though Chorin did not view his method as local preconditioning, his approach adds a time .... 101. 102. 103. Mach number g. Figure 2: Wavefront sti ness, g, versus Mach number for Euler (solid line), ...... explicit, upwind solutions over a NACA 0012 airfoil were obtained, showing that .... (Also, ICASE Report No. 80-6,. 1980).
Local Preconditioning: Manipulating Mother Nature to Fool Father Time David L. Darmofal Bram van Leery 1 Introduction A common strategy for solving steady equations is to march the associated unsteady equations in time until the solution converges to a stationary result. In the case of the compressible Euler equations, this approach has the important advantage of transforming the problem from a mixed hyperbolic-elliptic problem in the steady state to a strictly hyperbolic problem in the transient stages. Unfortunately, this approach also introduces any sti ness due to the unsteady equations into the convergence process for the steady equations. For example, in a nearly incompressible ow, the speed of sound is signi cantly faster than the local ow speed. As a result, acoustic disturbances propagate much faster than convective disturbances (such as entropy and vorticity) which only travel with the local velocity. For explicit time-marching codes, this sti ness from disparate propagation speeds can signi cantly slow convergence to a steady state: the fast modes set the maximum allowable timestep while the slow modes set the number of iterations needed for a disturbance to convect out of the computational domain. In order to accelerate convergence for time-marching methods, one may introduce a preconditioner into the unsteady equations, @u + P(u) r(u) = 0; @t
where u represents the state vector, r is the spatial residual which the calculation is attempting to drive to zero, and P is the preconditioner. In this review, we will concentrate on local preconditioners for the two-dimensional Euler equations. Local preconditioners are evaluated using purely local information from y
Aerospace Engineering, Texas A & M University, College Station, TX 77843-3141 Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109-2140
1
2 DARMOFAL, VAN LEER the cell or node where the spatial residual will be preconditioned. While the preconditioner modi es the unsteady equations, the steady solution of the analytic equations is unmodi ed assuming P is a positive de nite matrix. As we will discuss, discrete implementation of a preconditioner often does require a subtle modi cation to the spatial residual; fortunately, this modi cation usually has the bene t of increasing the accuracy of the discretization. In this paper, we will detail the process of designing a local preconditioner for the 2-D Euler equations. While the examples will be speci c to these equations, the basic design considerations and analysis techniques are applicable to many other systems of equations. Then, we will provide examples of what has been achieved with local preconditioning in practice. Finally, we will indicate remaining areas that still need development if local preconditioning is to achieve its full potential.
2 Design considerations The rst attempt at local preconditioning can probably be associated with Chorin's[6] method of arti cial compressibility for incompressible ows. Although Chorin did not view his method as local preconditioning, his approach adds a time derivative of pressure to the incompressible continuity equation which is equivalent to local preconditioning. Since this initial e ort at modifying unsteady ow equations to calculate steady ows, the design of local preconditioners has been continually extended and re ned. Presently, the design of a local preconditioner is a balance of widely di erent, sometimes competing critera. In the following, we describe the various design considerations that one faces when deriving a local preconditioner for the 2-D Euler equations. The analysis of local preconditioners for the 2-D Euler equations is considerably simpli ed by utilizing the symmetrizing variables[35, 1] which are de ned as, dw
T
"
#
= dp ; dq; dr; dS ; c
where dq and dr are the change in the streamwise and normal components of velocity, dp is the change in the static pressure, dS = dp ? c2 d is proportional to the change in entropy, is the density, and c is the speed of sound. Using this basis, the preconditioned Euler equations may be written as, @w + PA @ w + PB @ w = 0; @t @ @
where
A
2 6 = c 664
3
M 1 0 0 1 M 0 0 777 ; 0 0 M 0 5 0 0 0 M
B
2 6 = c 664
(1) 0 0 1 0
0 0 0 0
1 0 0 0
03 0 777 ; 05 0
LOCAL PRECONDITIONING
3 with the Mach number, M = q=c, the streamwise and normal coordinates, (; ), and q is the local speed.
2.1 Essential requirements An essential requirement on a local preconditioner is that it should not reverse the propagation direction of any waves. A preconditioner that violates this requirement would result in the incorrect application of boundary conditions since incoming and outgoing waves would reverse roles. This criterion can be met by forcing P to be positive de nite (although not necessarily symmetric). In this case, the preconditioned residual is a positively weighted sum of contributions, which does not reverse the time development. Although positive de niteness is an essential criterion, it does not place severe limitations on the preconditioner. The typical approach for assuring a positive de nite preconditioner is to check it after satisfying other desirable constraints. When preconditioning the Euler equations, the result should also be a wellposed, hyperbolic set of equations. A necessary condition for well-posedness is that the energy of the system remains bounded as time advances. A suf cient but not necessary condition to guarantee bounded energy is to force the preconditioned system to be symmetrizable. The preconditioned system is symmetrizable if a symmetric, positive de nite matrix, Q, exists such that,
Q @@tw + QPA @@w + QPB @@w = 0; and QPA and QPB are symmetric. If a Q exists which satis es these requirements, then the system is easily shown to be stable in the norm, wT Qw, ignoring the in uence of boundary conditions. Similar to positive de niteness for P, symmetrizability is usually veri ed at the end of the preconditioner design process. A number of important hyperbolic systems of conservation laws, including the Euler equations, are symmetrizable. Godunov[11] showed that this implies the existence of an entropy function satisfying an additional conservation law. The preconditioned Euler equations, however, have lost the conservation form, so Godunov's result is not applicable to these. Finally, the preconditioner should also satisfy certain smoothness conditions throughout all ow regimes, especially at the break point of the steady Euler equations, the sonic point (M = 1). In the development of preconditioners, the subsonic and supersonic regimes have to be treated separately because of the qualitative change in the propagation direction of the acoustic waves. A mismatch in the preconditioner branches across M = 1 could result in poor sonic point capturing abilities and stalled convergence.
