AIAA-85-0293 Multigrid Solution of the Euler Equations Using Implicit Schemes A.Jameson and S.Yoon Princeton University, Princeton,N. J.
AlAA 23rd Aerospace Sciences Meeting January 14-17, 1985/Reno, Nevada For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1633 Broadway, New York, NY 10019
MULTIGRID SOLUTION OF THE EULER EQUATIONS USING IMPLICIT SCHEMES by Antony Jameson and Seokkwan Yoon Department of Mechanical and Aerospace E n g i n e e r i n g P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , N J 08544
Abstract
A r m l t i g r i d method f o r i m p l i c i t schemes of t h e approximate f a c t o r i z a t i o n t y p e is d e s c r i b e d . The a p p l i c a t i o n of t h e method coupled w i t h a n A l t e r n a t i n g D i r e c t i o n I m p l i c i t scheme t o t h e Solut i o n of t h e E u l e r e q u a t i o n s f o r t r a n s o n i c flow o v e r a n a i r f o i l has r e s u l t e d i n v e r y r a p i d convergence. The number of time s t e p s r e q u i r e d t o r e a c h a s t e a d y s t a t e is reduced by a n o r d e r of magnitude by t h e i n t r o d u c t i o n of m u l t i p l e g r i d s . 1.
Introduction
M u l t i g r i d has emerged as a p a r t i c u l a r l y e f f i c i e n t t e c h n i q u e f o r a c c e l e r a t i n g t h e conv e r g e n c e of numerical c a l c u l a t i o n s . While t h e a v a i l a b l e theorems i n t h e t h e o r y of m l t i g r i d methods g e n e r a l l y assume e l l i p t i c i t y , i t seems t h a t i t ought t o be p o s s i b l e t o accelerate t h e e v o l u t i o n of a h y p e r b o l i c system t o a s t e a d y s t a t e by u s i n g large time s t e p s on c o a r s e g r i d s , so t h a t d i s t u r b a n c e s are more r a p i d l y e x p e l l e d t h r o u g h t h e It has been e s t a b l i s h e d t h a t t h e o u t e r boundary. m u l t i g r i d method can d r a m a t i c a l l y a c c e l e r a t e t h e convergence of t r a n s o n i c p o t e n t i a l flow c a l c u l a t i o n s , a l t h o u g h t h e governing e q u a t i o n s a r e of mixed e l l i p t i c and h y p e r b o l i c t y p e [ l ] . For t h e E u l e r e q u a t i o n s a m u l t i g r i d scheme proposed by Ni has w i d e l y been used 121. I n i t s p u b l i s h e d form h i s d i s t r i b u t e d c o r r e c t i o n scheme i n c l u d e s an a r t i f i c i a l d i s s i p a t i o n t e r m which r e s t r i c t s t h e r e s u l t s to f i r s t o r d e r accuracy. A new m u l t i g r i d method which is second o r d e r a c c u r a t e i n s p a c e was d e v e l o p e d f o r t h e m l t i s t a g e e x p l i c i t time s t e p p i n g scheme L31. S i n c e t h e t i m e s t e p of an e x p l i c i t scheme is l i m i t e d by the Courant-Friedrichs-Lewy (CFL) c o n d i t i o n , which r e q u i r e s that t h e r e g i o n of dependence of t h e d i f f e r e n c e scheme must a t l e a s t i n c l u d e t h e r e g i o n Of dependence of t h e d i f f e r e n t i a l e q u a t i o n , an implic i t scheme is p r e f e r r e d as t h e d r i v i n g scheme. TWO g e n e r a l t y p e s of i m p l i c i t schemes can be d i s t i n g u i s h e d , t h o s e u s i n g r e s i d u a l a v e r a g i n g , and t h o s e based on approximate f a c t o r i z a t i o n . R e c e n t l y t h e r m l t i g r i d method was s u c c e s s f u l l y i n t r o d u c e d f o r t h e i m p l i c i t m l t i s t a g e scheme 141 which b e l o n g s t o t h e r e s i d u a l a v e r a g i n g type. In t h i s p a p e r i t is shown t h a t t h e schemes of a p p r o x i m a t e f a c t o r i z a t i o n t y p e can be a d a p t e d f o r use i n c o n j u n c t i o n w i t h a m u l t i g r i d t e c h n i q u e t o produce a r a p i d l y convergent a l g o r i t h m f o r calcul a t i n g s t e a d y s t a t e s o l u t i o n s of t h e E u l e r equations.
