AIAA JOURNALVol. 33, No. 6, June 1995

Optimal Multistage Schemes for Euler Equationswith Residual Smoothing

Chang-Hsien Tai* and Jiann-Hwa Sheu1^Chung Cheng Institute of Technology, Tao-yuan, Taiwan 33509, Republic of China

andBram van Leer*

University of Michigan, Ann Arbor, Michigan 48109-2140

A numerical technique, composed of the Van Leer-Tai-Powell optimization and a modified procedure, is appliedto design multistage schemes that give optimal damping of high frequencies for given upwind-biased spatialdifferencing with implicit and explicit residual smoothing. The analysis is done for a scalar convection equationin one space dimension. The object of this technique is to make the schemes suited for multigrid acceleration.The optimal multistage schemes and their damping properties are presented in this paper. By keeping the multistagecoefficients from the one-dimensional analysis and simply redefining the Courant number, the schemes can beapplied to multidimensional problems. A fast Euler code is used to evaluate the suitability of the schemes formultidimensional multigrid computations. Numerical results show that the modified schemes effectively enhancethe convergence performance on both single and multiple grids.

I. Introduction

T HE computation of a steaty flow pattern induced by a specificfixed geometry has been done traditionally by implicit methods

in the past few decades. This is due to the relatively large reductionin residual that can be achieved in one iteration step. Once multi-grid relaxation was successfully implemented for solving hyperbolicequations, explicit methods became very effective and available formarching to steady-state solutions. Explicit methods have manyadvantages over implicit methods: they can be easily executed oncomputers with vector or parallel architectures, and they allow localgrid refinement. The last advantage is crucial, since adaptive gridrefinement seems to be the most promising way to efficiently obtainspatial accuracy in complex problems.

The explicit time-integration method proposed by Jamesonet. al.1 and Jameson,2 which is founded on separate time andspace discretizations, is a remarkable combination of componentsincluding convergence-acceleration techniques and multigrid relax-ation. Since the multigrid strategy acts to remove low-frequencycomponents of the error while marching, it suffices for the single-grid scheme used on each grid to damp only the high frequencies.Tai3 and Van Leer et al.4 have shown how to develop multistageintegration methods that yield optimal damping of high-frequencymodes. These schemes were derived for a scalar convection equa-tion in one space dimension, and used to solve one- and mul-tidimensional Euler equations. The analysis for optimizing thesmoothing properties of multistage scheme was extended to a two-dimensional scalar convection equation by Catalano and Deconick5

and, more satisfactorily, by Lynn and Van Leer.6 In the latterwork, the optimization method is extended to the system of Eulerequations, using the local preconditioning of Van Leer et al.7 tomake the system act more like a scalar equation. These devel-opments largely enhance the convergence performance of explicitmethods.

The convergence-acceleration techniques introduced byJameson2 are local time-stepping, enthalpy damping, and residualsmoothing. Among these techniques, implicit residual smoothing

(IRS) is particularly effective, since it increases the maximum al-lowable Courant number and smoothes the high-frequency com-ponent of the residual, as needed in multigrid relaxation. Jamesonsuggested some multistage schemes with good smoothing proper-ties and derived a formula to obtain new Courant number whileincreasing smoothing coefficients. Enander8 also tried to improvethe high-frequency damping properties of multistage schemes byusing the developed combined implicit-explicit residual smoother.More recently, Blazek et al.9 introduced a upwind-biased operatorfor the IRS method and showed this method allows substantiallyhigher Courant-Friedrichs-Lewy (CFL) numbers as compared tothe centered IRS method. In this paper, only a central differencingoperator for the implicit and explicit residual smoothing (ERS) isof interest.

In the present study we design optimally smoothing multistageschemes for given upwind-biased spatial discretizations10 usinga central-difference IRS and ERS operator for extra smoothingand stability. They are designed by means of the Van Leer-Tai-Powell optimization3'4 and a modified procedure. Since the spa-tial discretization dictates the final accuracy of the solution, themost natural way is to select the spatial discretization and thenselect the multistage coefficients in such a way that short wavesare effectively damped. This means these schemes are designedpurely for their properties of damping and stability, without re-gard to time accuracy. The search for the values of the parame-ters of the multistage schemes that yield optimal damping is donenumerically.