DARMOFAL, VAN LEER
4
2.2 Equalization of long-wave time scales
As discussed in the Introduction, the Euler equations have an inherent sti ness at low Mach numbers resulting from the disparate propagation speeds of the convective and acoustic modes. Similar arguments may be made for compressible ow near sonic conditions. Local preconditioning can be used to equalize these propagation speeds or time scales. Furthermore, for viscous ows, dissipation time scales now enter and could also be accounted for in preconditioning. Note, the analysis and design process to equalize time scales is performed at the continuous, partial di erential equation level. Since the discrete numerical scheme should correctly approximate long-wave behavior, we expect the long-wave properties of the preconditioner to be reproduced in the numerical solutions. However, short waves or high frequencies are generally not correctly represented by the discretization. Thus, the equalization of wavespeeds concerns long-wave time scales only. A rst step in analyzing the preconditioned wave properties is to calculate the propagation speeds, , for plane wave solutions of the form w(x; y; t) = w(x cos + y sin ? t), where is the angle of the wave propagation direction relative to the streamwise direction. Substitution of plane waves into Equation 1 results in an eigenvalue problem for , (?I + PA cos + PB sin ) r = 0; where r is the corresponding right eigenvector. Without preconditioning, the 2-D Euler equations have eigenvalues 1;4 = q cos c; 2;3 = q cos :
While preconditioning design can proceed directly from the eigenvalues, i, a caveat must be given with this approach. Speci cally, a typical measure of sti ness from wave speeds would be to use the ratio of maximum to minimum magnitudes, i.e., maxi max ji( )j : mini min ji( )j However, as is clear from the 2-D Euler equations, this approach is fruitless because 2;3 = 0 whenever = 90 indicating in nite sti ness regardless of the local Mach number. Similar troubles exist when considering the preconditioned system. To remedy this trouble, Van Leer et al[40] use the envelope of plane waves passing through one point at some instant; this in fact is the wavefront emitted by a point disturbance. The point-disturbance envelope is given by, g g
!
"
? sin = cos sin cos
#
!
( ) ; 0 ( )
LOCAL PRECONDITIONING
5 where (g ; g ) is the location of the envelope. In fact, this approach is equivalent to using the group velocity instead of the phase velocity (i.e. ) where (g ; g ) is the group velocity vector for a plane wave of direction . A well-known property of the group velocity is that the energy of a disturbance propagates at the group velocity. Thus, optimizing the point-disturbance envelope is equivalent to optimizing the speeds at which energy propagates. Following Van Leer et al, we can also use the point disturbance envelope to visually inspect the wave-propagation properties of the system. The wavefronts for the Euler equations are shown in Figure 1 for M = 0:1, 0:5, 0:9, and 1:3. Note: all plots have been scaled by the largest wavespeed. The convective waves corresponding to 2;3 are clearly seen in all plots as the isolated point disturbances. By contrast, the acoustic wavefronts propagate with the local convective velocity, q, while spreading radially with the acoustic velocity, c. At low Mach numbers, we can see the previously mentioned sti ness resulting from the fast acoustic modes and the slow convective modes. At the higher Mach numbers, the sti ness is also present but in this case, the fast/slow modes are the down/upstream propagatingq portion of the acoustic modes. Employing the group velocity magnitude, g = g2 + g2, an appropriate measure of wave speed sti ness is g , de ned as, g
maxi max jgi( )j : mini min jgi( )j
Without preconditioning, the 2-D Euler equations have a wave-propagation sti ness given by, g = (M +1)= min(M; jM ? 1j). A plot of g is shown in Figure 2. As expected, g indicates sti ness problems are present for M ! 0 and M ! 1. While not the rst preconditioner for compressible ows, the Van Leer-LeeRoe preconditioner[40] is the rst valid for all Mach numbers. Furthermore, for the class of symmetric preconditioners, P = PT , Van Leer-Lee-Roe can be proven to produce the optimal wavefronts[17] with the smallest sti ness, g , for all Mach numbers. Speci cally, this preconditioner is given by,
Pvlr
2 6 6 = 66 4
2 2 M 2 M
?
0 0
3
? M 0 0 7 2
2
+ 1 0 0 77 ; 0 0 75 0 0 1
q
where = j1 ? M 2 j and = min( ; =M ). For subsonic ows, the eigenvalues produced by this preconditioner are
p
1;4 = q 1 ? M 2 cos2 2;3 = q cos
)
for subsonic Pvlr :
DARMOFAL, VAN LEER 1
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(b) M = 0:5 (d) M = 1:3 Figure 1: Wavefronts for 2-D Euler. Wavefronts have been scaled by largest wavespeed. For supersonic ows, the eigenvalues are p
1;4 = c M 2 ? 1 cos sin 2;3 = q cos
)
for supersonic Pvlr :
The resulting wavefronts for Pvlr are shown in Figure 3 and the sti ness in Figure 2. From these plots, it is evident that the low-Mach-number sti ness has been completely removed. In subsonic ow, the acoustic wavefront is now an ellipse centered at the origin. The minor p axis of the ellipse is in the streamwise direction and has a semi-width of q 1 ? M 2 . Thus, as sonic conditions are reached, this ellipse collapses and the in nite sti ness still exists, although it is signi cantly reduced from the unpreconditioned case. In supersonic ow, the acoustic disturbances collapse on to two points that convect in the direction of the Mach angles relative to the ow, at a rate equal to the ow speed, q, meaning
LOCAL PRECONDITIONING
7
3
10
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10
0
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1
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3
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4
Figure 2: Wavefront sti ness, g , versus Mach number for Euler (solid line), optimal preconditioner (dashed line), block-Jacobi preconditioner (dash-dotted line), and Choi-Merkle/Weiss-Smith preconditioner (dotted line). Note: Van Leer-Lee-Roe preconditioner is an optimal preconditioner.
