Multi-stage schemes for the Euler and Navier-Stokes ... - Deep Blue

Multi-Stage Schemes for the Euler and Navier-Stokes Equations with Optimal Smoothing John F. Lynn* and Bram van Leert Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140

Abstract The effect of a recently derived local preconditioning matrix [I] on discretizations of the spatial Euler operator is a strong concentration of the pattern of eigenvalues in the complex plane. This makes it possible to design multi-stage schemes that systematically damp most highfrequency waves admitted by the particular discrete operator. The resulting schemes are not only preferable as solvers in a multi-grid strategy, they are also superior single-grid schemes, as the preconditioning itself already accelerates the convergence to a steady solution, and the high-frequency damping provides robustness. In this paper, we describe the optimization technique, use it t o obtain the optimal sequence of time-step values for upwind Euler discretizations, and present some convergence results for numerical integrations performed with the new schemes. Furthermore, the extension t o discrete Navier-Stokes operators is treated.

1

Introduction

there was a local preconditioning matrix that removes the spread among the characteristic speeds as much as possible. It achieves what can be shown to be the optimal condition number for the characteristic speeds, namely, 1/J1 . . - min(M2,M - 2 ) , where M is the local Mach number. This is a major improvement over the condition number before preconditioning, which equals ( M 1)/ min(M, IM - 11). The effect of the preconditioning on discretizations of the spatial Euler operator is a strong concentration of the pattern of eigenvalues in the complex plane. This, finally, makes it possible t o design multi-stage schemes that systematically damp most high-frequency waves admitted by the particular discrete operator. The design technique is an extension of the techniques used in [3, 6, 41. The resulting schemes are not only preferable as solvers in a multi-grid strategy, they are also superior single-grid schemes, as the preconditioning itself already accelerates the convergence to a steady solution, and the high-frequency damping provides robustness. In this paper we describe the optimization technique, use it to obtain the optimal sequence of time-step values for upwind Euler discretizations, and present some convergence results for numerical integrations performed with the new schemes. Furthermore, the extension to discrete Navier-Stokes operators is treated. The Euler preconditioning matrix on which this research is based is an improvement over the matrix presented in [I]; the NavierStokes preconditioner is new. Both are presented in a companion paper on preconditioning, simultaneously submitted to this conference [7].

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Explicit marching schemes for the Euler and NavierStokes equations must feature effective high-frequency damping in order t o be suited for use in multi-grid marching. Multi-stage schemes offer the flexibility to achieve the desired smoothing properties. Until recently, however, the design of optimally smoothing multi-stage schemes was based entirely on the scalar one-dimensional [2, 31 or twodimensional [4] convection equation. In [5] we presented, for the first time, a design approach for Euler schemes in which the Fourier trans2 Previous scalar analysis form of the full spatial operator is used. This approach has become possible owing to a breakthrough Tai's [3] procedure for optimizing the high-frequency in preconditioning algorithms [I], reported at the 10th damping in a one-dimensional convection scheme is a geAIAA CFD Conference, Honolulu, June 1991. Presented ometry exercise in the complex plane: putting- the zeros of the multi-stage amplification factor on top of the locus 'Doctoral Candidate, Aerospace Engineering and Scientific of the Fourier transform - the "Fourier footprint" - of the Computation discrete spatial operator. This can be achieved for one t Professor, Associate Fellow AIAA Copyright 0 1 9 9 3 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

