[american institute of aeronautics and astronautics 15th aiaa computational fluid dynamics...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization.

15th AIAA Computational FluidDynamics Conference

11-14 June 2001 Anaheim, CA A01-31061 AIAA-2001-2556

Multigrid Diagonalized-ADI Method forCompressible Flows

Chun-ho Sung * Soo-Hyung Park t

Jang Hyuk Kwon *Korea Advanced Institute of Science and Technology,

373-1 Kusong-Dong, Yusong-Gu, Taejon 305-701, KoreaTelephone : 82-42-869-3715, fax : 82-42-869-3710

An efficient multigrid diagonalized ADI(DADI) method for the compressible flowsis presented. The second order upwind TVD scheme in conjecture with the Roe's flux-difference splitting(FDS) is used for the numerical flux calculations. An implicit operator,which is consistent with the Roe's FDS, is derived from the numerical flux. A modifiedsaw-tooth cycle multigrid method is designed to accelerate wave propagations.

To show the efficiency of the present method, 2-D and 3-D transonic flows are com-puted and comparisons with the explicit method with the multigrid are presented.

IntroductionSolution methods for the compressible flows have

been an important topic in the CFD community.Many efforts were concentrated on the development ofefficient and accurate numerical methods for both theEuler and Navier-Stokes equations.1 The governingequations of compressible flow have convection termswhich have properties of hyperbolic partial differentialequations(PDEs). One of the most important featureof hyperbolic PDE is that it requires an additional ar-tificial dissipation to get a stable numerical solution.2'3

A widely used solution method for the compressibleflows is a time-marching method which uses a spatialdiscretization to produce semi-discrete equations andsubsequently applies a time-stepping method to findthe final solution. For steady state problems, the timeaccuracy of the solution is not required, therefore atime-stepping method can be modified for the conver-gence acceleration.2'4

The spatial discretization methods can be groupedas the central and upwind methods. The artificialdissipation, proposed by Jameson et a/, is the cen-tral method2 while the Roe's flux-difference split-ting(FDS)3 and van Leer's flux-vector splitting(FVS)5

are known as the upwind methods. For transonic flowcalculations, the Roe's flux-difference splitting (FDS)method, which was used in this work, shows bettershock resolution.

On the other hand, the time-stepping methods canbe divided into explicit and implicit methods. For theimplicit method, the approximate-factorization(AF)

* Post-doctoral research associate, Dept. of Mechanical En-gineering., Member AIAA

^Doctoral candidate, Dept. of Mechanical Engineering.* Associate Professor, Dept. of Mechanical Engineering, Se-

nior Member AIAA.>-Hyung Park and Jang HyukCopyright © 2001 by Chun-hi

Kwon. Published by the American ]Inc. with permission.

Sung, So

method by Beam and Warming6 and its diagonalizedversion by Pulliam and Chausse,7 have been widelyused. The diagonalized ADI(DADI) method is very ef-ficient for the compressible Euler equations. Since thediagonalization of implicit operator largely reduces thecomputing cost by replacing block tri-diagonal matri-ces to its scalar from, it shows faster convergence thanthe explicit multistage time-stepping(MST) method.

For the solution of the compressible Navier-Stokesequations however, the DADI method is no longer su-perior to the explicit multistage time-stepping method.While the explicit method - shows very fast conver-gence by using a multigrid method, a multigrid DADImethod is not as fast as the explicit MST method.8

The main difficulty of applying the multigridmethod for DADI is : DADI method does not have ad-justable parameters which improve the high-frequencydamping characteristics. Caughey applied DADImethod with multigrid to central differencing scheme.8

He introduced a coefficient for artificial dissipation toimprove high-frequency damping. Although the resultshowed improved convergence, the efficiency of multi-grid was not fully exploited so much as in the MSTmethod.