p
perfect preconditioning. Analytically, we nd that g = 1= 1 ? M 2 for M < 1 and g = 1 for M > 1. Clearly, a signi cant improvement in time-scale sti ness has been made by local preconditioning. Turkel[36] was among the rst researchers to generalize the method of Chorin. In his analysis, he proposed preconditioners utilizing the primitive variables, p, u, v , plus either entropy, S , or density, . In the symmetrizing variables, the general form of the Turkel preconditioner is, 3 2 t2 0 0 0 7 6 Pt = 664 ? 0tM 01 10 00 775 ; 0 0 0 1 where t and t are free parameters. Turkel's preconditioner is most widely used to remove sti ness at low Mach numbers[38]. In the limit as M ! 0, if t ! 1 and t ! M , then g ! 1. The Turkel preconditioner can also be extended to generate the optimal wavefronts over all Mach numbers similar to the Van LeerLee-Roe preconditioner. For subsonic ow, the optimal wavefronts are achieved with t = M and t = 1 + t2[17]. Unfortunately, when used in conjunction with an upwind or matrix dissipation scheme, the optimal form of Pt results in discrete eigenvalues which fall into the unstable right half plane for higher subsonic Mach numbers[7, 23]. Thus, this speci c form of the preconditioner cannot be used in practice for transonic ows. To avoid this problem, Turkel et
DARMOFAL, VAN LEER 1
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(b) M = 0:5 (d) M = 1:3 Figure 3: Wavefronts for optimal preconditioning, Popt (includes Pvlr ). Wavefronts have been scaled by largest wavespeed. al[38] de ne t and t to return to the unpreconditioned case above a cut-o Mach number. Although the wavefronts produced by Pvlr are known to be optimal for symmetric preconditioners, no proof exists that the optimality extends to general non-symmetric preconditioners. For supersonic ow, although other choices may exist, since g = 1, these wavefronts may justly be called optimal. In subsonic
ow, we are left with the possibility for improvement. However, all evidence points to these wavefronts as being optimal in the subsonic ow also. For now, we will label a preconditioner as optimal when it produces the same wavefronts as Pvlr . Since Pvlr and Pt can both be made optimal, optimal preconditioning is obviously non-unique. Another illustration of preconditioner non-uniqueness is that given a preconditioner P for the symmetrizing variables, the same eigenvalues are found when preconditioning by its transpose, PT . Thus, given an
LOCAL PRECONDITIONING
9
optimal preconditioner, its transpose is also optimal. Recently, we have been able to make signi cant progress on deriving the general form of all optimal preconditioners. Speci cally, two families of optimal preconditioners exist. One family is most closely related to the Van Leer-LeeRoe preconditioner. The inverse of this optimal family is,
Popt?
1
2 66 66 = 666 66 4
3 p C 07 A MA ? M 1 ? M C 77 1 1 0 77 M 77 ; p 0 1 ? M C 77 G M (C + G) 0 5 2
2
2
(2)
2
0 0 0 1 where A, C , and G may be chosen to satisfy other design constraints. Obviously, the transpose of this preconditioner is also optimal. We note that the symmetric form of this preconditioner is given by C = G = 0 and A = ( + 1)=M and results in the Van Leer-Lee-Roe preconditioner. An interesting point is that the optimal Turkel matrix is not a member of this family and, thus, must be a member of the other family of optimal preconditioners. Unfortunately, a clean algebraic description is not presently available for this second family of optimal matrices. Other preconditioners that have been developed include the D.Choi-Merkle inviscid preconditioner[4], the Y.Choi-Merkle viscous preconditioner[5], and the Weiss-Smith preconditioner[44]. These preconditioners have been developed only for low Mach number ows. Transforming the preconditioners to the symmetrizing variables gives 2 0 0 ?cmv 3 6 1 0 0 777 ; Pcm=ws = 664 00 0 1 0 5 cmi ( ? 1) 0 0 1 where is a user-de ned parameter. For the D.Choi-Merkle inviscid preconditioner, cmi = 1 and cmv = 0. For the Y.Choi-Merkle viscous preconditioner, cmi = 0 and cmv = 1. And, for the Weiss-Smith preconditioner, cmi = cmv = 0. In fact, the Weiss-Smith preconditioner is pa member of the Turkel family of preconditioners (speci cally, t = 0 and t = ). For subsonic inviscid ows, the recommended choice is = M . The eigenvalues for these three preconditioners are identical and given by q ) ; = q(1 + ) cos q (1 ? ) cos + c for subsonic Pcm=ws: ; = q cos Analysisp of the wavefronts shows that at low Mach numbers with = M , p g ! ( 5 + 1)=( 5 ? 1). Thus, while the Choi-Merkle/Weiss-Smith preconditioner does not achieve the optimal conditioning at low Mach numbers, it does 2
2
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(b) M = 0:5 (d) M = 1:3 Figure 4: Wavefronts for Choi-Merkle and Weiss-Smith preconditioning, Pcm=ws, with = min(M ; 1). Wavefronts have been scaled by largest wavespeed. 2
eliminate the in nite sti ness of the Euler equations at low speeds. At higher Mach numbers, the preconditioner reverts back to the eigenvalues of the unpreconditioned Euler equations, although, above Mach numbers of approximately 0:4, these preconditioners have worse wavespeed conditioning than the unpreconditioned Euler equations. Note, the Choi-Merkle inviscid and Weiss-Smith preconditioners become the identity matrix as M ! 1 but the Choi-Merkle viscous preconditioner does not. For supersonic ows, we assume that = 1, thus the eigenvalues return to the eigenvalues of the Euler equations without preconditioning. These conclusions are evident in the wavefronts and the values of g shown in Figures 4 and 2. A more classical suggestion for local preconditioning is the use of the Jacobi block[26, 27, 2, 3, 29] induced by upwind or matrix-dissipation discretizations. If an upwind scheme is used to discretize the spatial operator, the ux across a
11
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(b) M = 0:5 (d) M = 1:3 Figure 5: Wavefronts for block-Jacobi preconditioning, Pj . Wavefronts have been scaled by largest wavespeed. cell face can be written as, ^f = 1 (fright + fleft ) ? 1 A^ c (uright ? uleft ) ; (3) 2 2 where A^ c is the face value of the Jacobian matrix for the conserved uxes, and A^ c has the same eigenvectors as A^ c, but its eigenvalues are the absolute values of the eigenvalues of A^ c. The block-Jacobi preconditioner is the block found on the diagonal in the coecient matrix of the linearized spatial operator. For rectangular cells, this preconditioner is given by
Pj ? = tmax 1
!
jAcj + jBcj ; x y
where is a factor that may be used to scale the size of the spatial footprint depending on the particular discretization. The timestep, tmax , is included
12 DARMOFAL, VAN LEER for dimensional consistency but, upon implementation, cancels with the tmax used in the timestep. The main advantage of the block-Jacobi preconditioner is its excellent clustering of the discrete eigenvalues (see Section 2.3) and its robustness (see Section 2.4). Unfortunately, the block-Jacobi preconditioner does little to improve the long-wave propagation speeds. The wavefronts for Pj are shown in Figure 5. At low Mach numbers, the entropy mode has been relocated to (1; 0), however, the vorticity mode still sits near the origin, and, as M ! 0, the sti ness is still in nite. At higher Mach numbers, the block-Jacobi system improves on the sti ness by increasing the speed of the upstream-moving acoustic waves. Although not as evident at low Mach numbers, the wavefronts for the vorticity mode are now clearly seen to be non-physical, taking on a triangular shape. In contrast, the optimal wavefronts in Figure 3 appear quite plausible | a direct consequence of the continuum-based design. Regardless, while optimal preconditioners improve the wave propagation over that of the Euler equations, they are not a substantial improvement over block-Jacobi at higher Mach numbers.
