi_!(l-vi_~)(ui-lLi-dr· =, vi+!(l- Vi+!)(Ui+l - Ui)'

We shall adopt the more general notation

(8.2)

In the second configuratioll (Fig. 9(b)), rP2. tP4, and tP6 now make contrihutions to thesolution at element i, where now

(8.3)

tP2 = tP(r(ui-l,Ui,Ui2))'

tP4 = tP(r(ui-2,'U;-1,Ui)),

tP6 = tP(r(ui-l, Iti, Ui6))'

where h and i6 are the right-hand element numbers (Fig. 9(b)).


FIG. 9. Adaptive support of scheme ({ direction: v > 0).

In the third configuration (Fig. 9(c)), ¢2, ¢4, and ¢8 make contributions to elementi. In this case ¢2 is a function of both r( Ui4' Uj, Uj+d and r( Uis ' Ui, Ui+l)· As a generalrule, whenever the flux limiter is a function of two ratios (as in this case), we shalldefine the limiter to be a minimum of the two possibilities. Therefore,

(8.4)

¢2 = min (¢(r(ui4, Uj, ui+d). ¢(r(uis, Ui, ui+d)) ,

¢4 = ¢(r(ui4_1, Ui4' Ui)),

¢8 = ¢(r(uiS_IlUiS,Ui»'

In the fourth configuration (Fig. 9(d)), all of ¢2, ¢4, ¢6, and ¢8 make a contribu-tion to element i. As in the third case, ¢2 is chosen as

Similarly, since ¢6 is now a function of two rutios, then


(OEfIB · I · I I

(gJ[RE · I · I I

(hJ tBjj ·rn I

(i) EEIB i EE Im[RE · rn I

FIG. 9 (continued).

while the limiters rP4 and rPsare evaluated as in the third case.In the configurations in Figs. 9(e)-(g) the evaluation of rP2 is identical to the third

case described above, while for the configurations in Figs. 9(h)-(j) the evaluations ofrP2 and rP6 are identical to the fourth case described above.

In configurations (e)-(j), rP4 and/or rPsarc functions of two ratios and we continueto apply the above rule, for example. in Fig. 9(i),

rP4 = min(rP(r(uiQ,ui.,ui)). rP(r(uib,ui.,Ui))),

rPs= min(rP(r(uiclui8' Ui)), rP(r(uid' Ui8' Ui))).

A similar analysis is performed for a negative wave with each configuration (e)-(j) inverted with extra refinements now appearing in the right-hand element of thesupport. Finally, the same analysis is performed for both positive and negative wavesin the 1} direction, where the limiters 011 the upper side of the element are rP3 and rP7.while the limiters 011 the lower side of the element are rPl and rP5'


9. Grid movement. The grid movement considered in this work is strictly forsteady-state calculations and is induced via an equidistribution scheme constructedusing

(9.1)

e

where Le are the position vectors of the centers of gravity of the elements surroundingnode i (with position vector !:J, and the summation is performed over these elements.Similar approaches have been used in [9] and [261.

As noted in [26], the scheme (9.1) can be identified as an equidistribution tech-nique in the limit as the iterative cycle converges. Consider the x component of (9.1)written as

(9.2)

where We = IV'pie' In one dimension the centers of the elements which surround nodel are

(9.3)

and

(9.4)

Equations (9.2), (9.3). and (9.4) give

(9.5)

and as X~+l - xn -+ 0 we obtaint t ,

(9.6) ~X+IW'+l = ~X'_lW'_l = constant.12'2 12'2

After some experimentation it was found that a reasonably robust equidistributionscheme can be defined as

(9.7)

2: r~((JV' pJe/lIV'plloo)Aeo) + 13rn+l = e

L:((IV' pie/IIV'plloo)Aeo) + 13e

where 0 is a user-defined constant which determines the strength of the grid move-ment, and 13( = 10-4) is a regularization constant which prevents any singularity fromoccurring (in (9.7)). Ae is the Jacobian of the eth element.

10. Results. The adaptive schemes described above are used to solve the Eulerequations of comprcssible flow, written in conservation form

(10.1) Ut + j x + gy = 0,


where

u = (p, pu, pv, E)T.

(10.2) f= (pu, pu2+p, puv, U(E+P))T, ig= (pv, puv, pv2+p,v(E+P))T

,Here p, p, and E are the density, pressure, and energy per unit volumc for an idealgas, (u, v) are the cartesian components of velocity, and

"y being the ratio of specific heat capacities and q2 = u2 + v2.

Three kinds of physical boundary conditions are imposed, namely, (i) supersonicinflow, (ii) supersonic outflow, and (iii) solid wall.