In this work, a new multigrid DADI method is pre-sented for the upwind differencing schemes. To ex-ploit the convergence acceleration mechanism of multi-grid method, a modified saw-tooth cycle is introducedwhich is used by Caughey.8'9 The modified saw-toothcycle is designed to accelerate wave propagation dur-ing iterations. The high frequency errors do not fullydamp out in the computational domain, they can beexpelled out fastly by a modified saw-tooth cycle. Inaddition, a consistent implicit operator for Roe's fluxdifference splitting(FDS) is developed to improve theconvergence speed of DADI method.

The efficiency of present scheme is assessed by com-

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2001-2556


puting compressible 2-dimensional and 3-dimensionalproblems. A wide range of compressible flows - fromsubsonic to supersonic - is presented to show the ef-ficiency of the implicit operator and the multigridmethod. Comparisons with the explicit multistagetime-stepping method4 are also presented.

Governing EquationsThe 3-dimensional compressible Navier-Stokes equa-

tions in Cartesian coordinates (xi, #2, #3) can be writ-ten in the conservation form as

where

where

dqTt +

+ pSn

= o (1)

(2)

and p is the density, Ui is the velocity component forXi direction, cr^- is the component of stress tensor, qi isthe heat flux for X{ direction and p,E,H are the pres-sure, total energy and total enthalpy respectively. Therepeated index represents the tensor summation con-vention. The pressure is determined by the equationof state

(3)P = (7 - I)/

and the total enthalpy is

PP

(4)

where 7(=1.4 for air) is the ratio of specific heats. Thestress tensor a is defined as

— [ { _i_ J ) _i_ \A fc /c\duk-dxk

and the heat flux is

. dT7 - 1 Pr (6)

where JJL and A are the first and second viscosity coeffi-cients respectively, and Pr is the Prandtl number. Theviscosity can be calculated by the Sutherland's law forlaminar cases. For turbulent flows, the turbulent vis-cosity is added to the viscosity and it is obtained fromthe Baldwin-Lomax model.10

The governing equations in the physical coordinatessystem can be transformed onto the computational co-ordinates (£1,^23 £3)- In the computational domain,the governing equations yield

(7)

- (Sijfj) , Fvi = (SyfVJ.) ,Q - -q,

and J is the transformation Jacobian.

Spatial DiscretizationEq. 7 is discretized with the cell-centered finite

volume method. After cell-wise integration in the com-putational domain, the governing equation yields4

dt + Ry * - 0 (8)

where

(9)

Here F^ and F^^ are the inviscid and viscous parts ofnumerical flux at the cell face.

The viscous flux Fvi can be simply obtained fromthe central difference. However, the inviscid flux Fjrequires an artificial dissipation to prevent numericalinstabilities around discontinuities.

In present study, Roe's flux difference split-ting(FDS) is used for the upwind method and thesecond order upwind TVD scheme is adopted to im-prove the solution accuracy.3'4 For £*. direction, theinviscid fluxes of the second order upwind TVD schemeare constructed as follows:

(10)- 5 tfcnfc (ak -

Tfc is the right eigenvector matrix of flux Jacobian,£lk is the eigenvalue matrix and L& is an anti-diffusiveflux which satisfies the total variation diminishing con-dition. The symbol 'tilde' means the Roe's averagedvalue.

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The eigenvalue matrix, 12 & is a function of eigen-value of flux Jacobian matrix A. To prevent non-physical solutions, such as expansion shocks, it re-quires an entropy-fixing function. In this work,Harten's entropy function is used.11'12 The entropyfunction is

(12)

a.k is the spatial differences of characteristic vari-ables,4 which is defined as follows:

i+l/2(13)

where A is the central difference across cell face andT^1 is the left eigenvalue matrix.

Finally, tk is a limit er function of TVD scheme. Ifthe limit er function is set to zero, the resulting numer-ical method will be spatially a first order accurate Roescheme. The van Leer limit er is used in this work anddefined as follows:

0(14)

wherecr = sign(A^)

Implicit Time-steppingThe steady-state solution of Eq. 8 can be obtained

by any stable time integration method. Implicit meth-ods admit a larger time step compared with the ex-plicit methods, at a cost of computing time for matrixinversions. The expensive cost of matrix inversion canbe significantly reduced by using approximate meth-ods such as approximate factorization.