2.3 Discrete eigenvalue clustering
While long-wave properties of the preconditioned equations may be analyzed from the continuous partial di erential equations, short waves require analysis of the discrete equations because these small wavelengths are generally not well represented by numerical schemes. The common approach for analyzing the discrete properties of a scheme is via the Fourier footprint of the discrete spatial operator which is the locus of its eigenvalues in the complex plane. Speci cally, after linearization about a constant state and substitution of a discrete Fourier mode of the form, uj;k = u^ ei jx ky , the spectral ampli cation matrix of the numerical approximation may be found for any discrete wavenumber combination, (x ; y ). The discrete wavenumbers have a range from ? to . The use of the Fourier footprint to analyze the smoothing abilities of preconditioned Euler and Navier-Stokes equations can be found in several other references[42, 19, 2, 9, 20, 3, 29]. One goal of convergence acceleration is to quickly damp as many eigenmodes of the Fourier footprint as is numerically possible. For example, an e ective multigrid algorithm requires e ective smoothing of all modes that do not exist on subsequently coarser grids. For full-coarsening multigrid, this requires e ective smoothing of high-low, low-high, and high-high frequency combinations where a high-frequency wavenumber ranges from =2 jj . For semi-coarsening, only the high-high frequency components need e ective smoothing. As we will show, local preconditioning can often cluster the Fourier footprint, allowing signi cantly enhanced damping. In particular, we concentrate on the highhigh frequency locus and show that without appropriate preconditioning, many eigenmodes will be poorly damped. (
+
)
LOCAL PRECONDITIONING
13 When a continuum preconditioner such as Van Leer-Lee-Roe, Turkel, or Choi-Merkle/Weiss-Smith is applied to an upwind scheme, the dissipation must be modi ed such that ^ ^f = 1 (fright + fleft ) ? 1 M ^ ? P^ ? P^ A^ cos + B^ sin M (uright ? uleft ) ; (4) 2 2 ^ is the transformation matrix from the conserved state vector to the where M symmetrizing variables, and is the angle of the grid normal with respect to the streamwise direction. This modi cation is necessary for stability as well as to improve the accuracy of the discretization at low Mach numbers[40, 17]. For the Van Leer-Lee-Roe preconditioner, this modi ed dissipation matrix is too complicated to derive a simple, analytic representation; thus, common practice is to modify the dissipation even further, to ^ ^f = 1 (fright + fleft )? 1 M ^ ? P^ ? P^ A^ j cos j + P^ B^ j sin j M (uright ? uleft) : 2 2 (5) The high-high frequency footprint of the upwind discretized Euler equations at low Mach numbers is very poorly clustered. Figure 6 shows the high-high frequency content of the Fourier footprints for M = 0:1 with a cell-aspect ratio, AR = x=y = 1, and the ow aligned with the grid ( = 0). As an illustration, the ampli cation-factor contours are also overlayed for an optimal two-stage multistage damping scheme as derived by Lynn[18]. Without preconditioning, a group of high-frequency Fourier modes is seen to cluster near the origin where the ampli cation factor must approach one. This clustering is a direct consequence of the disparate propagation speeds of the Euler equations at low Mach numbers. By comparison, the Van Leer-Lee-Roe preconditioner has eigenvalues that are well removed from the origin. While the clustering has improved, better clustering can be achieved with only a small change in the original preconditioner. We note that two distinct clusters of eigenvalues are observed with the original Pvlr . In fact, these families of eigenvalues are due to the presence of acoustic and convective waves. Lynn[18] has shown that by scaling back the propagation speeds of the acoustic waves, the Fourier footprint may be perfectly clustered for the highest frequency waves, x = y = . This scaling, developed by Lee[15, 14], can be accomplished by rede ning as = =(ARq + ), where 1
1
1
1
P jux + vy j ARq = Pk jvx k ? uyk j : k k k 4
4
=1
=1
Physically, ARq can be interpreted as the cell-aspect ratio de ned in the ow direction, i.e. the ratio of the streamwise to normal cell lengths. Using this de nition, high-high frequency modes for all Mach and cell-aspect-ratio combinations are clustered at the same location in the complex plane. For semi-coarsening
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1
(d) Van Leer with modi ed Figure 6: High-high Fourier footprint for M = 0:1, AR = 1, = 0. Contours of ampli cation factor between 0 and 1 with increments of 0.1 are overlayed. (b) block-Jacobi
multigrid, this improved clustering is essential since cell-aspect ratios can be extremely large. The improved clustering can be veri ed in Figure 6. The highhigh frequency locations for the same conditions but with a cell-aspect ratio, AR = 100 are shown in Figure 7. At this high aspect ratio, the improvement over both the unpreconditioned Euler and the original Van Leer preconditioner is drastic and justi es the modi cation. Allmaras[2] has performed extensive analysis of the block-Jacobi preconditioner and found it has excellent high-high frequency clustering over all ranges of cell-aspect ratio, Mach number, and ow angle. The block-Jacobi footprints have been included in Figures 6 and 7. An important di erence to note is that the block-Jacobi preconditioner has a family of modes which cluster about the real axis. This e ect is also related to the large di erence in propagation speeds at low Mach numbers, which the block-Jacobi preconditioner does not alleviate.
15
2
2
1.5
1.5
1
1
0.5
0.5 Imaginary
Imaginary
LOCAL PRECONDITIONING
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2 −3
−2.5
−2
−1.5
−1 Real
−0.5
0
0.5
−2 −3
1
2
1.5
1.5
1
1
0.5
0.5 Imaginary
2
0
−0.5
−1
−1
−1.5
−1.5
−2.5
−2
−1.5
−1 Real
−0.5
0
0.5
−2
−1.5
−1 Real
−0.5
−1 Real
−0.5
0
0.5
1
0
−0.5
−2 −3
−2.5
(c) Van Leer with original
(a) No preconditioning
Imaginary
0
1
−2 −3
−2.5
−2
−1.5
0
0.5
1
(d) Van Leer with modi ed Figure 7: High-high Fourier footprint for M = 0:1, AR = 100, = 0. Contours of ampli cation factor between 0 and 1 with increments of 0.1 are overlayed. (b) block-Jacobi
In contrast, the Van Leer-Lee-Roe preconditioner spreads these modes away from the real axis such that they become better clustered with the other error modes.