Since the scheme is cell-centered, a band of dummy nodes is required at all bound-aries to complete the definition of the discretization of the scheme.

In condition (i) all conscrvation variables arc spccified at thc dummy nodcs withtheir free-stream values

U dummy = U free-stream'

In condition (ii) the dummy values are found by assuming that the normal gradi-ents of the flow variables are zero at the outflow boundary, so that ay) an = 0, whichresults in

U dummy = U interior'

In condition (iii) at a solid wall, the normal velocity is set equal to zero whilcreflection conditions are used for the density p, energy E, and tangential velocity v'with

ap = aE = av' = 0an an an 'where the tangential velocity v'is defined by v' = q . t and q is the velocity vector.A second layer of dummy values are introduced (;ia reflection) at a solid wall tocomplete the support of the second-order scheme.

We present the results obtained for three test cases. In all cases the computeddensity contours will be shown for 30 uniform intervals both for uniform and adaptivegrid computations, respectively.

For the first two cases involving steady-state flow, three kinds of adaptive strategyare tested: (i) grid movement using equidistribution; (ii) grid refinementjunrefinement;and (iii) grid refinement and movement.

10.1. Supersonic flow over a wedge. The wedge has an inclination of 200 tothe horizontal. Initially the flow is assumed to be uniform throughout the field witha free-stream Mach number Moo = 3.0. An exact solution for this problem can beobtained [27], which consists of a uniform shock wave extending from the compressioncorner into thc flow field at an angle of 37.5°.

The result obtained using a uniform 32 x 16 grid Fig. lO(a) is shown in Fig.lO(b). The exact solution lies within the band of density contours.


--

(a)

FIG. 10. Wedge-regular grid.

203

Using the uniform grid solution as an initial data for the equidistribution scheme(9.7), the solution and grid shown in Fig. 11 are obtained by applying (9.7) for threeiterations, each iteration being performed after 300 steps, wit.h Q = 4.3. A consid-erable improvement in shock resolution is obtained which may be due not only tothe local reduction in the size of thc clcment in the shock region, but also to theimprovement in the local orientation of the grid relative to the shock.

Ncxt we consider the h-refinement strategy of §7. An initial solution is obtainedon a 16 x 8 grid (Fig. 12). The solution (and grid) obtained after two applications ofthe h-refinement strategy of §7 is shown in Fig. 13, and after a third application, inFig. 14. Each application was made after 200 steps. The refinement and unrefincmenttolerances are er = 0.16 and eu = 0.16, respectively. The final solution is well alignedwith the exact solution, and very good resolution of the shock has been obtained.

Finally, the combination refinement and movement (h - 1') is tested using thesolution of Fig. 13 as an initial data; the result shown in Fig. 15 is obtained afterthree movements, one every 300 steps, with Q = 5.

10.2, Blunt body. The second steady-state problem that we consider is a bluntbody placed in a supersonic flow field with free-stream Mach number Moo = 6.57 andI = 1.38 at 00 angle of attack. The larger (extended body) version of this problemhas been studied by Oden, Strouboulis, and Dcvloo [91 and Bey et al. [281.

An initial skeleton grid of 5 x 5 elemcnts is generated and globally refincd twiceto produce a 20 x 20 uniform grid, which is smoothed by using (9.7) with the densitygradient removed.


\.-.I-

.;--vk;< ..-~

trl -I-t:.l--~Yfy

FIG. 11. Wedge-regular grid with node relocation.

FIG. 12. Wedge-initial coone grid.


FIG. 13. Wedge-with five grid levels of h-refinement.

FIG. 14. Wedge--with six grid levels of refinement.

205


FIG. 15. Wedge-h-r method.

The solution obtained on the uniform 20 x 20 smoothed grid is shown in Fig. 16.The exact shock location of this problem is found following Billig 1291 and lies at thecenter of the computed shock location.

As in the previous case, the uniform grid solution (Fig. 16) is used as an initialdata for the equidistribution scheme (9.7). After using (9.7) for two iterations, oneevery 200 steps, the result in Fig. 17 is obtained. The effect of grid distortion uponthe solution error is not analyzed here. However, in both of the steady-flow problems,(9.7) proves to be an extremely effective simple tool for enhancing the initial uniformgrid solutions.

Next, the h-refinement strategy of §7 is applied. The refinement algorithm isapplied (with er = 0.007, eu = 0.44) using the solution in Fig. 16 as an initial data,then applied again after 100 steps and 500steps. The solution obtained at convergenceis shown in Fig. 18 together with the final grid. The shock resolution is much improvedand is aligned with the exact solution.