In this work, the DADI method is used to findsteady-state solutions. As previously mentioned, thediagonalized-ADI method proposed by Pulliam andChausse is the most efficient one, especially for solv-ing compressible flows. However, most of previouswork used a van Leer's FVS type implicit operator,regardless of the spatial discretization method. Sincethe Roe's FDS is used in present study, a consistentimplicit operator should be developed. The more con-sistent operator will result in faster convergence. Theconsistent implicit operator can be derived from thefollowing procedure.

The semi-discretized equations can be expressed ina backward Euler implicit method :

Qn+1 - Qn (15)

The residual vector Rn+1 can be linearized in time asfollows:

Q-p \

where

By using Eq. 16, Eq. 15 yields as

(17)

|S is the Jacobian matrix of residual. It differsaccording to the spatial discretization method. For thepresent scheme, by assuming that the Roe's average is/ \constant, (f§) of £1 direction can be written asfollows:

5R

9Q9Qijk

1 = 1 + 1

9Qnk "-* *(18)

where AI will be '0' when / > i + 2 or / < i - 2.It should be noted that the contribution of viscous

term is neglected in the above equations for DADImethod, since the viscous term is not simultaneouslydiagonalizable with the inviscid one. The contributionof viscous term will be considered only by a spectralradius scaling, proposed by Coakley.13 The compo-nent of (|B J along £2 and £3 can be derived in thesame manner*

By using ADI method, Eq. 17 can be factorized asfollows:

(19)The resulting implicit method requires the solution

of block penta-diagonal matrix. It can be reducedto a tri-diagonal matrix by removing the contributionof ant i- diffusive flux L in the implicit operator. Thenumerical flux F$ without anti-diffusive flux can bewritten as follows:

(20)i+l/2

The flux Jacobian of ̂ L can be diagonalized by localsimilarity transformation.7

(22)

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where A is the diagonal matrix of eigenvalues. Thesecond term of Eq. 21 should be modified to be diag-onalizable with Eq. 22 . Therefore, Tk and T~l willbe used instead of T^ and TjJ"1.

The diagonalizable form of AI can be summarizedas follows:

(23)

where

(24)The diagonal implicit operator, D& represents theeigenvalue matrix of the Roe's FDS method. It is con-sistent with the FDS since the Roe's average valuesand the entropy fix function are accounted. However,the contributions from the cell face metrics are ignoredfor the diagonalization.

Finally, by using Eq. 23 , Eq. 19 can be expressedas diagonalized AD I form :

(25)

where

Note that M and N are constant coefficient matriceswhich are independent on the flow variables.

The solution procedure is similar to that of theblock ADI scheme. During one iteration, 15 scalartri-diagonal inversions and 4 matrix-vector multipli-cations are required for 3-D calculations. Even withadditional matrix-vector multiplies, the diagonalizedalgorithm is more efficient than the solution procedureof the block algorithm. For the Navier-Stokes calcula-tions, the spectral radius of viscous term is added to

The local time step is applied to At, for the conver-gence acceleration.

Multigrid algorithmAs mentioned in the introduction, the DADI al-

gorithm does not have any adjustable coefficient toimprove the high-frequency damping characteristics.To overcome this deficiency, Caughey8 introduced anartificial dissipation to implicit operator which has ad-justable coefficients. This strategy can improve thedamping characteristic, however, it results in a penta-diagonal matrix problem due to the dissipation term.

The acceleration mechanisms of the multigridmethod are known as the low frequency damping im-provement and the fast wave propagation in a coarse

grid. The former mechanism can be useful whena time-stepping method has a good high frequencydamping characteristic. Thus, the present study con-centrates on the latter mechanism, an accelerated wavepropagation on the coarse grid.

The 4 level saw-tooth multigrid cycle used by Jame-son14 is also adopted in this work. The general saw-tooth algorithm is described in Ref.4 A coarse gridcan be generated by eliminating every second line ofthe fine grid and denoted by the subscript 2/i.