Based on the finite volume formulation,2'11 a high-efficiencymultigrid code for solving the Euler equations was programmedto simulate the axisymmetric external flowfield about a projec-tile. This code has been used for the actual multidimensionalapplication of the optimal multistage schemes. Besides resid-ual smoothing the code includes local time-stepping to speedup the rate of convergence and to make the smoothing analy-sis more appropriate. Convergence performance of these schemesand a comparison with the classical Runge-Kutta schemes arepresented.

Received Aug. 30, 1994; revision received Nov. 21, 1994; accepted forpublication Dec. 20,1994. Copyright © 1995 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

* Associate Professor, Department of Mechanical Engineering, Ta-shi.^ Ph.D. Candidate, Department of System Engineering, Ta-shi.* Professor, Department of Aerospace Engineering. Member AIAA.

II. Optimization of the Multistage SchemeA. Van Leer-Tai-Powell Optimization

Consider the linear ordinary differential equation

du—dt (1)

1008

TAI, SHEU, AND VAN LEER: SCHEMES FOR EULER EQUATIONS 1009

and use a predictor-corrector integration method to discretize thetemporal term as

u = un+a\Atun

un+l = un + XAtu = [1 + AAf + a(AA02K (2)

where the time-step ratio or is a free parameter. Apparently, thestability and damping properties are determined by the complexpolynomial:

P2(z, a) = 1 + z + cez2, z = h-At (3)

This polynomial has two complex-conjugate zeros which movealong a circle when varying a.

When a partial differential equation is interpreted by the methodof lines, A, represents the Fourier transform of the spatial differ-encing operator and depends on the spatial frequency f or, morespecifically, on the spatial wave number ft = 2n%Ax. For instance,consider the one-dimensional linear convection equation

du 8uc> 0 (4)

and discretize the space term by first-order upwind differencing.After inserting harmonic data u(x) = UQe~27ri^x, Eq. (4) reduces toan equation of the form (1)

At—=-v(l -at (5)

where the nondimension time step v = c A?/ AJC is the CFL number.All information about the spatial differencing operator is includedin the complex function

z(/8) = -v(l - «-") (6)

Similarly, the complex function for the kappa10 scheme becomes

+ (1 +*)(«" -1)]}. (7)

The parameter K of the high-order upwind-biased differencing regu-lates the upwindedness: K = I yields central differencing, K .= — Isecond-order accurate fully one-sided differencing, K = 1/3 third-order accurate upwind-biased differencing. If a certain frequency ftneeds to be perfectly damped, one inserts z(ft) into P(z) and de-termines the values of v and a that make the amplification factorvanish.

Applying a sequence of m predictor-corrector methods generatesschemes with an even number of stages (2m). To get a schemewith 2m H- 1 stages, a single application of the forward Euler timeintegration scheme

un+l = (1 + A.AOM" (8)

should be added to the string of m predictor-corrector methods. Theforward-Euler scheme has amplification factor

P1(z) = l+z (9)

This polynomial has one zero at z = -1. For first-order upwinddifferencing, the spatial frequency that can be perfectly damped isft = 7t, and the required CFL number is 1 /2.

A general £-stage scheme with final marching time step Af canbe written as

(10)

where yk is the £th coefficient of the £-stage scheme, and yt = 1.The scheme has the amplification factor

(H)

In the optimization procedure of Van Leer-Tai-Powell, a string ofpredictor-corrector methods, which damp different Fourier modesin the high-frequency range [n/2, n} and, possibly, a single forward-Euler step, are merged into the multistage scheme. Therefore, theamplification factor of the scheme can be written as

(12)

The coefficients and CFL number of the multistage scheme can befound by multiplying out and comparing Eqs. (11) and (12). But thiscombined multistage scheme may not be an optimal smoother. Theoptimization method of Tai3 and Van Leer et al.4 can now be im-plemented to minimize the maximum of the amplification factor inthe high-frequency range. A set of perfectly damped frequenciesis perturbed by means of Newton's iterative method, until all localmaxima of the amplification factor in the high-frequency range areequalized within a specified margin. The parameters of optimizedschemes for certain spatial discretizations, without use of residualsmoothing, are shown in Table 1. In the table, v is the CFL number, %the kth coefficient of the optimal multistage scheme, and |P|max theequalized minimal maximum of the amplification factor of £-stagescheme in the range [jr/2, TT]. The optimized schemes designed inthis paper are for the first-order upwind scheme and a range of kappaschemes (excluding central differencing K = 1). From the table itis observed that the higher is the number of stages, the greater isthe CFL number and the better is the damping of high-frequencyerrors.