642

I

Four~erFootprtnt of n Six-Stage Scheme

= 1/3

Magnitude of Amplification Factor for a

5 0 ,

= 1/3

I

Figure 1: Fourier footprint (dashed line) of the third- Figure 2: Modulus of the amplification factor as a function order upwind-biased spatial discretization of the one- of spatial frequency, for the case of Figure 1. dimensional convection operator, and level lines (solid) of the amplification factor of Tai's optimal six-stage scheme. pends on the flow angle. Catalano and Deconinck repeat the optimization for each flow angle; this makes all paramspecific value of the time-step, the finding of which is part eters dependent on the flow angle, which is less desirable. of the design process. An example of the result of this Moreover, the alignment problem causes the the parameprocedure is shown in Figures 1 and 2. ters to vary strongly when the flow angle becomes small. Catalano and Deconinck [4] relaxed the condition We have redone the optimization for two-dimensional that the zeros must lie exactly on the Fourier locus, convection using the full two-dimensional footprint. For thereby achieving a further reduction of the maximum small flow angles, though, we excluded the low-high freamplification factor for the high frequencies. quency combinations which would render the optimizaFor a two-dimensional discrete convection operator tion meaningless. The optimal coefficients derived using the Fourier footprint no longer is a single curve, but covers spatial operators associated with the two-dimensional conan area; the location and shape of this area vary greatly vection equation are very close to the coefficients obtained with the convection direction. Figures 3 and 4 show the with the Modified Roe operator for the preconditioned Eulocus for the first-order upwind-differencing operator, for ler equations. Figure 21 shows the Fourier footprint of the convection directions of 10" and 45". The frequencies in- first-order upwind discretization (high-frequency compo& E [O,T]. High- nent) for the two-dimensional convection equation supercluded in the footprint are @, € [O, frequency damping by a fixed multi-stage scheme (coef- imposed upon level lines of the multi-stage scheme based ficients independent of flow direction) is easily achieved upon the modified Roe operator for the preconditioned for modes propagating in the physical convection direc- Euler equations. As can be seen in the figure, these coeftion, but is fundamentally difficult for modes varying in ficients are close to optimal for the two-dimensional conthe normal direction, especially if the convection is almost vection discrete operator as well. It is important to make in the grid direction. This is what we may call the single- an appropriate choice of the length scale used in defining grid alignment problem; its solution lies beyond the scope the Courant number v (cf. Section 5). of this paper. The alignment problem is evident in Figure 3 from the low-high frequency combinations found near the ori- 3 Euler equations: effect of pregin. To damp these, zeros must be put close to the origin; conditioning to benefit from these zeros, a large time-step would be needed; this works against numerical stability. For the two-dimensional Euler equations the situaFor two-dimensional convection Tai as well as Cata- tion gets even worse, because there now are different kinds lano and Deconinck use a one-dimensional optimization: of physical signals propagating in all possible directions at they only consider high-frequency plane waves moving different speeds; these are more or less accurately reprein the flow direction. Tai accepts the optimal sequence sented by the discrete operator and produce different conof time-step ratios for one-dimensional convection and centrations of eigenvalues in its Fourier footprint. Figures merely adjusts the final Courant number; the latter de- 5 to 8 show the Fourier footprint of the first-order up-

TI,

Fourier Footprint of First-Order Upwind Scheme

1

Fourier Footprint of First-Order Upwind Scheme

Figure 3: Fourier footprint of the first-order upwind ap- Figure 4: Fourier footprint of the first-order upwind approximation of the two-dimensional convection operator; proximation of the two-dimensional convection equation; convection angle 4 = 45'. convection angle 4 = 10'. wind scheme for the Euler equations, based on Roe's [8] upwind-biased flux formula, for a range of Mach numbers. All figures are for the case when the flow speed is aligned with the grid. The different sizes of the different concentrations in the footprint make it impossible t o place the zeros of a multi-stage amplification factor a t fixed locations in the complex plane and still achieve good high-frequency damping for all Mach numbers (even disregarding flow angles) . The next sequence of figures, 9 to 12, show the Fourier footprint of the preconditioned first-order upwind scheme for the Euler equations, again based on Roe's flux formula, for a range of Mach numbers. The frequencies The included in the footprints are ?/, E [0, n],By E [O, preconditioning matrix is the one presented in [I]. A comparison with the previous sequence shows that removing the variation among the characteristic convection speeds has resulted in a thorough clean-up of the footprint. Especially impressive is the job it does for small M. For M T 1 a growing separation of two regions of concentration of eigenvalues is observed; this corresponds to the growing disparity between the acoustic speeds in the flow direction (= q d m , q flow speed) and normal direction (= q). For M > 1 the footprint starts looking very much like one for scalar convection (cf. Figure 3); this is because all signals in supersonic flow move downstream.

TI.