2.4 Healthy eigenvector structure A very important aspect of preconditioner design is maintaining a healthy eigenvector structure. The rst study of the importance of eigenvectors on preconditioner design was performed by Darmofal and Schmid[7]. In this work, they show how the eigenvector orthogonality directly relates to the possibility of transient ampli cation for small disturbances even in systems where all eigenvalues indicate stability. Speci cally, consider the evolution of the following
DARMOFAL, VAN LEER
16 linear problem,
du + iLu = 0; dt
where L is an N N matrix. In preconditioning, this would represent the Fourier transformed, preconditioned spatial operator. The solution of the above equation can be written compactly using the matrix exponential as, u(t) = exp (?itL) u ; where u is the initial condition. The maximum energy ampli cation G(t) at a given time can be expressed as the norm of the matrix exponential, G(t) = jj exp (?itL) jj = jjR exp(?it )R? jj; where R is the matrix of right eigenvectors of L and is the diagonal matrix of eigenvalues. The ampli cation matrix can be bounded for all times by, 1 < Gmax < (R) sup j exp(!max t)j; where (R), the condition number of the eigenvector matrix, is de ned as (R) = jjRjjjjR? jj, and !max is the largest positive imaginary part of all eigenvalues. For hyperbolic systems such as the Euler equations, the latter is usual zero since the eigenvalues are strictly real. Thus, the bound on the largest ampli cation is given by (R). For an orthogonal eigenvector structure, (R) = 1, and no energy growth is possible. Greater departure from orthogonality results in larger (R) and larger possible transient growth. Without preconditioning, the symmetrizing variables are an orthogonal system since the matrices A and B are symmetric. However, after preconditioning, PA and PB are no longer guaranteed to be symmetric; thus, in general, preconditioning introduces the possibility of transient growth where previously none existed. Unfortunately, Darmofal and Schmid found that continuum-based preconditioners such as Turkel, Van Leer-Lee-Roe, and Choi-Merkle are all subject to severe transient growth as M ! 0. In fact, they found that the nondimensionalized energy growth rate at t = 0 is proportional to 1=M for these preconditioners as M ! 0. Previously, many researchers had discovered the extreme lack of robustness at stagnation points; however, the exact cause of this instability had not been known. For example, Turkel[38] employs a cut-o in his t de nition similar to t = max M ; Mref where Mref is some reference of freestream Mach number, and is a user-de ned parameter typically between 1 and 3. This e ectively avoids t going to zero which was shown by Darmofal and Schmid to limit the non-orthogonality. However, this x is problem dependent and, furthermore, complicates the analysis of preconditioners by introducing a non-local parameter. Another avenue to improve the eigenvector structure of the preconditioned equations is to enforce some measure of orthogonality from the beginning of the 0
0
1
1
+
2
2
2
LOCAL PRECONDITIONING
17 design. The strictest requirement would be to enforce orthogonality under all conditions. This is equivalent to requiring PA and PB to be symmetric. The preconditioner that results from requiring complete orthogonality is too limited and cannot signi cantly alter the long-wave time scales. A weaker restriction is to require orthogonality only in the streamwise direction. Lee and Van Leer[16, 14] have pursued this approach and found an optimal subsonic preconditioner with orthogonal eigenvectors in the streamwise direction. Speci cally, this preconditioner has the form 3 2 M2 M ? f 0 0 66 ? M f + 1 0 0 77 Pvl = 664 0 0 0 775 ; 0 0 0 1 where ? : f = 21M+ M The preconditioner has been coined Pvl , i.e. Van Leer 96, as it was developed by Van Leer during the summer of 1996. This preconditioner smoothly connects with the Van Leer-Lee-Roe preconditioner as M ! 1, thus, the supersonic Van Leer preconditioner can be used for M > 1. Note, Pvl is a member of the Popt family of preconditioners; furthermore, Pvl is the only positive de nite member of Popt which is orthogonal in the streamwise direction. 96
2
2
96
96
96
2.5 Accuracy preservation
As M ! 0, a well-known problem with many compressible nite volume techniques is an extreme loss of accuracy in addition to slow convergence[43]. The problem can be traced to a poorly balanced dissipation matrix at low Mach numbers[8]. In this analysis, the symmetrized Euler equations are rst nondimensionalized by a reference density and ow speed. Gustafsson and Stoor[12] have shown that using this non-dimensionalization, dp c = O(M ), as the Mach number decreases. Thus, the entire symmetrizing state vector behaves as w = [ O(M ); O(1); O(1); O(1) ]T : Similarly, the A and B matrices are found to be of the following order, 2 0 2 O(1) O(1=M ) 0 0 O(1=M ) 0 3 0 3 7 7 6 6 A = 664 O(10=M ) O0(1) O0(1) 00 775 ; B = 664 O(10=M ) 00 00 00 775 : 0 0 0 0 0 0 0 O(1) Combining these results it is easy to show that
A @@w + B @@w = [ O(1=M ); O(1); O(1); O(1) ]T :
18 DARMOFAL, VAN LEER For a rst-order upwind or matrix dissipation scheme, similar analysis shows that jAj @
2
w + jBj @ w = [ O(1); O(1=M ); O(1=M ); O(1) ]T :
@
2
@
2
2
Thus, except in the entropy equation, the Euler and dissipative terms are mismatched as M ! 0. When a local preconditioner is applied, a balance can be achieved between the preconditioned ux and dissipation terms. Fiterman et al[8] have shown that a proper match occurs if
2 O(1=M ) O(1=M ) O(1=M ) O(1=M ) 3 6 (1=M ) O(1) O(1) O(1) 777 : P? ; P? jPAj; P? jPBj = 664 OO(1 =M ) O(1) O(1) O(1) 5 O(1=M ) O(1) O(1) O(1) 2
1
1
1