The contours of the amplification factor of the five-stage schemeoptimized for the third-order spatial differencing, K = 1/3, togetherwith the locus of the Fourier transform z(ft) of the spatial operator(dashed line), are indicated in Fig. la. The contour levels for themagnitude of the amplification factor are | P \ — 1,0.9,0.8,0.7,0.6,0.5, 0.4, 0.3, 0.2, 0.1, 0.05, and 0.01. A graphic representations ofthe amplification factor as a function of ft is given in Fig. Ib forfive different multistage schemes, all optimized for third-order dif-ferencing without residual smoothing. Note that the CFL numberachieved in an £-stage scheme is considerably lower than the maxi-mum stable CFL number for that ^-stage scheme, which makes theoptimal multistage schemes robust marching schemes.

B. Modified ProcedureThe residual smoothing, using central differencing, is applied

implicitly in the following form

(l-s82)R = R (13)

where R is the residual^ R is the averaged residual, £ is the smooth-ing coefficient, and 82 is the discrete Laplace operator. Smoothingis applied in each stage. The Fourier transforms of the averagedresiduals corresponding to Eqs. (6) and (7) are

(14)

and

-1)1}+ 2s -

(15)

Using the optimization procedure of Sec. II. A, four-stage schemesfor the third-order spatial differencing, with various smoothing co-efficients, were obtained and are defined in Table 2. The dampingand stability properties of these schemes are indicated in Figs. 2aand 2b. Although the maximum of the amplification factor in thehigh-frequency range has been minimized for each scheme, the mag-nitude of the amplification factor in the low-frequency range lies

1010 TAI, SHEU, AND VAN LEER: SCHEMES FOR EULER EQUATIONS

Table 1 Optimally smoothing multistage schemes for given spatial differencing without residual smoothing

Stage V Xi X2 X3 X4 X5 X6 l^lmax

First-order upwind scheme

23456

1.01.52.02.53.0

0.33330.14810.08340.05330.0370

10.40000.20710.12630.0851

10.42670.23750.1521

10.44140.2562

10.4512

0.33330.14140.05940.0244

1 0.0100

0.91342.03362.32403.04373.5851

0.82771.32561.73162.16662.5981

0.70291.05591.39991.74842.0975

0.60540.89491.1.8841.48411.7805

0.52940.78081.03701.29511.5537

1.12830.35570.23800.14640.1038

0.66120:28830.16680.10670.0742

0.54090.24690.14000.08970.0623

0.48240.22090.12490.08000.0556

0.44750.20380.11510.07380.0512

Kappa scheme, K =

10.37390.28500.18760.1415

10.56290.32930.2303

Kappa scheme, K =

10.50090.30280.19780.1393

10.52760.32330.2198

0.48950.3227

10.52010.3301

Kappa scheme, K = 0

10.52100.29390.18660.1288

10.52520.31520.2106

0.52160.3251

Kappa scheme, K — —

10.51490.28090.17570.1201

10.51930.30570.2018

10.51740.3178

Kappa scheme, K =

10.50380.26940.16700.1135

10.51210.29710.1944

,0.51200.3109

Kappa scheme, K = —I

10.5129

10.5178

10.5185

10.5150

10.5106

0.80940.65270.43040.30340.2077

0.70160.46720.29520.18490.1153

0.66330.42100.25760.15590.0938

0.64290.40060.24110.14330.0848

0.62880.38840.23140.13610.0797

23456

0.46930.69360.92141.15071.3806

0.42430.19190.10840.06940.0482

10.49300.26010.16030.1085

10.50510.28980.1884

10.50670.3049

10.5062 1

0.61780.37990.22480.13130.0763

Table 2 Four-stage schemes modified by the Van Leer-Tai-Powelloptimization without the modified procedure, for the third-order

kappa scheme with IRS

Xi X2 X3 X4

00.10.20.30.40.50.6

1.73162.28582.81793.33313.83674.33254.8227

0.16680.17120.17550.17950.18300.18610.1888

0.30280.29260.28680.28300.28020.27810.2764

0.52760.51680.51340.5129 .0.51360.51480.5162

StableStableStableStable

UnstableUnstableUnstable

outside the stability region, especially for the smoothing coefficientsgreater than 0.3. It is seen that the instability becomes worse whenincreasing the smoothing coefficient in the IRS method. When us-ing the Van Leer-Tai-Powell optimization, the resulting multistageschemes for all given spatial discretizations show the same unstablebehavior. Clearly, the procedure has to be modified.