2. Semi-Discrete F i n i t e Volume Scheme The E u l e r e q u a t i o n s i n i n t e g r a l form can he w r i t t e n as:
a at
n
wdn +
If p an
- ds
0
f o r a f i x e d r e g i o n n w i t h boundary 2% Here w r e p r e s e n t s t h e conserved q u a n t i t y and 9 is t h e c o r r e s p o n d i n g f l u x term. I f we m i t e p, P, u, v , E and H f o r t h e p r e s s u r e , d e n s i t y , C a r t e s i a n veloc i t y components, t o t a l energy and t o t a l e n t h a l p y , t h e n e q u a t i o n (1) r e p r e s e n t s mass c o n s e r v a t i o n if we p"t w = P,
p
= (PU, PV).
For c o n s e r v a t i o n of mmentum i n t h e x d i r e c t i o n we have
w = pu, p =
(P"
2
+
p.
PUV)
in e q u a t i o n (1). C o n s e r v a t i o n of momentum i n t h e y d i r e c t i o n is s i m i l a r l y d e f i n e d . F i n a l l y energy c o n s e r v a t i o n is g i v e n by
w = pE,
= (pHu, pHv).
-
These e q u a t i o n s are t o be s o l v e d f o r a s t e a d y state awlat 0 where t d e n o t e s time. A c o n v e n i e n t way to a s s u r e a s t e a d y s t a t e s o l u t i o n independent of t h e time s t e p 18 t o s e p a r a t e t h e s p a c e and time d i s c r e t i z a t i o n procedures. I n s e m i - d i s c r e t e f i n i t e volume scheme one b e g i n s by a p p l y i n g a s e m i - d i s c r e t i z a t i o n i n which o n l y t h e s p a t i a l d e r i v a t i v e s are approximated. I n o r d e r t o d e r i v e a s e m i - d i s c r e t e = d e l which c a n be used t o treat complex g e o m e t r i c domains, t h e c o m p u t a t i o n a l domain is d i v i d e d i n t o q u a d r i l a t e r a l cells. Assuming t h a t t h e dependent v a r i a b l e s are known at t h e c e n t e r of each c e l l , a s y s t e m of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s is o b t a i n e d by a p p l y i n g e q u a t i o n (1) s e p a r a t e l y t o e a c h c e l l . These have t h e form
where Sij is t h e cell area. and Qij is t h e n e t . f l u x o u t of t h e c e l l . T h i s can be e v a l u a t e d as
where fk and gk denote values of the flux vectors f and g on the kth edge, A+ and AYk are the increments of x and y along the edge with appropriate signs, and the sum is over the four sides of the cell. The flux vectors are evaluated by taking the average of the values in the cells on either side of the edge:
where f and g are flux vectors. A prototype implicit scheme far a system of nonlinear hyperbolic equations such as the Euler
equations can be formulated as w
"+'w "- BAt{Dxf(w "f
+ Dy g(w
'+j}
- (l-B)At[Dxf(wn)+
for example. The scheme constructed in this manner reduces to a central difference scheme on a Cartesian grid, and is second order accurate in space provided that the mesh is smooth enough. It also has the property that uniform flow is an exact solution of the difference equations.
3g(wn)] (10) are difference operators that where D, and D approximate and aiay. In this form the scheme is too expensive, since it calls for the solution of coupled nonlinear equations at each time step. Let the Jacobian matrices be
3.
and let the correction be
Adaptive Dissipation
In order to suppress the tendency for spurious odd and even point oscillations, and to prevent unsightly overshoots near shock waves, the scheme is augmented by a dissipative term 60 that equation ( 2 ) becomes
adx
6w
-
w n+f
-
After linearization and dropping terms of the second and higher order, the scheme becomes
[I + BAt(D&
+ DyB)]&w
+ At R
-
0
(11)
where R is the residual Here Dij is the dissipation, which is constructed so that it is of third order in smooth regions of the flow. For the density equation Dij(p) has the form %+l/Z,j
-
di-li2,j
+
di,j+l/2 - d i,j-112 ( 6 )
Both coefficients include a normalizing factor Si+l/?,j/~~roportional to the length of the cell side, and E(4)i+l/2,j is also made proportional to the normalized second difference of the pressure
in the adjacent cells.