Finally, the combination of h-refinement and movement is tested with six itera-tions of grid movement, one every 200 steps, with Q = 5.0. The resulting solutionand grid are shown in Fig. 19. The residual IIpn+1 - pn 1100 is reduced to 0 (10-5)

by 2000 timesteps. All of the above results were computed with the non-time-splitscheme using the van Leer flux limiter 116], 14] with

(10.3) ¢(r) = (r + Irl)/(1 + Irl).

The superbee limiter 14] wa...c;also tried but gave some convergence problems for thesecases.


FIG. 16. Blunt body, regular grid.

FIG. 17. Blunt body, regular grid after node relocation.

10.3. A Mach-3 wind tunnel with a step. This problem, used by Woodwardand Colella [81 for comparing a variety of methods, is used here for a test of thetransient behavior of the method and solution. The problem begins with a uniformMach-3 flow throughout the field impinging on a step [8]. The sequence of results


FIG. 18. Blunt body, h-refinement.

FIG. 19. Blunt body, h-r method.

obtained using the time-split scheme, together with the resulting adaptive grids, isshown in Figs. 20-24 for output times t = 0.5, 1,2,3,4. These results were obtained bystarting on a uniform (Ax = Ay = 1{20) grid and applying the refinement algorithmin the sequence given in §7 after every 25 timesteps.


FIG. 20. t = 0.5, fully second-order time-split scheme.

209


l

The condition (iii)(a) of §7 was particularly important in ensuring consistcntlygood results. The tolcrances used are Cr = 0.061 and eu = 0.44. The superbce fluxlimiter was used in all three formulations.

Excellcnt agreement is ohtained between thc adaptive- and fixed-grid solutions,e.g., at time t = 2 compare Fig. 22 with Fig. 25(a). In the final stage of the computa-tion from t = 3 to t = 4 the fixed-grid method is unable to sustain the shock reflection




on the uppcr wall of the tunnel (Fig. 25(b)), in striking contrast to the adaptive gridresult of Fig. 24.

At transition cclls (e.g., Fig. 7) the approximation (6.4a) induces truncation errorsof ±O(l) in the two ncighboring cells i2 and i6, respectively. However, since disconti-nuities nevcr cross transition zones (due to adaptivity) this potential source of erroris avoided. Also this error cancels in the global sum with respect to conservation.


FIC. 24. t = 4.0, fully second-order time-split scheme.

FIC. 25. Uniform grid computation (time split). (a) Time t = 2.0; (b) Time t = 4.0.


The current version of the computer code has not been optimized in any wayand has been written mainly from the point of view of convenience. However, largesavings in computer time are obtained in all cases. A typical comparison between anh-adaptive computation and a uniform fine grid (with the same level of refinement,using the same computer code) shows the adaptive method to be about four timesfaster. This factor tends to increase with grid level, thus, the gains obtained dependconsiderably upon the adaptive strategies employed, particularly for the steady-statecases.

Timestepping. A uniform timestep (fixed D.t throughout the grid) has been usedin all of the work presented here. A point for further investigation would be tocombine this method with the adaptive timestepping procedure of Berger [30], wherea variable-size timestep is used according to the local level of grid refinement.

There is a striking contrast between the adaptive method of Berger and Oliger[11] and that employed here. Their method for refining the grid involves overlayingsequences of finer grids on the coarse-grid areas which contain steep flow gradients.This method inevitably involves using more elements than are required. Conversely,in the adaptive method used here, only the actual elements which detect large flowgradients are flagged for refinement, although in practice some transition elements arecreated (see §7). While the former approach is logically more simple than the latter,it would appear that it has a greater need for adaptive timestepping.

In Table 1, we give the grid level and number of elements for the adaptive gridsshown in Figs. 14, 18, and 24. An inspection of Table 1, together with the correspond-ing grids and results, demonstrates that near optimal use has been made of the gridrefinements and unrefinements, with the majority of elements packed into the regionsof high flow gradients in each case.

TABLE 1Grid levels and element c01mts.

Gridlevel

Number of ]elements Total

Wedge 4x2 1Ca.sel 8x4 10

16 x 8 3532 x 16 7164 x 32 161128 x 64 588 866

Blunt body 5x5 4Case 2 10 x 10 37

20 x 20 11940 x 40 276 436

Step 15 x 5 14Case 3 30 x 10 66

60x20 265120 x 40 836 1181

From the table it is possible to deduce the effective advantage of adaptive timestep-ping. Assuming that the timestep is halved for each fine-grid level created, then forevery eight minimum timesteps, Case 3 of Table 1 reveals a potential saving of 16percent of the run time of the adaptive code. This assumes no extra overhead inperforming the adaptive timestepping procedure. For the nine-point schemes usedhere the support can be across four interfaces simultaneously in one direction (§8),which complicates the logic of adaptive timestepping in two dimensions.