On the other hand, it is observed that the multistage coefficientsobtained by applying the Van Leer-Tai-Powell optimization do notchange significantly while the smoothing coefficients increase, evenfor s = 0.5 and s = 0.6, see Table 2. This observation moti-vated the following modified procedure: keep the coefficients fromthe Van Leer-Tai-Powell optimization without residual smoothing,then search for the optimal CFL number that minimizes the areaunder the high-frequency amplification-factor curve while satisfy-ing the stability condition for all low frequencies. After the modi-fied procedure, the CFL number for the first-order upwind schemeand the kappa schemes (K = 1/3, —1) with various smoothingcoefficients become those listed in Table 3. It is seen that theCFL number now increases with increasing smoothing coefficient.For instance, the three-stage scheme obtained by the Van Leer-Tai-Powell optimization for the third-order kappa scheme withoutresidual smoothing has a CFL number of 1.3256, as seen in Table 1.Applying the modified procedure, a scheme results with a CFL num-ber of 3.0029. Figure 3a shows the amplification factor for the twooptimal schemes and also for the smoothed scheme using the opti-mal CFL number of nonsmoothed scheme.

4.0

2.0 -

lm(Z)

.0 .-

-2.0 -

-4.0

TAI, SHEU, AND VAN LEER: SCHEMES FOR EULER EQUATIONS

4.0

1011

Optimal 5-stage scheme____CFL=2.1666 ""

-6.0 -4.0

Fig. 1 Optimal multistage schemes, for the third-order scheme withoutresidual smoothing: a) locus and contours of optimal five-stage schemeand b) amplification factor.

Residual smoothing can also be implemented explicitly (ERS),to suit parallel computing and unstructured grids. In this case theresidual R is replaced by the average

(16)

at each stage of multistage scheme. Using central differencing in theERS method, the complex function z(p) corresponding to Eqs. (6)and (7) become

- 2s (17)

and

+ (1 + K)(eift - !)]}[! - 2s + B(j* + e-* (18)

It is seen that for s > 1/4, there are Fourier modes such that Rcould be zero when R is not. Applying the modified procedure,the CFL number for the first-order upwind scheme and the kappaschemes (K = —1,1/3) with various smoothing coefficient wereobtained and are listed in Table 4. It is seen that the CFL numberincreases with increasing smoothing coefficient. For instance, thethree-stage scheme obtained by the Van Leer-Tai-Powell optimiza-tion for the third-order scheme without residual smoothing has aCFL number of 1.3256, as seen in Table 1. Applying the modi-fied procedure for s = 0.2, a scheme results with a CFL number of

2.0 -

lm(Z)

.0

-2.0

-4.0

Optimal 4-stage scheme [Table 2]without the modified procediwith IPS, e =0.6____CFL=4.8227

-6.0 -4.0

a)

-2.0Re(Z)

2.0

1.60

1.20

.40 -

.00

b)

Optimal 4-stage scheme [Table 2]without the modified procedurewith IPS

71/4 71/2 371/4

Fig. 2 Optimal four-stage scheme without the modified procedure, forthe third-order scheme with IRS: a) locus and contours and b) amplifi-cation factor.

2.3180. Figure 3b shows the amplification factor for the two optimalschemes and also for the smoothed scheme using the optimal CFLnumber of nonsmoothed scheme. From Tables 3 and 4, it is observedthat the CFL number for the ERS method is larger than one for theIRS method with the same smoothing coefficient. It is seen that thedamping and stability are greatly improved by the modified proce-dure. Thus, the procedure makes the schemes a good "smoother" asmultigrid strategy requires.

HI. Extension to Multidimensional ProblemApplying the multistage schemes optimized by one-dimensional

analysis to multidimensional or axisymmetric convection prob-lems is not so straightward. Fortunately, as shown by Tai,3 a two-dimensional minimax technique for a two-dimensional convectionleads to optimal multistage coefficients that are not significantlydifferent from those obtained by one-dimensional optimization, al-though the optimal CFL number may differ somewhat. Thus, thepractical strategy proposed in this paper is keeping the coefficientsfrom one-dimensional analysis and redefining a two-dimensionalCFL number.

