Appendix C RECENT DEVELOPMENTS IN MULTIGRID EFFICIENCY IN COMPUTATIONAL FLUID DYNAMICS A. Brandt
Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76 100, Israel
Very high efficiency has been attained by multigrid solvers for some types of problems, such as general uniformly elliptic problems. Our objective is to attain such an optimal performance for general fluid dynamics problems. A set of obstacles to achieving that goal is tabled below, along with a list of possible ways for overcoming each obstacle, their current state of development and references. The table includes staggered and nonstaggered, conservative and nonconservative discretizations of viscous and inviscid, incompressible and compressible flows at various Mach numbers, as well as a simple (algebraic) turbulence model and comments on chemically reacting flows. The listing of associated computational barriers involves: nonalignment of streamlines or sonic characteristics with the grids; recirculating flows; stagnation points; discretization and relaxation on and near shocks and boundaries; far-field artificial boundary conditions; small-scale features not visible on some of the coarse grids; large grid aspect ratios; boundary layer resolution; and grid adaption. C. I INTRODUCTION
The table below does not refer to a vast literature on multigrid methods in CFD (see for example [203]), in which enormous improvements over previous (single-grid) techniques have been achieved, but without adopting the systematic top multigrid efficiency (TME) approach. This approach insists on obtaining basically the same ideal efficiency to every problem, by a very systematic study of each type of difficulty, through a carefully chosen 573
574
MULTlGRlD
sequence of model problems. Several fundamental techniques are typically absent in the multigrid codes that have not adopted the TME strategy. Most important, those codes fail to decompose the solution process into separate treatments of each factor of the PDE principal determinant, and therefore do not identify, let alone treat, the separate obstacles associated with each such factor. Indeed, depending on flow conditions, each of those factors may have different ellipticity measures (some are uniformly elliptic, others are nonelliptic at some or all of the relevant scales) and/or different set of characteristic surfaces, requiring different combinations of relaxation and coarsening grocedures. The table deals only with steady-state flows and their direct multigrid solvers, i.e. not through pseudotime marching. Time-accurate solvers for genuine time-dependent flow problems are in principle simpler to develop than their steady-state counterparts. Using semi-implicit or fully implicit discretizations, large and adaptable time steps can be used, and parallel processing across space and time is feasible. The resulting system of equations (i.e. the system to be solved at each time step) is much easier than the steady-state system because it has better ellipticity measures (due to the time term; cf. Section 2.8.2), it does not involve the difficulties associated with recirculations, and it comes with a good first approximation (from the previous time step). A simple multigrid “F-cycle” at each time step can solve the equations much below the discretization errors of that step [78]. It is thus believed that fully efficient multigrid methods for the steady-state equations will also yield fully efficient and highly parallelizable methods for time-accurate integrations. C.2 TABLE O F DIFFICULTIES AND POSSIBLE S O L U T I O N S
Throughout the table, wherever appropriate, pointers to the subject and sections in the body of the book are provided. When no comment is made in the “Status” column of the table it usually means that the discussed “Possible Solution” is not known to have been tried.
Difficulty
0 Uniformly
elliptic scalar equation on uniform grids in general domains 0 Nonlinearity
Possible Solutions Multigrid cycles, guided by local mode analysis FMG (see Chapters 2 and 4)
Status
+
(1) FAS cycles (Section 5.3) (2) Continuation processes (to obtain good initial approximations), integrated into one-shot FMG algorithm (Section 10.2)
0 Fluid dynamics: general
Basic ideas are reviewed in Sec. 2 of [500]; see also Sections 8.6-8.9 above.
0 Nonscalar PDE systems
(1 ) General rules for the order of the intergrid transfer
(see Chapter 8)
operators are given in Sec. 4.3 of [66] with some more details in Sec. 3.3 [69] (2) A general approach to the design of relaxation is based on the operator principal matrix L and on the factors of det L Secs. 3.4 and 3.7 in [66]. In this approach a distribution matrix M and a weighting (or “preconditioning”) matrix P are constructed so that PLM is triangular, containing the factors of det L on the main diagonal (separated from each other as much as possible, to avoid the complication with “product operators” discussed next). This (if necessary-together with the technique described next), leads to decomposing relaxation into simple schemes for the (scalar) factors of det L . Many specific examples are given below (3) For systems of PDE which are of mixed type (elliptic-hyperbolic) another possibility is to sometimes introduce new unknowns in terms of which elliptic and hyperbolic parts are separated
TME demonstrated 1971 [56, 581 and rigorously proved [68,69]
( I ) Demonstrated 1975 [58, 3691. (2) Described in Sec. 8.3.2 of [66]
TME demonstrated in a number of cases (see many details below). TME proved for uniformly elliptic systems [68,69]
TME demonstrated for incompressible and compressible cases [381-3841
Difficulty 0
Product operator: an equation LU = f ,where L = L2L . Assume a relaxation process for L j is given, with the amplification factor p (Q) and the smoothing factor p i , ( j = 1, 2)
Possible Solutions Two alternative approaches: (1) Throughout the multigrid algorithm (not just in the relaxation sweeps), introduce an explicit new unknown function V , replacing the equation with the pair of equations L I U - V = 0 and L2V = f . The resulting smoothing factor is j i = max(bl, ,&). See Section 8.3 (2) Use V only as an auxiliary function in relaxation. That is: starting with u = L l u , where u is the current approximation to U , perform u2 sweeps on the equation L2V = f, yielding a new value 5. Then perform uI sweeps on the equation L l u = 6. The resulting amplification factor is p(Q) = p,(Q)"l + [l p~( Q ) ' l ] i ~(Q)-'p2(Q)'z/l( Q ) , where the Fourier symbols are defined by L j ( Q )= e-is.*/hLje'"x/h. Hence in scalar cases fi < by' b?