In order to implement multigrid method on DADImethod, some modifications are applied to the saw-tooth cycle algorithm. The modifications are focusedon maximizing the wave propagation speed. In thecoarse grid, the cell size is larger than the fine grid,consequently a larger time step can be allowed. Thecomputing cost on the coarse grid is cheaper than thaton the fine grid. Therefore, a different number of timestep is applied according to the grid level. It results inthat the correction on the fine grid reflects the acceler-ated wave propagations on the coarser grids. For theinviscid case, only two time steps are applied for thefinest level, while L + 2 time steps in the coarse levels.L denotes the grid level and L — 1 for the finest level.For the viscous case, 2 x L time steps are performedin the coarse levels. In addition, the CFL number isincreased according to the grid level. The numericalexperiments show than a larger CFL number can beused for the coarse grid. When the CFL number ofthe finest level is CFL\, that of coarse levels are de-termined by

This strategy results in a kind of multistage time-stepping, while the number of time steps differs inthe grid level. However, it is different to the explicitmultistage time-stepping of Jameson et al2 and Parkand Kwon.4 The time advance of a multistage time-stepping in explicit MST is only At, regardless of thenumber of stage. For the present method, it is n x Atwhere n is the number of stage.

The resulting multigrid algorithm is as follows:

1. Obtain the solution at the finest grid level.Following procedures, (ii)-(viii), are iterated untilthe coarsest grid level.

2. Calculate the Residuallevel.

again at the present

3. Using the forcing function P^, obtain a new resid-ual R? defined as

Rft = + P,h- (26)

At the finest grid level the forcing function trans-ferred from the finer grid does not exist, so R£ isreplaced with R^.

4 OF 8



4. The residual R£ and conservative variables aretransferred to the next coarser level as follows:

= EShwh/S2h (27)(28)

5. Using transferred variables, calculate the residual

6. Calculate the forcing function as follows:

7. The residual

(29)

at the qth stage is defined as

_ r>(9) i T3 f^C\\h — ̂ 2h +-^2/i- (6{J)

, obtain the solution at the8. Using the residual2h level.

9. Finally, using a bi- linear interpolation, the com-puted corrections at each grid level are passedback to the next finer level.

When the multigrid is applied, the initial guess offlow field is computed by the mesh-sequencing method.The solution is obtained at the coarsest grid level, thentransferred to the finer grid by a bi-linear interpola-tion.

Numerical ResultsThe present method is applied to viscous and in-

viscid cases. Firstly, two-dimensional inviscid flowsover NACA 0012 airfoil and RAE 2822 are presented.The convergence histories are compared with the re-sult of Park and Kwon.4 The numerical methods of thepresent study and Park and Kwon are identical, exceptin the time-stepping method. Park and Kwon used aexplicit multistage time-stepping with the saw-toothcycle multigrid while the multigrid DADI is adoptedin the present work. The coefficients of the multistagetime-stepping are optimized to the second order up-wind TVD scheme. Therefore, the comparison canbe used for the performance assessment of proposedmultigrid DADI method.

Four level multigrid is used for both cases and thecomputations are performed on the SGI octane withMIPS R10000. Figs. 1 and 2 show the surface pres-sure distributions of NACA 0012 and RAE 2822. Theresults are absolutely identical.

Figs. 3 and 4 represent the convergence histories.The density L2 norm is used for the error and it is nor-malized by the value of the first iteration. When theL2 norm is reduced to 10 orders of magnitude for theNACA 0012 case, the multigrid DADI method takes150 cycles and 24.9 CPU seconds while the multigridexplicit method takes 1221 cycles and 281.7 seconds.For the RAE 2822 case, the multigrid DADI requires

- 1

-0.75

-0.5

-0.25

0'°

0.25

0.5

0.75

1

• • • " * " "

Park/ * present

_ *

« *•

_ •

8

- : "X- • \- ' *

aa

1 1 1 1 1-0.5 -0.25 0.25 0.5

Fig. 1 Pressure distributions of NACA 0012 (In-viscid, M=0.8, a = 0)

-1.5

-1.25

-1

-0.75

-0.5

6*.25

0

0.25

0.5

0.75

1

Parkpresent

-0.5 -0.25 0.25 0.5

Fig. 2 Pressure distributions of RAE 2822(Invis-cid, M—0.73, a = 2.79)

only 225 cycles and 38.5 seconds while the multigridexplicit converges to 3 x 10~9 after 2000 cycles and 447seconds. The computing costs of the multigrid DADImethod are less than 10% of the explicit method.