It is obvious that the schemes should be optimal with regardto errors propagating in the convection direction; this only requiresextending the definition of Courant number to the case of convectionthrough a two-dimensional grid. Consider a two-dimensional cellABCD, as shown in Fig. 4; set the convection speed in the x andy directions equal to a and b. From a consideration of numerical

1012 TAI, SHEU, AND VAN LEER: SCHEMES FOR EULER EQUATIONS

1.00

.75

IPI

.50

.00

a)1.00

.75

IPI

.50

.25

.00

b)

Opt. 3-stage scheme:(with IRS)___"CFU1.3256, £=0without the modified procedure___ CFU1.3256, e=0.5with the modified procedure___ CFL=3.0029, 8=0.5

71/4 7C/2p 37C/4

Opt. 3-stage scheme:(with ERS)____ CFL=1.3256, e-0without the modified procedure___ CFL=1.3256,. e=0.5with the modified procedure___ CFL=2.3180, £=0.5

7C/4 7C/2p

3n/4

Fig. 3 Amplification factor of optimal three-stage scheme with andwithout the modified procedure for the third-order scheme with a) IRSand b) ERS.r

Ay

iAx

Fig. 4 Geometry of two-dimensional cell.

stability, the characteristic cell width in the convection directionshould not be greater than AG. From geometric relations, the lengthof AG can be found to be

|AG| =bAx 4-flAv-

and the CFL number based on this length would be redefined as

(19)

AG A>(20)

Table 3 CFL number of optimally smoothing multistage scheme for_________IRS with various smoothing coefficient_______________

Stagee 2 3 4 5 6

First-order upwind scheme

0.10.20.30.40.50.60.70.80.91.0

0.10.20.30.40.50.60.70.80.91.0

1.28471.57201.84432.10232.34352.56992.78322.98353.00383.0120

1.10141.20381.37301.52671.65551.14141.82251.89961.97342.0442

1.90542.26732.62332.96873.28883.58563.86304.12074.36824.6020

2.58763.01823.42623.84034.23364.58894.91785.22505.51735.7940

Kappa scheme, K = 3

1.66831.99072.28042.54192.78043.00293.15623.28423.40523.5206

2.16332.57482.95093.28873.59713.88274.14784.40084.64014.8695

3.28883.78794.22994.69095.13815.53705.90356.24836.57846.8902

2.69253.18043.64304.05644.43174.77995.10415.4.1275.70465.9862

3.99524.58915.07045.57196.04486.46756.86377.24277.60177.9502

3.23063.78544.32334.80885.27475.65266.03426.39546.73957.0690

Kappa scheme, K = — 1

0.10.20.30.40.50.60.70.80.91.0

0.25520.25620.25720.25810.25920.26010.26110.26210.26300.2640

0.82670.83030.83380.83740.84090.84390.84710.85030.85360.8567

1.16741.40021.62051.83542.04932.26152.47412.68592.89293.0956

1.48511.77252.03212.28082.52582.77013.01453.25793.50403.7487

1.80822.16192.46542.74463.02193.29643.56913.84214.11804.4016

robust in the sense of reproducing the good damping propertiesof the optimized one-dimensional schemes for two-dimensionalconvection.

IV. Fast Euler SolverA fast Euler solver was developed to prove the applicability of

the optimally smoothing multistage schemes to an axisymmetricEuler flow problem. An axisymmetric secant-ogive-cylinder-boat-tail projectile (SOCBT) is used as the testing model. The compress-ible inviscid external flow of the projectile is described by the Eulerequations in conservation form. A finite volume method was chosen,i.e., the integratial formulation of the equations is directly discretizedin physical space. A system of ordinary differential equations intime is obtained by applying the spatial discretization in each cell.First-order upwind differencing and the Van Leer kappa schemeswere chosen to be the spatial discretization. Roe's flux-differencesplitting,12 modified to satisfy the entropy condition,13 was adoptedto evaluate the fluxes through the control surfaces. The Van Albadalimiter was imposed on the kappa schemes to prevent numericaloscillations.

The residual smoothing is applied on the finest and coarser gridlevels. In the two-dimensional case, the implicit and explicit stepsare applied in product forms

and

(21)

(22)

where v is the CFL number obtained from one-dimensional analysis.This may not be the perfect choice, but it has been shown to be very

i.e., approximation factorization is applied to obtain the aver-aged residual. It is not at all clear that a two-dimension anal-ysis for sx = sy would yield an optimal CFL number that is

TAI, SHEU, AND VAN LEER: SCHEMES FOR EULER EQUATIONS 1013

Table 4 CFL number of optimally smoothing multistage scheme for________ ERS with various smoothing coefficients _______