A quick check shows that preconditioners designed for low-speed convergence acceleration generally possess this accuracy property. Speci cally, Pvlr , Pt, Pcm, and Pvl all have the proper limits as M ! 0 while block-Jacobi preconditioning does not (block-Jacobi preconditioning does not even require the dissipation be modi ed). Finally, we also note that Reed[30, 31] has performed a more restrictive (but also more straightforward) truncation-error analysis and shown that the truncation error of the Choi-Merkle preconditioned equations is a factor of M lower than the truncation error without preconditioning for some terms. Regardless, the general result remains the same | local preconditioning with the associated modi ed dissipation can cure low-Mach-number accuracy problems of common computational methods. 96
2
2.6 Separability
Preconditioning is also useful when attempting to separate equations into independent blocks that would remain hyperbolic or become elliptic if the time derivatives were to be dropped. For example, given the steady Euler equations,
A @@w + B @@w = 0;
these equations may be transformed into independent hyperbolic and/or elliptic parts. For supersonic ow, the steady Euler equations can be separated into a set of four convection (i.e. hyperbolic) equations,
2 0 0 0 3 0 dp + u dv 66 0 0 0 77 @ BB dp ? u dv 64 0 0 1 0 75 @ B@ H S 0 0 0 1
1 2 1 0 0 0 3 0 dp + u dv CC 66 0 ?1 0 0 77 @ BB dp ? u dv CA+64 0 0 0 0 75 @ B@ H 0 0 0 0 S
1 CC CA = 0;
LOCAL PRECONDITIONING 19 where S and H are the entropy and enthalpy. The rst two equations are the convection of acoustic disturbances along Mach lines while the last two equations are the convection of enthalpy and entropy along streamlines. In subsonic
ow, the convection of enthalpy and entropy remains; however, the acoustic subsystem is now elliptic and equivalent to the Cauchy-Riemann equations,
2 1 0 0 0 3 0 dp 1 2 0 ? 0 0 3 0 66 0 ?1 0 0 77 @ BB udv CC 66 ? 0 0 0 77 @ BB 64 0 0 1 0 75 @ BB H CC + 66 0 0 0 0 77 @ BB @ A 4 5 @ 0 0 0 1 S 0 0 0 0 1
1
1 dv CCC = 0: H CA S
u dp
The advantage of this splitting is that di erent discretization methods can now by employed on the elliptic and hyperbolic parts. This separation is not possible for the unsteady Euler equations. However, the introduction of a local preconditioner o ers the freedom to separate the parts of hyperbolic and elliptic origin while keeping the time derivatives and allowing time-marching algorithms to be utilized. Roe[33] has developed a general strategy for performing this separation of elliptic and hyperbolic parts. For the 2-D Euler equations, the general form of the preconditioner allowing this separation is,
Psep
?1
2 66 = 66 4
1
M
2
?a 1
M
a 3
0
1
2 M
a
0 M 1 0 0 0 ?a M 0 0 0 1 1
2
4
3 77 77 ; 5
where the ai 's are free parameters. Combining this result with the family of optimal preconditioners, Popt, we can nd the family of preconditioners which permit elliptic/hyperbolic separation with optimal wavefronts,
p ? 2 a2 3 2 a 2 07 66 M 2 M M 77 66 77 1 0 0 77 ; = 666 pM 2 2 66 ? a 0 a 0 775 M 4 1+
Psep=opt
?1
1
1
0
0
1
1
0
1
where a is a free parameter. In particular, we note that the Van Leer-LeeRoe matrix is a member of the Psep=opt family (with a = 1= ). The hyperbolic/elliptic splitting based on the Van Leer-Lee-Roe preconditioner has been used to formulate genuinely multidimensional upwind schemes for steady
ows[25, 24, 28]. These schemes o er improved accuracy over standard gridaligned upwind methods; however, they also appear to be susceptible to transient ampli cation of disturbances.
20
3 Current status
DARMOFAL, VAN LEER
In the previous section, we focused on the design of a local preconditioner, hightlighting the di erent design criteria from which one might choose. In this section, we will concentrate on the current status of local preconditioning | speci cally, what has local preconditioning achieved in practice?
3.1 Sti ness removal for long waves In principle, the removal of disparate time scales for long waves should result in accelerated convergence, as all errors in the solution can now propagate at the same rate without slowly-propagating error modes staying behind. This removal of sti ness was rst demonstrated for low-Mach-number ows. Merkle et al[4, 5] showed signi cant convergence acceleration for a variety of low-speed internal ows using an Alternating Direction Implicit algorithm with central di erencing. The rst example of the bene t of local preconditioning across all Mach numbers was for the Van Leer-Lee-Roe preconditioner[40]. In this work, explicit, upwind solutions over a NACA 0012 airfoil were obtained, showing that in all cases, the application of Pvlr results in a signi cant speed-up compared to the use of the unpreconditioned Euler scheme. Subsequently, many author investigators have found similar results[15, 10, 9, 30, 14, 31]. In this section, we will demonstrate the convergence-acceleration e ect for a single-grid calculation of the ow over a bump at a M1 = 0:1, 0.3, 0.5, and 0.8. We will only test two preconditioners, Pvlr and Pj , and compare the results to unpreconditioned Euler calculations. The basic ow solver consists of a high-resolution upwind scheme[39] with Roe's[32] approximate Riemann solver. The integration in time is performed with a 2-stage optimally damping scheme from Lynn[18]. As shown in Figure 8, convergence rates for the M1 =0.1 and 0.8 cases are lower than for the two moderate Mach numbers for the unpreconditioned Euler calculations. Using block-Jacobi preconditioning, all but the M1 = 0:1 case converge very similarly, dropping approximately six orders of magnitude in about 1000 cycles (each cycle contains 4 multi-stage integrations so this is actually 4000 multistage iterations). In contrast, however, the Van Leer-Lee-Roe preconditioner performs almost independently of Mach number with six orders of magnitude drop in approximately 1000 cycles for all cases. Compared to the unpreconditioned Euler results, the Van Leer-Lee-Roe preconditioner is approximately 1.5 to 3.0 times faster (in terms of cycles to converge six orders). These results agree well with the wavefront analysis discussed in Section 2.2 which showed that block-Jacobi preconditioning does not aid long wave propagation at low Mach numbers. Thus, a properly-designed local preconditioner can successfully remove long wave time scales and, in practice, this e ect accelerates convergence for single-grid calculations.