4

step values a d k ) , k = 1, .., m , of an m-stage algorithm. When updating the solution of Ut = Res(U) from time level tn to tn+' = tn the form ~ ( 0 )=

un,

u ( ~ )=

u(O)

(1)

+ At, the method takes (2)

+ A ~ ( ~ ) (u(~-')) R~s , k = I, .., m,(3)

with At = adrn). According to linear theory, one step with the full scheme multiplies each eigenvector of the operator Res(U), with associated eigenvalue A , by a factor of the form m

where z

= AAt

generally is complex. The m - 1 coefficients crk relate to the time-step ratios a k = a t c k ) / a t ; the actual time step At is the mth parameter. The optimization procedure starts out by computing, for a fixed combination of M and 4 (Z flow angle), a discrete set of eigenvalues for wave-number pairs (P,, Py) in the high-frequency range, i.e.

Optimization procedure

The procedure for optimizing high-frequency damping aims a t minimizing the maximum of the modulus of Assuming a set of starting values for the m-stage scheme, the scheme's amplification factor over the set of high- for instance Tai's values, the value of IP(z)l is computed frequency eigenvalues. The input parameters are the time- for all eigenvalues previously obtained, and its maximum

Fourier Footprint of First-Order Roe Scheme = 0 . 1 , 4 = oO, v = 1

Fourier Footprint of First-Order Roe Scheme M = 0 . 9 , 4 = 0°, v = 1

M

0.90

I I I I

I I

I

Figure 5 : Fourier footprint of the first-order upwind ap- Figure 7: Fourier footprint of the first-order upwind approximation of the spatial Euler operator, for M = 0.1, proximation of the spatial Euler operator, for M = 0.9, and flow angle 4 = 0'. The time-step chosen corresponds and flow angle 4 = (10. to a Courant-number value of 1.

,

Fourier Footprint of First-Order Roe Scheme 2,10, M

= 0 . 5 , 4 = 0°, u = 1

2.101

Fourier Footprint of First-Order Roe Scheme M = 2.0, 4 = 0°, u = 1

1

Figure 6: Fourier footprint of the first-order upwind ap- Figure 8: Fourier footprint of the first-order upwind approximation of the spatial Euler operator, for M = 0.5, ~roximationof the spatial Euler operator, for M = 2, and flow speed aligned with the grid. and flow angle 4 = 0'.

First-Order Modified Roe Scheme

First-Order Modified Roe Scheme = 0.1, q5 = 0°, v = 1

M

2'1

Figure 9: Fourier footprint of the preconditioned first- Figure 11: Fourier footprint of the preconditioned firstorder upwind approximation of the spatial Euler operator, order upwind approximation of the spatial Euler operator, for M = 0.9, and flow angle 4 = OO. for M = 0.1, and flow angle 4 = 0'.

2.10

First-Order Modified Roe Scheme

First-Order Modified Roe Scheme M = 0.5,q5 = 0°, v = 1

i

Figure 10: Fourier footprint (symbols) of the precoxidi- Figure 12: Fourier footprint of the preconditioned firsttioned first-order upwind approximation of the spatial Eu- order upwind approximation of the spatial Euler operator, ler operator, for M = 0.5, and flow angle 4 = 0'. for M = 2, and flow angle 4 = 0'.

is found. This is our functional a ( A d l ) , .., Adrn);M , 4); it must be minimized by varying the m parameters. (Dependence on M and 4 will be considered later.) The optimal (in the L, sense) m-stage scheme may hence be obtained as the solution to the following minmax problem:

The optimization procedure that appeared to be most robust is Powell's method (see, e.g., [9]). The optimization procedure is not without its problems. Specifically, there are two points of concern: 1. The alignment problem makes the optimization meaningless for flow angles near 0 (or ~ / 2 ) ,since the amplification factor for low-high (or high-low) frequency combinations tends to 1. Our solution is to filter out these frequency combinations, e.g. for q5 = 0 we optimize only over the eigenvalues with 1 5 IP, I, a wedge-like region in the frequency plane. When 4 increases the wedge rotates with the flow angle and opens up, until for 4 = ?r/4 the entire domain (Eqn. 7) is used. When 4 increases further, the optimization domain shrinks again, until for 4 = 1r/2 only those frequency pairs are included satI/?y ( As said before, the alignment isfying !@, I problem has to be dealt with separately, for instance by semi-coarsening [lo].