For steady-state problems, nonconservative local timestepping could be used.


However, Table 1 suggests that only a comparatively small reduction in computertime would be obtained in the steady-state cases 1 and 2.

11, Conclusions. A two-dimensional version of the higher-order approximateRiemann solver of Roe [14\ is presented in finite-volume form for application to un-structured quadrilatcral grids.

A one-dimensional analysis of the solution error indicates that convergence isformally O(~x) for an arbitrary nonuniform grid .

However, the quality of thc results and the very fine resolutioll obtained by thehigh-order scheme on an adaptive grid is comparable with that obtained by the samescheme on a fixed uniform grid, with a cell size corresponding to that of the finestadaptive grid level.

Great savings in computer time are obtained; the adaptive code is a factor of 4times faster than the same code run on the corresponding fixed uniform grid.

Appendix A. Question of accuracy on a nonuniform grid. In this ap-pendix we will show that for a sufficiently smooth solution, when the cell-centeredlow-order upwind scheme is applied to the linear advcction equation on a nonuniformgrid, the error in the solution is of O(h) while the local truncation error is 0(1). Whenthe error in the solution is bettcr bchavcd than the local truncation error thc schcmeis supraconvcrgent [22].

The low-order upwind scheme for linear advection (3.6), (3.8) can be written as

(A ) n+1 n a~t ( n n) lal~t (n 2 n n).1 uj = uj - 2~Xi ui+l - ui-l + 2~Xi ui+l - Uj + Uj_l .

Substituting thc exact solution u(x, t) into (A.l) and performing Taylor series expan-sions about U(Xi, tn), the leading truncation error is found to be

(~Xi+! + ~Xi_! - 2~xj) (~Xi+! - ~Xi_!)(A.2) T = aux; A - laluXi A + O(~t, ~x).

2~Xj 2~Xi

where

~Xj+! = ~ (~Xi + ~xi+d

and ~Xi is the ith element length.The local truncation (A.2) is 0(1), which suggests that the numerical solution will

not converge to the physically correct solution. However, by performing an analysissimilar to Kreiss et al. [22] we can show that the error in the solution is of O(~x).

First, we obtain the discrete error equation by subtracting the grid differenceequation (A.1) from thc truncation error which results in

(A.3)

Multiplying (A.3) by ~Xi and summing over i = 1 to j gives

(A.4)

- ~(en + en) + ~(en - en) = ~ T~X"2 1 0 2 1 0 ~. .,i=1

214

using (A.2) gives

EDWARDS, ODEN, AND DEMKOWICZ

i iLTiAX; = 4 L(aA_(AXi+1 - Ax;) -lal(Axi+! - Ax;_!»)uZj'

i=1 ;=1

Performing a summation by parts results in

i i(A.5) LT;AXi = -~ L((AXi+l - AXi)a - AXi+! lal)(UXi+1 - Ux.).

;=1 ;=1

For U sufficiently smooth,

hence (A.5) is of O(Ax).Using the initial data at time t = 0 (n = 0), u? = u(x;, 0) or

(A.6) °-0e; - .

From (A.4), (A.5), and (A.6) it follows that

(A.7)i 1 ~Xi

" e· - = O(Ax).~ I At;=1

Since (A.7) is true for any j, then

(A.B)

(A. 10)

and the CFL condition (3.21) ensures that

e~ = O(Ax).

By induction it follows from (A.4) that

(A.9) ef+1 = O(Ax) Vi, n,

hence the solution converges with O(~x).Observations. The same analysis caD be applied to the higher-order scheme to

show first-order convergence. For example, the Lax-Wendroff scheme used here (3.6),(3.7) has a leading truncation error of

(Axi+! + AXi_1 - 2~x;)2 aUZI + O(Ax, At),

which is due to approximating the physical flux component of (3.7) by

(A.l1)

Second-order convergence can be recovered in two ways: (i) Replace (A.ll) bythe second-order approximation

(A.12)AXi+lJ; + Axd;+l

(AXi+l + Axd

AN h-T-ADAPTIVE RIEMANN SOLVER FOR EULER EQUATIONS 215

which would reduce the truncation error to O(6.x). However, the introduction of(A.12) into (3.7) results in a scheme which cannot be shown to be TVD, and theexact shock resolution property of the Roe scheme is lost on an arbitrary nonuniformgrid.