Stagee

First-order upwind scheme

0.050.100.150.200.25

0.050.100.150.200.25

1.16431.41501.75802.10482.2368

0.92771.08621.25121.37841.4018

1.74482.03232.41012.89213.1791

2.37802.71303.10893.66734.0907

Kappa scheme, K = |

1.52331.78782.07442.31802.4442

1.97992.30192.67073.00143.2158

3.01653.44983.86134.47145.0179

2.4739.2.84203.28773.70313.9956

3.70334.19514.64505.30205.9530

2.90753.30473.80164.29384.6588

Kappa scheme, K = — 1

0.050.100.150.200.25

0.25470.25520.25570.25620.2567

0.79200.96081.17251.19361.1949

1.06381.25101.47811.80102.1868

1.35251.59121.83772.18022.6208

1.64351.94472.22502.58543.0573

close to the value derived from one-dimension analysis. How-ever, we have already redefined a two-dimensional CFL numbersuited for two-dimensional convection term. It has been shownthat using the redefined CFL number will not make the opti-mal multistage schemes unstable. Therefore, the smoothing coef-ficients used in the different directions are set to be equal in thispaper.

Recall that the multigrid method effectively removes the low-frequency components of error on the coarser grids in the iter-ative procedure but requires a good single-grid smoother. Thus,the use of a multistage scheme with good high-frequency damp-ing on each grid may effectively improve the performance of themultigrid strategy.2-14 Therefore, the optimally smoothing multi-stage scheme was considered as the time-integration method. Themultigrid strategy used in the test code is a saw-tooth cycle,2 inwhich first-order upwind differencing is used on all coarse gridsand the kappa schemes only on the finest grid. A powerful tech-nique is to perform some preliminary iterations on a coarser grid,then use the resulting approximation as an initial guess on thefiner grid; this so-called "nested iteration" was used in the fast-solver to get a better initial quess on the finest grid. In addi-tion, one more acceleration technique was used, i.e., local timestepping which allows each cell to advance in time at maximumrate and makes the scalar, constant-coefficient analysis locallyapplicable.

V. Application of the Optimally SmoothingMultistage Schemes

The numerical application of the optimal multistage schemes wascarried out for the SOCBT projectile in supersonic flow, M^ = 1.2,at zero angle of attack. All computations were performed on anHP-750 workstation and achieved a drop of residual to 10~6. Thecomputational work required for convergence, denoted as WU, isexpressed in terms of finest-grid residual calculations. Most of thetest cases were performed by using a 64 x 64 mesh which wasgenerated by O-type elliptic solver. Figure 5 is the expanded viewof the grid.

A. Grid AnalysisThe quality of the solution and the effect of mesh size on con-

vergence can be inferred from Figs. 6-8. Figure 6 shows the pres-sure coefficient [defined as Cp = (p - /?00)/l/2p00f/(

2<)] on the

projectile, obtained by using the optimal five-stage scheme for thefirst-order upwind scheme with the IRS method, s = 0.5, on single-and five-level grid calculations (64 x 64 cells). The surface pres-

Y/D

-9.0-6.0 12.0

Fig. 5 Expanded view of computational grid near the model.

1.80

1.00 -

Cp

-.60

By using the optimal 5-stage with CFL=5.1381(for the 1st-order upwind scheme with IRS, £=0.5)

o o Experiment——— 1-level Calculation

A 5-level Calculation

.0 2.0 4.0 6.0X/D

Fig. 6 Surface pressure distribution of the test model.

5.0

2.5 -

Y/D

.0 -

-2.5 -

-5.0-1.0 1.5 4.0 6.5 9.0

X/D

Fig. 7 Pressure contours of the test model.

sure distributions of the SOCBT projectile for both calculations areidentical and in agreement with the experimental data.15 Figure 7is the pressure contour plot for the projectile at M^ = 1.2. In thisfigure, the flow pattern of projectile, such as an attached shock, ex-pansions, and a recompression shock can be found. Meshes withdifferent numbers of cells, 64 x 64,128 x 64, and 128 x 128, were

1014 TAI, SHEU, AND VAN LEER: SCHEMES FOR EULER EQUATIONS

-3.00

Optimal 5-stage scheme with CFL=5.1381(for the 1st-order upwind scheme with IRS, e=0.5)

1-level Calculations:

-4.00

Log(Res).

-5.00 -

5-level Calculations : ___ 64X64____ 128X64..„.„.„. 128X128

-6.00

____ 64X64____ 128X64_ _ _ _ _ _ 128X128

-3.00

7000. 14000. 21000.WU

Fig. 8 Convergence history of optimal five-stage scheme on single- andfive-level grids with different grid size.