status (1) TME demonstrated for L = A' [77, 2391
+
0 Smoothingfor special CFD systems 0
Cauchy-Riemann on staggered grid
(; 2,)
0
= Stokes on staggered grid
M = distribution operator P = preconditioner (see discussion above) Two alternatives: (1)M=L, P = I (2) P = L , M = I (1) See Section 8.7 1 0
(1) TME demonstrated [72, 1301
(2) TME validated
(1) TME demonstrated [72, 1301
-a, ( 2 ) TME validated
(2)
a,
a,
-A
0
M=h;
Stokes, nonstaggered (1) Quasi-elliptic discretization
with averaging of the resulting pressure (2) h-elliptic discretization, e.g. -A
0
0
Nonconservative incompressible Euler, whose principal operator in 2D is U ' V
0
No modifications of the FMG algorithm is required, even in the quasi-elliptic case (as explained in Sec. 18.6 of [66]). In generalization to Navier-Stokes, pressure averaging is required of coarse-level results before their interpolation to the next finer level (whenever the coarse-level employs the quasi-elliptic discretization) (1) Employ cycle index y = 2 P , where p is the order of discretization, with 1 0 1
.=(.
1;)
0 (similarly 3D), on staggered grid, second- (or higher) order discretization
0
Low-Reynolds incompressible Navier-Stokes, staggered or not
(2) TME demonstrated (see Section 8.8)
(1) TME for first-order discretization using W-cycles shown in [72, 1301
u.v
(2) With the same M , for each of the momentum equations employ a relaxation scheme which is fast converging for the advection operator u . v (i.e. converging fast not only for high frequency, but also for smooth characteristic components; see discussion of advection below) (3) Use canonical variable ( u , u , P ) on staggered grid, where P = (u2 v2)/2 p . Upwind only P , use central discretization for ( u . u ) . Relaxation is marching for P , and weighted (preconditioning) for ( u , u ) Fully analogous to Stokes solvers: just replace A in L by Q = - R - ' A + u . v. Seealso Sec. 8.7
+
0
( 1 ) In a quasi-elliptic approach, TME demonstrated (Sec. 18.6 of [66], and [83]
Analogous to the staggered case; e.g. 1 o -a,'k
+
(2) TME demonstrated for 2D entering flows with second-order discretization [86] and for recirculating flows with first-order discretization [87] (3) TME in [381-3831
TME demonstrated 1978 [72, 1301
Difficulty 0
0
High-Reynolds incompressible Navier-Stokes, staggered
Compressible Euler, nonconservative, on staggered grid: the subprincipal operator on (u1, u ~u3, , p , 6 , p ) is
L=
v)*
det L = p 5 ( u .v ) ~ ( ( u . - a 2 A ) , a = ( a p / a p ( p / p * ) ( a p / a s ) ) ' / *is the sound speed, p , E , p defined at cell centers, u i-at center of cell faces perpendicular to the i th coordinate
+
0
0
2D Compressible Euler, nonconservative and conservative, staggered grid, using canonical variables ( u , u , S, H ) . Structured and unstructured grids
2D/3D incompressible and compressible Euler: canonical variables in which velocities are replaced by vector potential representation. Nonstaggered structured and unstructured grid
Possible Solutions Fully analogous to incompressible Euler (outside boundary layers: see discussion on such layers below): just replace u . v everywhere with Q . See also Sec. 8.8
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0
0 0
0 0
0 0
1 0
Status TME demonstrated for firstorder discretization on staggered [72, 1301 and nonstaggeredgrids (Sec. 19.5 in [66]), and for second-order staggered discretization [86]
-p(u'v)a,
-p(u.v)a* -p(u'v)a3
P*(U'V)*
The advection and full potential operators are each relaxed by one of the approaches described for them below (in the section on nonelliptic operators. The semicoarsening described there would then be used as an inner multigrid cycle for relaxing one factor of the determinant, to be distinguished from the outer multigrid cycle, which can usefull coarsening) Use ( u , u ) at cell edges, H at middle of cell, S at vertices. Upwind only S at momentum equations. Relax S, H by marching. ( u . u ) by a weighting relaxation. Crocco's form is used here to define relaxation All variables at cell nodes. Relax hyperbolic quantities using marching. Relax vector potential using point Gauss-Seidel
TME in [382-3841
TME achieved (unpublished) for interior and exterior flows in 2D, interior in 3D
0
Compressible Nmier-Stokes, nonconservative. The subprincipal operator on (u,, u2. u,, p , E , p ) is
hall -ha2] -ha31
Q,
-
-hal2 Q , - ha,, -ha32
L, =
(1) Where 1,k , K (2) Otherwise use
1
-ha23 - ha33
Q,
PZ&
o
o o
o o
a3
Qo
0
0
ha,
+
+
where Qa = - a A ,u, A = pu . V, 1 = A ( 2 1 3 ) K~ ~= k / c , (coefficient of thermal conductivity divided by the specific heat at constant volume), det L, = Q’, det L,, where L , is the “core operator”
0
At standard conditions of laminar air flow the Prandtl number y k / K x 0.72; for turbulence y p / K x 0.9, with y = c p / c , = 1.4 Nonconservative nonstaggered Euler and NS