The second test case is a turbulent transonic flowover RAE 2822. The experimental data is availablefrom the AGARD report.15 The 383 x 65 C-type gridis used, which can be obtained from the NPARC al-liance homepage.16 Five level multigrid is used andthe computation is performed on the PC-cluster with4 Pentium II-400MHz. Fig. 5 shows the pressure dis-tributions of the present method, WIND16 code andthe experiment. The present result agrees well withthe experiment and the result of WIND. For this case,the result of Park and Kwon is not available, there-fore a comparison with single grid DADI is presented.

5 OF 8



100 200CPU time(seconds)

Fig. 3 Convergence histories of NAG A 0012 (In-viscid, M=0.8, a = 0)

100 200 300CPU time(seconds)

400

Fig. 4 Convergence histories RAE 2822(Inviscid,M=0.73, a = 2.79)

The convergence histories are presented in Fig. 6. The5 level multigrid DADI requires 234 cycles and 259.1seconds until L2 norm reduced to 4 orders of magni-tude, while that of single grid DADI are 2806 cyclesand 1787.7 seconds. By the multigrid, the computingtime is reduced to a factor of 6.9.

It should be noted that the convergence of multigridmethod becomes slower in the asymptotic region. Inthe asymptotic region, the dominant numerical errorshave very low frequency and its propagation speeds arealso slow. The convergence slow down is inevitablesince the present multigrid cycle is not designed todamp out these errors. Even though, the presentmultigrid cycle is very useful for the initial stage ofiterations and it shows fast convergence until 5 ordersof error reductions.

0.5

-0.5

0.25 0.5X

0.75

Fig. 5 Pressure distributions of RAE2822(M=0.729, a = 2.31, Re=6.5 x 106)

10°

10'1

10'2

510'3

5E1Q-4

10-5

10-6

1C'7 -

single gridmultigrid

1000 2000 3000CPU seconds

4000 5000

Fig. 6 Convergence histories2822(M=0.729, a = 2.31, Re=6.5 x 106)

of RAE

The last test case is a transonic flow over the ON-ERA M6 wing. Both the inviscid and viscous flows arecomputed and the multigrid efficiency is also demon-strated by comparisons with the single grid DADI.

For a inviscid case, 129 x 33 x 33 O-H type gridsystem is used and 193 x 49 x 33 C-O for a viscouscase.

Fig. 7 shows the results of inviscid, viscous compu-tations and experiment.17 The numerical results showgood agreements with the experimental data. Fig. 8shows the convergence histories of both the viscous andinviscid computations. For the inviscid case, only 40multigrid cycles are sufficient to reduce the L2 normdown to 4 orders of magnitude. When the L2 normis converged to 10 orders of magnitude, 1192 cyclesand 1920 seconds are requires for the single grid while

6 OF 8



0.5

-0.5

0.25 0.5X

0.75

a) 44% span

10°

1C'1

10'2

10'3

2'QC

o-6

io-7

10'8

10"9


200 400 600 800iterations

1000

a) Inviscid

0.5

O

as ExperimentInviscid

o Viscous

10°

0.25 0.5X

0.75

b) 65% span

Fig. 7 Pressure distributions of ONERA M6 atthe selected wing sections( M—0.8395, a = 3.06, Re= 11.7 x IO6 )

only 281 cycles and 723 seconds for the multigrid. Forviscous case, the effect of multigrid is more impres-sive. The single grid solution converges very slowlyand 1824 cycles and 13330 seconds are required to re-duce the L2 norm 4 orders of magnitude. It takes only193 cycles and 1970 seconds with the multigrid DADI.The computing time of the multigrid is reduced to 15%of single grid.