-3.00

Optimal schemes(for the 1st-order upwimd scheme with IRS.e =0.5)

-4.00

.og(Res) -

-5.00 -

___ 2-stage; CFL=2.3435____ 3-stage; CFL=3.2888____ 4-stage; CFL=4.2336____ 5-stage; CFL=5.1381

-6.005000. 10000. 15000.

WUFig. 9 Convergence history of optimal schemes with different numberof stage.

used to test convergence dependence on cell size. Figure 8 showsresults of the optimal five-stage scheme for the first-order upwindscheme with the IRS method, s = 0.5, implemented on a single-and a five-level grid. It is seen that the multigrid convergence isnot completely independent of the number of cells in the finest

-grid.

B. Covergence Effect of the Optimal Multistage StagesThe effect of the number of stages on convergence can be seen

in Fig. 9. The results in this figure are obtained by using the op-timal two-, three-, four- and five-stage schemes for the first-orderupwind scheme with IRS, s = 0.5, on single- and five-level grids.It is shown that the convergence of these schemes is effectivelyenhanced by using the multigrid strategy. The convergence of thetwo-stage scheme is superior to the others for a single-level calcu-lation. This is because it uses fewer flux calculations per unit CFLnumber. However, for the five-level calculations, the convergencehistories of all of the schemes are similar. This is because in multi-grid relaxation the damping properties become important, and theseare better for the many-stage schemes. The effect of the number ofgrid levels can be seen in Fig. 10, showing the convergence histo-ries of calculation for various numbers of levels, by using for theoptimal five-stage scheme and the first-order upwind scheme withthe IRS and s = 0. The convergence history of the single-grid cal-culation illustrates that initially the residual decreases sharply, cor-responding to the quick elimination of the high-frequency modes,

-4.00

Log(Res).

-5.00 -

-6.00

Optimal 5-stage scheme with CFL=5.1381(for the 1st-order upwind scheme with IRS, £=0.5)

___ 1 level___ 2 levels___ 3 levels___ 4 levels___ 5 levels

5000. 10000.WU

15000.

Fig. 10 Convergence history of optimal scheme on grids with differentnumber of levels, first-order upwind scheme.

-3.00

-4.00 -

Log(Res).

-5.00 -

-6.00

Optimal 5-stage scheme with CFL=2.5258(for the 2nd-order upwind scheme with IRS, £=0.5)

___ 1 level___ 2 levels___ 3 levels___ 4 levels___ 5 levels

V/\ ,' ^M^V .-4\ /x\/y\^v-^%;\>10000. 20000.

WU30000. 40000,

Fig. 11 Convergence history of optimal scheme on grids with differentnumber of levels, second-order upwind scheme.

-3.00

-4.00 -

Log(Res) -

-5.00 -

-6.00

Optimal 5-stage scheme with CFL=4.4317(for the 3rd-order scheme with IRS, £=0.5)

___ 2 levels3 levels

___ 4 levels___ 5 levels

30000.10000. 20000.WU

Fig. 12 Convergence history of optimal scheme on grids with differentnumber of levels, third-order scheme.

followed by a slow decrease due to the low-frequency modes. Thefive-level calculation displays the best convergence performance,illustrating that the multigrid code based on the optimal schemeeffectively benefits from the damping of the low-frequency errors.Figures 11 and 12 show the multilevel convergence behavior of

TAI, SHEU, AND VAN LEER:

-3.00

-4.00

Log(Res)

-5.00

-6.00

Optimal 5-stage scheme(for the Ist-order upwind scheme with IRS)

____ £=0.0, CFL=2.58=0.2, CFL=3.78798=0.5, CFL=5.13818=0.8, CFL=6.2483

SCHEMES FOR EULER EQUATIONS

-3.00

1015

6000. 12000. 18000.WU

Fig. 13 Convergence history of optimal scheme with different smooth-ing coefficients, IRS.

1.00

.75 -

IPI

.50 -

.25 -

.00

Optimal 5-stage scheme(for the Ist-order upwind scheme with IRS)

___ e =0.0, CFL=2.5___ 8=0.2, CFL=3.7879___. 8=0.4, CFL=4.6909___. e =0.6, CFL=5.5370___ 8 =0.8, CFL=6.2483........... e=1.0,CFL=6.8902

71/4 71/2 37C/4

Fig. 14 Amplification factor of optimal five-stage scheme with differentsmoothing coefficients, IRS.

optimal five-stage schemes for the kappa scheme (K — —1,1/3)with the IRS method, s = 0.5.