LOCAL PRECONDITIONING
21
0
−1
log10 RMS
−2
−3
−4
−5
−6
−7 0
500
1000
1500 cycles
2000
2500
3000
0
0
−1
−1
−2
−2
log10 RMS
log10 RMS
(a) No preconditioning
−3
−4
−3
−4
−5
−5
−6
−6
−7 0
500
1000
1500 cycles
2000
2500
3000
−7 0
500
1000
1500 cycles
2000
2500
3000
(b) block-Jacobi (c) Van Leer-Lee-Roe Figure 8: Mach number independence for convergence of uncon ned bump ow with a single grid of 64 32 cells. Solid: M1 = 0:1, Dash: M1 = 0:3, Dashdot: M1 = 0:5, Circles: M1 = 0:8. Note: each cycle contains 4 two-stage integrations.
3.2 Improved multigrid convergence In addition to accelerating single grid methods, local preconditioning can have a favorable e ect on multigrid methods. In fact, local preconditioning has a double bene t when used in multigrid methods: long waves are accelerated, thus improving the ne grid solver, and discrete eigenvalues of the spatial operator are clustered, thus improving the smoothing of high frequency error modes. Many authors have shown the bene ts which local preconditioning has on multigrid performance[34, 19, 20, 18, 29, 38, 21, 13, 23, 22]. An important contribution in this area is the work of Tai[34] in 1-D and Lynn[18] in 2-D who showed that the combination of local preconditioning and multigrid can accelerate convergence better than either technique applied independently. To demonstrate this
DARMOFAL, VAN LEER
22 0
−1
log10 RMS
−2
−3
−4
−5
−6
−7 0
50
100
150
cycles
0
0
−1
−1
−2
−2
log10 RMS
log10 RMS
(a) No preconditioning
−3
−4
−3
−4
−5
−5
−6
−6
−7 0
50
100 cycles
150
−7 0
50
100
150
cycles
(b) block-Jacobi (c) Van Leer-Lee-Roe Figure 9: Mach number independence for convergence of uncon ned bump ow with full coarsening multigrid. Fine grid 64 32 cells; coarse grid 8 4 cells. Solid: M1 = 0:1, Dash: M1 = 0:3, Dash-dot: M1 = 0:5, Circles: M1 = 0:8. result, we have applied full and semi-coarsening multigrid[26, 27] methods to the previous bump ow test cases. Convergence histories are shown in Figure 9 and 10 for full-coarsening and semi-coarsening, respectively. Comparing the unpreconditioned and Van Leer-Lee-Roe preconditioned results, we see that the preconditioned results are at least ve times faster than the unpreconditioned results. Recalling that without multigrid, the observed speed-ups were between 1.5 and 3.0, we conclude that the combination of multigrid and local preconditioning enhances the performance of the two acceleration techniques. An interesting observation is that the block-Jacobi preconditioner is signi cantly slower than the Van Leer-Lee-Roe preconditioner for the M1 = 0:1 case. This di erence can be attributed to the improved long wave propagation of the Van Leer-Lee-Roe preconditioner. Figure 11 are contour plots of the error in
LOCAL PRECONDITIONING
23
0
−1
log10 RMS
−2
−3
−4
−5
−6
−7 0
50
100
150
cycles
0
0
−1
−1
−2
−2
log10 RMS
log10 RMS
(a) No preconditioning
−3
−4
−3
−4
−5
−5
−6
−6
−7 0
50
100 cycles
150
−7 0
50
100
150
cycles
(b) block-Jacobi (c) Van Leer-Lee-Roe Figure 10: Mach number independence for convergence of uncon ned bump ow with semi coarsening multigrid. Fine grid 64 32 cells; coarse grid 8 4 cells. Solid: M1 = 0:1, Dash: M1 = 0:3, Dash-dot: M1 = 0:5, Circles: M1 = 0:8. the solution. The error is calculated by rst solving the equations to machine precision. Then, the simulations are re-run and the error is the magnitude of the di erence between the current solution and the nal solution. Initially, the error is distributed over the bump. After the rst three cycles, the errors for block-Jacobi and Van Leer-Lee-Roe are similar with both exhibiting grid-aligned errors along the solid wall on and downstream of the bump. In the following cycles, the Van Leer-Lee-Roe preconditioner propagates these grid-aligned error modes out of the computational domain while the block-Jacobi preconditioner does not. Thus, the block-Jacobi preconditioner must rely on smoothing to remove these results; however, since these modes are grid-aligned, they cannot be smoothed and convergence su ers.
24
DARMOFAL, VAN LEER
Van Leer block-Jacobi (a) Initial condition. 16 error contours from 0 to 0.015.
Van Leer block-Jacobi (b) 3 cycles. 16 error contours from 0 to 0.002.
Van Leer block-Jacobi (c) 6 cycles. 16 error contours from 0 to 0.002.
Van Leer block-Jacobi (d) 9 cycles. 16 error contours from 0 to 0.002.
Van Leer block-Jacobi (e) 12 cycles. 16 error contours from 0 to 0.002. Figure 11: Error contours for full coarsening multigrid convergence of M1 = 0:1 bump ow with block-Jacobi and Van Leer preconditioners.
LOCAL PRECONDITIONING
3.3 Low Mach number accuracy
25
As discussed in Section 2.5, local preconditioning can also improve the accuracy of low Mach number ow calculations as a result of the modi ed dissipation matrices. This e ect was rst observed numerically by Van Leer et al[40], and, subsequently, by many other investigators[37, 38, 31]. A typical example of the improved accuracy at low Mach numbers is the M1 = 0:01 ow around a NACA 0012 airfoil at 1:25 angle of attack as shown in Figure 12. In these plots, we compare unpreconditioned and Weiss-Smith preconditioned Euler simulations to incompressible panel methods. As can be clearly seen from both the Cp contours and surface plots, the unpreconditioned results su er from signi cant inaccuracies. However, the preconditioned results have clean Cp contours and the surface pressures match well with panel computations.