ConclusionsA new multigrid DADI method for the compressible

flows is presented. The high-resolution shock captur-ing scheme and the second order upwind TVD withRoe's flux-difference splitting is used for the spatialdiscretization. The diagonalized-ADI method is used

10'3


100 200 300 400 500 600iterations

b) Viscous

Fig. 8 Convergence histories of ONERA m6 wing( M=0.8395, a = 3.06, Re = 11.7 x IO6 )

for the time integration. For the convergence and sta-bility improvement of the present method, an implicitoperator is derived form the numerical flux function ofthe Roe's FDS. A modified saw-tooth cycle multigridis proposed. The multigrid cycle is constructed onlyfor the fast wave propagations.

The present method is applied for both the viscousand inviscid transonic flows. The comparisons withthe multigrid explicit method show that the presentscheme can obtain the same numerical result withinonly a 10% of computing cost. The comparisons withthe results of single grid DADI reveal that the con-vergence can be accelerated by a factor of 3-7 by thepresent multigrid strategy.

7 OF 8



References1MacCormack, R., "A perspective on a quarter century of

CFD research," AIAA Paper 93-3291, 1993.2Jameson, A. and Schmidt, W., "Numerical Solution of The

Euler Equations by Finite Volume Method Using Runge-KuttaTime-Stepping Schemes," AIAA Paper 81-1259, 1981.

3Roe, P., "Approximate Riemann Solver, Parameter Vectorsand Difference Schemes," J. Comput. Phys., Vol. 43, No. 2,1981, pp. 357-372.

4Park, T. S. and Kwon, J. H., "An Improved MultistageTime Stepping for Second-Order Upwind TVD Schemes," Com-puters and Fluids, Vol. 25, No. 7, 1996, pp. 629-645.

5van Leer, B., "Flux-vector splitting for the Euler equa-tions," 8th International Conference on Numerical Methods inFluid Dynamics, Vol. 170 of Lecture notes in Physics, Springer-Verlag, June 1982.

6Beam, R. and Warming, R., "An Implicit Factored Schemefor the Compressible Navier-Stokes Equations," AIAA Paper 77-645, 1977.

7Pulliam, T. and Chaussee, D., "A Diagonal Form of anImplicit Approximate-Factorization Algorithm," J. Comput.Phys., Vol. 39, 1981, pp. 347-363.

8Caughey, D., "Diagonal Implicit Multigrid Algorithm forthe Euler Equations," AIAA J., Vol. 26, No. 7, 1988, pp. 841-851.

9Caughey, D., "Implicit Multigrid Euler Solutions withSymmetric Total-Variation-Diminishing Dissipation," AIAAPaper 93-3358-CP, 1993.

10Baldwin, B. and Lomax, H., "Thin layer approximationand algebraic model for separated flows," AIAA Paper 78-257,1978.

11Harten, A., "High Resolution Schemes for Hyperbolic Con-servation Laws," Journal of Computational Physics, Vol. 49,1983, pp. 357-393.

12Yee, H., "Implicit Total Variation Diminishing(TVD)Scheme for Steady-State Calculation," Journal of Computa-tional Physics, Vol. 57, 1985, pp. 327-360.

13Coakley, T., "Implicit Upwind Methods for the Compress-ible Navier-Stokes Equations," Tech. Rep. 3, 1983.

14Jameson, A., "Solution of the Euler Equations by a Multi-grid Methods," Appl. Math. Comput., Vol. 13, 1983, pp. 327-356.

15P.H. Cook, M. M. and Firmin, M., "Aerofoil RAE2822pressure distributions, boundary layer and wake measure-ments," Advisory Report 138-A6, AGRAD, 1979.

16 "http://www.grc.nasa.gov/www/wind/valid," .17Schmitt, V. and Charpin, F., "Pressure distributions of

the ONERA-M6-Wing at transonic Mach numbers," AdvisoryReport 138-B2, AGRAD, 1979.


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