This quantity is of second order except in regions containing a steep pressure gradient. The fourth differences provide background dissipation throughout the domain. In the neighborhood of a shockwave, v i is order one and the second differences become tie dominant dissipative terms. The dissipative term for the other equations are Constructed from similar formulas with the exception of the energy equation where the differences are of PH rather than PE. The purpose of this is to allow a steady state solution for which H remains constant.
4 . Implicit Schemes
In differential form the Euler equations are
-
1/2 or 1 the scheme is second or first With B order accurate in time. Note that if B = 1 the scheme reduces to a Newton iteration in the l i m i t At*-. Written for an n x n mesh, the equation (11) has a block tridiagonal form, requiring O h 4 ) operations for inversion. Because of this rapid growth of the operation count vith n, the Newton method appears to be uncompetitive except on very coarse meshes. The scheme is reduced to an Alternating Direction Implicit (ADI) scheme by replacing the operator of equation (11) by a product of two one-dimensional operators [SI, L61. (1 + BAt DxA)(I
+ $At DyB)bw + AtR
= 0
(12)
Equation (12) can be inverted in two steps as
and the operation count is now O(n2). The scheme is unconditionally stable for the linear scalar case if B > 112.
In transonic flow calculations, artificial dissipative terms m s t be added explicitly to capture shock waves without spurious oscillations. Since a constant coefficient dissipation formed from fourth differences cannot prevent wiggles near shock waves in fine meshes.171 it is necessary to use adaptive dissipation of blended second and fourth differences described in Section 3 unless one relies on upwind or total variation diminishing schemes. Although linear stability test shows that the scheme is unconditionally stable, explicit addition of dissipation restricts the stahilitv bound. In order to relieve this stability bound. implicit dissipation can be ineerted. After adding all dissipative term, the scheme takes the form in one dimension ~~
~
(I + 6 At DxA
-
At D2X
E:’)
+
E?)
4
Calculate the correction and update the solution on the coarse grid.
iv)
At DX ) 6w =
For the next coarser grid the residual is recalculated as
4
R4h
2 4 where Dx and D are difference operators that approximate a ZX /ax2 and a‘/ax‘ and E?), E(~) i
are coefficients of second and and c(4) e e fourth difference dissipations. One can construct various implicit schemes by appropriate selection of the coefficients of dissipation.
-
‘6
+
’4h +
PZh)
(18)
E(’)
where u means the updated value. On the first coarse grid, R& is replaced by IRh with the result that the evolution on the coarse grid is driven by the residuals on the fine grid. The evolution on the next coarser grid is driven by an estimate of whet the fine grid residuals would have been as a result of the correction on the first coarse grid.
-
I. With c ( 2 ) i const. and E(~), = 0, the scheme uses implicit dissipation of second differences with constant coefficient and requires block tridiagonal inversion.
-
-
11. With E ( z ) i E~(E(~)~) and s(4)i E2(cc4),) where El and E2 are functional relationships to be determined, the scheme uses implicit dissipation of blended second and fourth differences which match explicit dissipation with variable coefficients and requires block pentadiagonal inversion.
The entire process is repeated on successively coarser grids. Finally, the correction calculated on each grid is passed back to the next finer grid by bilinear interpolation. Since the evolution an a coarse grid is driven by residuals collected from the next finer grid, the final solution on the fine grid is independent of the choice of boundary conditions on the coarse grids. The surface boundary condition is treated in the same way on every grid, by using the normal pressure gradient to extrapolate the surface pressure from the pressure in the cells adjacent to the wall. The far field conditions can either be transferred from the fine grid, or recalculated by the procedure described in Section 2.
111. With ~ ( 4 =) E(&) ~,
= 0, and ~ ( 2 = ) ~ the scheme uses second difference dissipation in both explicit and implicit parts and requires block tridiagonal inversion.
5.