(ii) Remove the 0(1) truncation error in (A.IO) by defining 6.xi to be the meancell length

This definition effectively removes the distinction between cell-centered and node-based (cell vertex) schemes. However, it is not practical to carryover into two di-mensions for the grids used here. (Note that the scheme (3.6), (3.7) would, therefore,achieve second-order convergence if it had been applied using cell vcrtices as opposedto cell centers in one dimension.)

Appendix B. The Roe decomposition for the Euler equations in twodimensions. In order to satisfy (6.3b). wc follow a similar procedure to that pre-sented by Roe and Pike [31] for the one-dimensional ca.r;e and by Baines [321 in twodimensions. For the Euler equations (§1O) the matrix A(u) defined in (6.3a) takesthe form

(B.2)

el.2 = [1, u ± ac, v ± as, H ± Ua]T,

e3 = [0, as, -ac, -aV]T,

e4 = [1, u, v, q2f2]T,

where H is the enthalpy (H = q2f2+"Ipf p(-y -1)), a is the sound speed (a = V"Ip/ p),

(B.3)

with corresponding eigenvalues

(B.4)

U == uc+vs,

V == -us +vc,

..\1.2 = U ± a,

..\3 =U,

..\.1 =U.


Following [311and [32], OJ (increments of the characteristic variables with respect tothe primitive variables (P, u, v, p)) are found to be

(B.5)

O} = [Ap+ap(cAu+sAv)1I2a2,

02 = [Ap - ap(cAu + sAv)]/2a2,

03 = p(sAu - cAv)ja,

04 = Ap - /::ip/a2.

Equation (6.3b) is satisfied provided that u, v, h, p, and a appearing in (B.1)-(B.5)are defined by

(B.6)

URJjiii + UL.J7iiu= ,Jjiii + .J7ii

vR.[iiR + vL ..(Piv= ,

.[iiR + ..(Pi

H = HRffn + HL.Jjii,ffn +.J7ii

P = ..jjiR Viii,a = J(H - q2/2)("f -1),

respectively, and the difference operator is defined such that

(B.7) Ap= PR - PL·

Acknowledgments. Thanks are given to Professor Earl Thornton for bringing[29] to our attention, Olivier Hardy of the TIC OM Computing Laboratory who as-sisted with some computing difficulties, and to Jan Shrode for her dedication (andpatience) in the typing of this report. Thanks are also due to the referees for theirhelpful comments and criticisms and the rcfercnces [22], [3D].

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[3] A. HARTEN, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys .. 49(1983), pp. 357-393.

[4] P. SWEBY, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAMJ. Numer. Anal., 21 (1984), pp. 995-1011.

[51 A. HARTEN AND J. M. HYMAN, Self adjusting grid methods for one-dimensional hyperbolicconservation laws, J. Compllt. Phys., 50 (1983), pp. 235-269.

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[7] H. C. YEE, Upwind and symmetric shock capturing schemes. NASA Technical Memorandum,89464, NASA Ames Research Center, Moffet Field, CA, 1987.

18] P. WOODWARD AND P. COLELLA, Review article-The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys, 54 (1984), pp. 115-173.


[9] J. T. ODEN, T. STROUBOULIS,AND P. DEVLOO, Adaptive finite element methods for theanalysis of inviscid compressible flow: Part I-Fast refinement, unrefinement and movingmesh methods for unstructured meshes, Comput. Methods Appl. Mech. Engrg .. 59 (1986),pp. 327-362.

(10) It. LOHNER. K. MORGAN, J. PERAIHR, O. C. ZIENKIEWICZ,AND L. KONG, Finite elementmethods for compressible flows, in Numerical methods for fluid dynamics II. K. W. Mortonand M. J. Baines, eds., Proc. Con£. Numerical Methods Fluid Dynamics, Academic Press,NY, 1985.

[H] M. J. BERGER AND J. OLIGER, Adaptive mesh refinement for hyperbolic partial differentialequations, J. Comput. Phys., 53 (1984), pp. 484-512.

(12] K. MILLER AND R. N. MILLER, Moving finite elements part 1, SIAM J. Numer. Anal., 18(1981), pp. 1019-1031.

(13] J. N. BRACKBII.LANDJ. S. SALTZMAN.Adaptive zoning for singular problems in two dimen-sions, J. Comput. Phys .. 46 (1982). pp. 342-368.

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[21] J. PIKE, Grid adaptive algorithms for the solution of the Euler equations on irregular grids, J.Com put. Phys .. 71 (1987), pp. 194-223.

[22] H. O. KREISS. T. A. MANTEUFFEL. 13. SWARTZ. B. WENDROFF. AND A. B. WIIITF.. JR.,Supra-convergent schemes on irregular grids. Math. Comp., 47 (1986). pp. 537-554.

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