4 Future developments While local preconditioning holds signi cant promise, the only preconditioning approach that appears robust for a wide-range of problems is block-Jacobi. However, the block-Jacobi approach does not provide e ective convergence acceleration or good accuracy at low Mach numbers. Thus, a major stumbling block which must be resolved is the lack of robustness at stagnation points for preconditioners with good low Mach number performance. While the work of Darmofal & Schmid[7] and Van Leer et al[41] has resulted in a better understanding of the causes of the robustness problem, adequate solutions are still lacking. While we have concentrated on inviscid ows, the need for convergence acceleration is most strongly felt in high-Reynolds-number, viscous- ow calculations. A signi cant amount of work has already been accomplished in extending and applying local preconditioning to the discretized Navier-Stokes equations[5, 10, 2, 30, 18, 44, 29, 13, 14, 21]. Further developments are also needed in extending the results to other systems of equations such as those of ideal magnetohydrodynamics.
Acknowledgements The work described above has been achieved only with the collaboration of many people. Speci cally, the authors would like to acknowledge the contributions of Dohyung Lee, John Lynn, Barrett McCann, and Phil Roe. The rst author would like to acknowledge the support of the NSF through NSF CAREER Award (ACS-9702435) and The Boeing Company.
DARMOFAL, VAN LEER
26
(a) No preconditioning
Cp contours
(c) Preconditioned Euler
0.6
0.6
0.4
0.4
0.2
0.2
0
0 −Cp
0.8
−Cp
0.8
Cp contours
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
Panel: Solid
Panel: Solid
−0.8
−0.8
Euler: Dashed −1 0
0.1
0.2
0.3
Preconditioned: Dashed 0.4
0.5 x
(b) Surface
0.6
Cp
0.7
0.8
0.9
1
−1 0
0.1
0.2
0.3
0.4
0.5 x
(d) Surface
0.6
Cp
0.7
0.8
0.9
1
Figure 12: Comparison of low Mach number accuracy for unpreconditioned Euler and Weiss-Smith preconditioned solutions and comparison with panel solution. NACA 0012, M = 0:01, = 1:25 degrees. 31 Cp contours are plotted from ?0:7 to 0:7.
LOCAL PRECONDITIONING
References
27
[1] S. Abarbanel and D. Gottlieb. Optimal time splitting for two and three dimensional Navier-Stokes equations with mixed derivatives. Journal of Computational Physics, 41:1{33, 1981. (Also, ICASE Report No. 80-6, 1980). [2] S.R. Allmaras. Analysis of a local matrix preconditioner for the 2-D NavierStokes equations. AIAA Paper 93-3330, 1993. [3] S.R. Allmaras. Analysis of semi-implicit preconditioners for multigrid solution of the 2-d Navier-Stokes equations. AIAA Paper 95-1651, 1995. [4] D. Choi and C.L. Merkle. Application of time-iterative schemes to incompressible ow. AIAA Journal, 23(10):1518{1524, 1985. [5] Y.H. Choi and C.L. Merkle. The application of preconditioning in viscous
ows. Journal of Computational Physics, 105:203{223, 1993. [6] A. J. Chorin. A numerical method for solving incompressible viscous ow problems. Journal of Computational Physics, 2:12{26, 1967. [7] D.L. Darmofal and P.J. Schmid. The importance of eigenvectors for local preconditioners of the Euler equations. Journal of Computational Physics, 127:346{362, 1996. [8] A. Fiterman, E. Turkel, and V.N. Vatsa. Pressure updating methods for the steady-state uid equations. AIAA Paper 95-1652, 1995. [9] A.C. Godfrey. Steps toward a robust preconditioning. AIAA Paper 94-0520, 1995. [10] A.C. Godfrey, R.W. Walters, and B. van Leer. Preconditioning for the Navier-Stokes equations with nite-rate chemistry. AIAA Paper 93-0535, 1993. [11] S.K. Godunov. An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR, 139:521{523, 1961. [12] B. Gustafsson and H. Stoor. Navier-Stokes equations for almost incompressible ow. SIAM Journal of Numerical Analysis, 28(6):1523{1547, 1991. [13] D. Jespersen, T. Pulliam, and P. Buning. Recent enhancements to OVERFLOW. AIAA Paper 97-0644, 1997. [14] D. Lee. Local preconditioning of the Euler and Navier-Stokes equations. PhD thesis, University of Michigan, 1996.
28 DARMOFAL, VAN LEER [15] D. Lee and B. van Leer. Progress in local preconditioning of the Euler and Navier-Stokes equations. AIAA Paper 93-3328, 1993. [16] D. Lee, B. van Leer, and J. Lynn. A local Navier-Stokes preconditioner for all Mach and cell Reynolds numbers. AIAA Paper 97-2024, 1997. [17] W.T. Lee. Local preconditioning of the Euler equations. PhD thesis, University of Michigan, 1991. [18] J. Lynn. Multigrid solution of the Euler equations with local preconditioning. PhD thesis, University of Michigan, 1995. [19] J.F. Lynn and B. van Leer. Multi-stage schemes for the Euler and NavierStokes equations with optimal smoothing. AIAA Paper 93-3355, 1993. [20] J.F. Lynn and B. van Leer. A semi-coarsening multigrid solver for the Euler and Navier-Stokes equations with local preconditioning. AIAA Paper 951667, 1995. [21] D. Mavriplis. Multigrid strategies for viscous ow solvers on anisotropic unstructured meshes. AIAA Paper 97-1952, 1997. [22] B. McCann. Evaluation of local preconditioners for multigrid solutions of the compressible Euler equations. Master's thesis, Texas A&M University, 1996. [23] B. McCann and D.L Darmofal. Evaluation of local preconditioners for multigrid solutions of the compressible Euler equations. AIAA Paper 972028, 1997. [24] L.M. Mesaros. Multi-dimensional Fluctuation Splitting Schemes for the Euler Equations on Unstructured Grids. PhD thesis, University of Michigan, 1995. [25] L.M. Mesaros and P.L. Roe. Multi-dimensional uctuation splitting schemes based on decomposition methods. AIAA Paper 95-1699, 1995. [26] W.A. Mulder. A new approach to convection problems. Journal of Computational Physics, 83:303{323, 1989. [27] W.A. Mulder. A high resolution Euler solver based on multigrid, semicoarsening, and defect correction. Journal of Computational Physics, 100:91{104, 1992. [28] H. Paillere, H. Deconinck, and P.L. Roe. Conservative upwind residualdistribution schemes based on the steady characteristics of the Euler equations. AIAA Paper 95-1700, 1995.
LOCAL PRECONDITIONING [29] [30] [31] [32] [33]
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