The interpolation of corrections back to the fine grid will introduce errors which cannot be rapidly expelled from the fine grid, and ought to be locally damped, if a fast rate of convergence is to be attained. Thus it is important that the driving scheme should have the property of rapidly damping out high frequency modes. The success of a multigrid method is critically dependent on the shape of the amplification factor. To see how the schemes can be adapted to meet this requirement we consider B model problem. w + w + w + E (Ax3w + Ay3w ) = 0 (19) t x y e xxxx YYYY Using the Yon Neumann stability test the amplification factor G ( E , n ) of B scheme is calculated. E and n denote wave numbers. h e can derive a functional relationship between the coefficients of dissipative terms which minimize the growth factor at the high frequency modes. For the above two dimensional model problem the relation is given by
Multigrid Method
In order to adapt the implicit schemes for a multigrid algorithm, auxiliary meshes are introduced by doubling the mesh spacing. Values of the flow variables are transferred to a coarser grid by the rule
I sh Wh/S2h
W2h =
(15)
where the subscripts denote v a l v e s of che mesh
spacing parameter, S is the cell area, and the sum is over the four cells on the fine grid composing each cell on the coarser grid. The rule conserves mass, mmentum and energy. The solution on a coarse grid is updated as follows. i) Calculate the correction and update the solution on the fine grid ii) Transfer the values of the variables to the coarse grid
Ei
-
Rt2h
+
P2h = IRh
(20)
-
(16)
-
Figure l(a) shows the amplfication factor of the scheme with 6 1 and without any dissipation. Note that the scheme is dissipative while it is not with 6 = 112. However, the scheme does not have good damping characteristic for the high frequency components. This can be improved by the addition of dissipation as shown in Figure l(b).
where R is the residual and t means the transferred value. The residual on the coarse grid is given by R2h = Rt2h
- 4)
2(2-m)A(-e
where A is the CFL number and m is the order of implicit dissipation. With this choice of coefficients the amplification factor G(S,n) should be zero at F n = n.
Collect the residual on the fine grid for the coarse grid. A forcing function is then defined as iii)
P2h = IRh
-
(17)
3
The terms added to the mass and mmetum equations are op(H-fL), apu(H-H.) and apv(H-H,), while that added to the energy equation is ap(H-&) to avoid a qusdrstic term in H which can be destabilizing. In mltigrid calculations an effective strategy is to include these terma only on the fine grid, and to increase the parameter a.
In this case the scheme damps out the very high frequency modes only. With the coefficients given by the equation ( 2 0 ) the scheme provides heavy damping over the broad range of the high frequency modes as shown in Figure l(c) with B = 112 and Figure l(d) with B 1. The scheme has better damping characteristic with B 1 than B 112. If the steady state solution is desired, one m y 1 without altering the final steady state. use 8
-
-
-
-
A further way of accelerating the convergence rate to a steady state is the use of a variable time step. The purpose of the variable time step is to increase the speed at which disturbances are propagated through the domain.
Several time steps might be used at each level of the mltigrid cycle, but it turns out that an effective mltigrid strategy is to use a simple saw tooth cycle (as illustrated in Figure 2), in which a transfer is made from each grid to the next coarser grid after a single time step. After reaching the coarsest grid the corrections are then successively interpolated back from each grid to the next finer grid without any intermediate Euler calculations. On each grid the time step is varied locally to yield a fixed Courant number, and the same Courant number is generally used on all grids, so that progressively larger time steps are used after each transfer to a coarser grid. In comparison vith a single time step on the fine grid, the total computational effort in one multigrid cycle is 1
+
1
+
7.
At a solid boundary the only contribution to the flux balance comes from the pressure. The normal pressure gradient aplan at the wall can be estimated from the condition that alat (pq,,) = 0, where qn is the normal velocity component. The pressure at the wall 1s then estimated by extrapolation from the pressure at the adjacent cell centers, using the known value of aplan.
* + ... < T 1
Boundary Conditions
4
The rate of convergence to a steady state will be impaired if outgoing waves are reflected back into the flow from the outer boundaries. The treatment of the far field boundary condition is based on the introduction of Riemann invariants for a one dimensional flow normal to the boundary. Let subscripts and e denote free stream values, and values extrapolated from the interlor cells adjacent to the boundary, and let qn and c be the velocity component normal to the boundary and the speed of sound. Assuming that the flow is subsonic at infinity, we introduce fixed end extrapolated Riemann invariants
plus the additional work of collecting residuals and interpolating the corrections. However, it has been found to be effective to use scheme I1 of section 4 an the fine grid, and to use scheme I11 on the coarse grids. Since the scheme on the coarse grids needs tridiagonal inversion while the scheme on the fine grid requires pentadiagonal inversion, the additional computational effort is less than 30L. The effective time step of a complete cycle using N grid levels is approximately (ZN-l)At where At is the time step on the fine grid.