A convergence comparison of the optimal five-stage schemesfor first-order upwind differencing with IRS, s = 0, 0.2, 0.5, and0.8, on single- and five-level calculations is shown in Fig. 13. It isseen that the speed of convergence increases while the smoothingcoefficient increases, as expected, because of the better dampingproperties and the larger Courant number. It is also seen thatthe convergence efficiency does not improve dramatically by in-creasing the smoothing coefficient beyond about 0.5. Figure 14shows that the high-frequency damping properties worsen whilethe smoothing coefficient increases. This also holds for the kappaschemes.

Figure 15 shows the convergence histories of the optimal five-stage schemes for first-order differencing with ERS, s = 0, 0.1,and 0.2 on single- and five-level calculations. It is seen that thefive-level calculation with s = 0.2 yields the best convergence effi-ciency. From Fig. 13 and 15, it is seen that the ERS method yieldsbetter convergence performance while using the same smoothingcoefficient. In our experiment, using the ERS method with s > 1/4may make the calculation diverge, unless a very small CFL numberis employed.

C. Comparison with the Runge-Kutta SchemeTwo cases with five-stage Runge-Kutta schemes, one with the

CFL number 2.5 and one with the allowable maximum CFL

Optimal 5-stage scheme(for the 1st-order upwind scheme with ERS)

____ 8=0.0, CFL=2.5

-4.00

Log(Res).

-5.00

____ 8=0.1, CFL=3.44988=0.2, CFL=4.4714

-6.006000. 12000. 18000.

WUFig. 15 Convergence history of optimal scheme with different smooth-ing coefficients, ERS.

1.00

.75 -

IPI

.50 -

.25 -

.00

(for the 1st-order upwind scheme with IRS)^RK 5-stage method :____ CFL=2.5, 8=0.5___ CFL=4.3, 8=0.5

71/4 7C/2p 371/4

Fig. 16a Loci and contours of Runge-Kutta five-stage scheme.1.00

IPI

.50

.00

(for the 1st-order upwind scheme with IRS )RK 5-stage method : ___ CFL=4.3,8=0.5

a ____CFL=2.5,8=0.5V ^ OPT. 5-stage method :___CFL=5.1381,8=0.5

i \

71/4 7C/2 37C/4

Fig. 16b Amplification factor of optimal and Runge-Kutta five-stageschemes.

number 4.3, were used to compare with the optimal five-stagescheme. The implicit residual smoothing, such that s = 0.5, is con-sidered in these schemes. The damping and stability of the Runge-Kutta four-stage scheme can be judged from Figs. 16a and 16b.The comparisons were implemented both on a single- and a five-level grid by utilizing the five-stage schemes for first-order upwind

1016 TAX, SHEU, AND VAN LEER: SCHEMES FOR EULER EQUATIONS

-3.00

-4.00 -

Log(Res).

-5.00

-6.00 -

5-stage schemes(for the 1st-order upwind scheme with IRS, £=0.5)

1-level Calculations:___ OPT. scheme, CFL=5.1381___ RK scheme, CFL=2.5___ RK scheme, CFL=4.35-level Calculations:___. OPT. scheme : CFL=5.1381

_ RK scheme, CFL=2.5.. RK scheme, CFL=4.3

0. 5000. 10000.WU

15000. 20000.

Fig. 17 Comparison of convergence history of optimal five-stagescheme and five-stage Runge-Kutta scheme.

differencing with the IRS method, s = 0.5, as shown in Fig. 17. Theoptimal scheme performs somewhat better than the Runge-Kuttascheme, presumably because of its large CFL number.

VI. ConclusionsIn this paper, a technique has been introduced for develop-

ing a multistage scheme suited for multigrid use. Parameters formultistage schemes for certain one-dimensional spatial discretiza-tions, with or without residual smoothing, implicit or explicit, arepresented. The damping and stability properties of some optimallysmoothing multistage schemes are studied. The extension to anyspecified spatial discretization with or without residual smooth-ing should be easily achieved with little computational cost. Themultistage coefficients from one-dimensional analysis directly ap-ply to multidimensional computations by redefining the CFL num-ber. In numerical applications of the optimally smoothing schemesthis significantly accelerates the convergence to a steady-statesolution.

AcknowledgmentThe authors gratefully acknowledge the financial support of the

National Science Council of the Republic of China under GrantNSC-82-0401-E-014-009.

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