-
and 6.
Enthalpy Damping
Re
Provided that the enthalpy has a conetanc value H, in the far field, it is constant everywhere in a steady flow, as can be seen by comparing the equations for conservation of mass and energy. If we set H = fL everywhere in the flow field throughout the evolution, then the pressure can be calculated from the equation
-
(22)
2c
qne
+
corresponding to incoming and outgoing waves. These may be added and subtracted to give
and c =
(Re
- RJ
(23)
where qn and c are the actual normal velocity c o w ponent and speed of sound to be specified in the far field. At an outflow boundary, the tangential velocity component and entropy are extrapolated from the interior, while at an inflow boundary they are specified a8 having free stream values. These four quantities provide a complete definition of the flow in the far field. If the flow is supersonic in the far field, all the flow quantities are specified at an inflow boundary, and they are extrapolated from the interior at an outflow boundary.
This eliminates the need to integrate the energy equation. The resulting three equation model still constitutes a hyperbolic system, which approaches the same steady state as the original system. An alternative modidfieation is to retain the energy equation, and to add forcing terms proportional to the difference between H and H, [SI. Since the space discretization scheme of Section 2 is constructed in such a way that H = fL is consistent with the Steady state solution of the difference equations, these terms do not alter the final steady state. Numerical experiments have confirmed that they do assist convergence. 4
8.
However, it seem that it should be possible to take larger time steps than those used in present calculations. Even though the implicit scheme is fast, the block inversions are expensive. The computational work may be able to be decreased by using a diagonalization of the blocks which needs scalar inversions [71.
Results
Two dimensional calerraltions have been performed to test the rmltigrid method with the Implicit schemes. The airfoil calculations were carried out using an +mesh, which was generated by the method of conformal mapping, with 128 intervals in the direction around the airfoil and 32 intervals in the radial direction. The mesh is shown in Figure 3.
Conclusion
It is shown that a multigrid method combined with the AD1 scheme can substantially accelerate the convergence of a time dependent hyperbolic system to a steady state. The solution is converged in less than 50 cycles and the achieved acceleration factor of 15 agrees well with the theoretical prediction. With the formulations here used, the steady state is entirely determined by the space discretization scheme on the fine grid. The discretization error of the present fine grid scheme is of second order in smooth regions of the flow. Further acceleration seems to be possible by achieving the potential of the implicit scheme.
Figure 4 shows the pressure distribution for the NACA 0012 airfoil at Mach 0.8 and zero angle of attack, while figure 5 shows Mach number contours of the flow field. The dashed line means the sonic line. Figure 6(a) and Figure 6(b) show the convergence histories for scheme I1 in section 4 on a single grid and with multiple grids. Convergence histories far scheme I are shown in Figure 7(a) and Figure 7(b) for comparison. In each figure two indicators of the convergence rate are presented. One is the decay of the logarithm of the error where the error is measured hy the root mean square rate of change of density on the fine grid. The other is the build-up of the number of points in the supersonic zone. For a transonic flow this indicates how quickly the supersonic zone develops and is a useful measure of the global convergence of the flow field. With 4 grid levels the flow field is fully converged in less than 50 cycles, starting from a uniform flow. Based on the build-up of the supersonic zone and the drag coefficient the nultigrid method needs 40 cycles for convergence while the single grid method requires 600 cycles. The number of cycles is reduced by a factor of 15. This dramatic acceleration of convergence is achieved by the use of the scheme I1 which solves the pentadiagonal system in the fine grid and the tridiagonal system in the coarse grids. With scheme I which uses the tridiagonal solver on all grids the acceleration due to rmltiple grids is less impressive. The slower convergence results from the use of a constant coefficient in the implicit dissipation. These results explain the importance of high frequency damping in rmltigrid through the matching of implicit and explicit dissipation.
References 1 . Jameson, A., "Acceleration of Transonic Potential Flow Calculations on Arbitrary Meshes by the Multiple Grid Method", AIAA 4th Computational Fluid bnamics Conference, Williamsburg, VA., AIAA Paper 79-1458, July 1979.
2.
Ni, R.H., "A Multiple Grid Scheme for Solving the Euler Equations", AIAA Journal, Vo1. 20, 1 9 8 2 ,
pp- 1565-1571.
3 . Jameson, A., "Solution of the Euler Equations by a Uultigrid Method", Princeton University, MAE Report No. 1613, June 1983. 4.
Jameson, A. and Baker, T.J., "Multigrid Solution of the Euler Equations for Aircraft Configurations', AIAA 22nd Aerospace Sciences Meeting, Reno, Nevada, AIAA Paper 84-0093, Jan. 1984.
Briley, W.R. and McDonald, H., "Solution of the Multidimensional Compressible NavierStokes Equations by a Generalized Implicit Method", J. Comp. Physics, Vol. 24, No. 4 , Aug. 1977. 5.
Other examples include lifting c a s e s . Pigvre 8 shows the transonic lifting case at Mach 0.8 and an angle of attack 1.25O. Figure 9 shows the solution of a subsonic flow over NACA 0012 airfoil at Mach 0.5 and an angle of attack of 3O. Figure 10 shows the case of KORN airfoil at a shock-free condition which is numerical verification of i'brawetz theorem that a shock free transonic flow is an isolated point. and that arbitrary small changes in boundary conditions w i l l lead to the appearance of shock waves. The drag be z e r o in shock free flows like subcritical flow or KORN airfoil c a s e , and the small calculated of drag coefficient is an indication of the level of discretization error. Finally the solution of the transonic flow 8t Mach 0.85 is shown in Figure
6. Beam, R., and Warming, R.F., "An lmplicit Factored Scheme for the Compressible Navier-Stokes Equations", AIAA Journal, Vol. 16, 1 9 7 8 , pp. 393-402. 7 . Pulliam, T.H., "Euler and Thin Layer NavierStokes Codes: ARCZD, ARC3D". CFD User's Workshop, The Univ. of Tennessee Space Institute, Tullshoms, Tennessee, Mar. 1984. 8 . Jameson, A., "Steady State Solution of the Euler Equations for Transonic Flow", Proc. Symposium on Transonic, Shock, and Multidimensional Flows, Madison, 1 9 8 0 , edited by R. E. Meyer, Academic Press, 1 9 8 2 , pp. 37-70.
11.
I n Practice the factored schemes have a stability bound. Possible sources of this are the interactions of the factorization error at very large time steps, the nonlinearity of the problem, and the treatment of boundary conditions.
5
1
6
?
r
€ Figure l ( a ) Amplification Factor without Dissipation.
6
-
I
Figure I(c) Amplification Factor with Dissipation and Optimal Coefficients. 8 112
-
\
Figure I(b) Amplification Factor with Dissipation.
6 = 1.
Figure I(d) Amplification Factor with Dissipation and Optimal 1 Coefficients. E
-
4
NRCQ IliiCH
CL
GRID
Figure 2 Multigrid Cycle
FL055
00i2 D.800
0.0
RLPHO CO
328x32
NCIC
0.0 0.0086 15
C* D.@ RESD.25iC-33
Figure 4 Surface Pressure Distribution
Saw Tooth
NRCR 0012
GRID
128 X
32
Flgure 5 Mach Number Contours
Figure 3 +Mesh for NACA 0012 7
a
I
I .
t
N M R 0032
IRCH CL
GRID
0.800
%PMR
0.3513 128x32
CD NClC
1.250 0.0230
150
Cn - C L:
[email protected]:-[>
Figure 8
Figure 10
$
a-
.......
, . .+
YRCP OD12 MRCn 0.500
CL
GRID
0.V269 128x32
RLPHR
CD NClC
3.000 0.0006 200
"(I
CM
Rncn
-D.DC:E
RfSO.2252-CI
0012
0.850
CL -0.0000 GRID 128x32
Figure 9
ntpm
0.0
LO NLX
O.OV71
150
Figure 11
9
CII 0.00i: RfS0.660:-:,i
