Analysis of a local matrix preconditioner for the 2-D Navier-Stokes equations

A local matrix preconditioner based on block-Jacobi is analyzed for the linearized 2-D compressible NavierStokes equations discretized using firstand secondorder accurate upwinding. The purpose of the matrix preconditioner is to tightly cluster residual eigenvalues for high-frequency Fourier modes. It is intended for use in an explicit relaxation scheme optimized to damp these modes. Analysis is first conducted for the scalar advection/diffusion equation to establish an optimal envelope of residual eigenvalues for the preconditioned Navier-Stokes system. The block-Jacobi preconditioner is then demonstrated and measured against this optimal envelope. Its effect on residual eigenvalue clustering is also compared to the system with scalar preconditioning and to a matrix preconditioner developed for the Euler equations. A numerical test case is also presented that demonstrates the block-Jacobi preconditioner in a scheme that effectively damps high-frequency error modes. Introduction and Motivation Mulder [I, 21 has developed an efficient solver for the steady 2-D Euler equations based on a novel semicoarsening multigrid algorithm. Essential to the success of this solver is effective smoothing of local highfrequency (i.e., short wave length) errors on each grid. Mulder has shown that effective smoothing of these errors can be accomplished using fully explicit relaxation schemes. The classic grid alignment problem that has plagued explicit-relaxation based multigrid solvers is resolved through semi-coarsening. This semi-coarsening multigrid algorithm should be extendible to solution of the Navier-Stokes equations as well. Indeed, semi-coarsening is currently being implemented in a few Navier-Stokes codes with promising results [3, 41. The problem with current Navier-Stokes re*Principal Engineer, Member AIAA Copyright @American Institute of Aeronautics and Astronautics, Inc., 1993. All rights reserved. laxation schemes, particularly explicit-based methods, ) is poor smoothing characteristics for many flow and grid conditions. Thus, true multigrid convergence rates will only result from the development of relaxation schemes that effectively damp high frequency errors for all flow conditions on grids with arbitrarily large cell aspect ratios. In his Euler algorithm, Mulder analyzed and implemented several relaxation schemes based on variations of classic block-Jacobi and block-Gauss-Seidel. Van Leer, et a1 [5, 6, 7, 81 have taken a somewhat different approach to the development of explicit relaxation schemes for the Euler equations. They break the development of the scheme into two components. The first is a local matrix preconditioner to reduce the condition number (the ratio of the largest to smallest eigenvalues) of the Euler equations, making them behave more like a scalar advection equation. The second component of the relaxation scheme is a multi-stage Runge-Kutta time-marching update with stage coefficients optimized to damp the high frequency modes of the discretized advection equation. Our approach to development of an explicit relaxation scheme for the Navier-Stokes equations is similar to van Leer's-combine a local matrix preconditioner with an optimized Runge-Kutta scheme. The subject of the present paper is development and analysis of a matrix preconditioner for the discrete system. Optimization of a multi-stage Runge-Kutta update will not 1 be addressed here. i Local matrix preconditioners have previously been introduced to alter the condition number, or eigenvalue spread, of the Euler equations. Turkel's preconI ditioner [9] was primarily targeted for curing condition, number problems in low Mach number flow. The work of van Leer, e t al, achieved the goal of optimizing the condition number of the Euler equations for all Mach numbers. However, their preconditioner has several drawbacks for present purposes. First, van Leer's matrix preconditioner is formulated for grids with cell aspect ratios near unity, and its extension to high aspectratio grids is not clear. Second, the preconditioner was developed for the Euler equations based on the concept of equating wave speeds in the system. It is not clear how to extend this approach to include the diffusive (i.e., non-wave) effects of the Navier-Stokes terms. Third, when used in conjunction with Roe's flux-difference splitting [lo], the dissipation matrix must be altered to avoid instabilities in the discretized system. Thus, to effectively precondition the discrete system, the steadystate discretization must be altered. In principle, one should not have to alter or sacrifice steady-state accuracy to achieve good convergence of a iterative solver; but then again, maybe this is being overly optimistic. Our development of a local matrix preconditioner for the 2-D Navier-Stokes equations is based on a somewhat different approach. Rather than work with the analytic equations, we will instead work with the discretized system directly. In addition, the objective is not to lower condition number, but t o tightly cluster eigenvalues of the residual operator for modes that are high frequency in both coordinate directions. These error modes must be eliminated for good performance of the semi-coarsening multigrid. We will show that these objectives can be met by classic block-Jacobi. We expect that the best we can possibly achieve for the preconditioned Navier-Stokes system is the envelope of all possible eigenvalues resulting from the discretized scalar advection/diffusion equation. Hence, this paper begins by analyzing preconditioning of this scalar model equation. The eigenvalue envelopes produced by the proposed matrix preconditioner will be presented and compared with scalar preconditioning and the Euler preconditioner of van Leer et a1 [7, 81 for selected flow conditions. The effectiveness of the present matrix preconditioner will also be demonstrated by convergence studies of a numerical test case. 1-D Scalar Advection/Diffusion The Navier-Stokes equations contain advective and diffusive mechanisms. Hence, preconditioning of the Navier-Stokes equations should cause the system of equations to behave more like a scalar advection/diffusion equation. Therefore, we start by examining the envelope of residual eigenvalues for the scalar advection/diffusion equation and analyze how preconditioning can improve cluster of the eigenvalues. The 1-D advection/diffusion equation is given by, The discretized form of the equation is, where r is the residual operator, &uj is the change in the solution at grid point j, At is the time step, and Ax is the grid spacing (uniform grid). The advection term is discretized by firstor second-order upwinding, and the diffusion term is discretized by second-order central differencing. uj ~ j 1 , first-order sxuj = (3uj 4uj-1 + uj-2)/2, second-order (3) szxuj = U j + l 2uj + uj-1. (4) Consider a Fourier mode Gj = eije with -T < 0 < T and i = a. The corresponding Fourier residual operator is, with the Fourier difference operators, 1 e-ie, first-order 8, = (3 4e-ie + eW2")/2, second-order (6) The time step At, which serves the same purpose as a scalar preconditioner, is chosen to satisfy an explicit CFL constraint, where X and a are constants. For example, point-Jacobi iteration gives X = 1, a = 2 for the first-order discretization, and X = 312, a = 2 for second-order. The choice of the constants X and a affects clustering of the 1-D advection/diffusion residual operator eigenvalues over the range of cell Reynolds number (aAt/v) and high-frequency Fourier modes (a12 < 6 < T). Figures 1 and 2 show the distribution of eigenvalues for the residual operator discretized by firstand second-order upwinding, respectively. The choice of constants X and a for each of the three plots in the two figures is given in Table 1. Each plot covers the full range of cell Reynolds number and high-frequency Fourier modes. Note that the eigenvalues are symmetric about the real axis, so only the upper half-plane is displayed. Figures l a and 2a show the eigenvalue distributions for equal weighting of advective and diffusive time steps, which is a common practice in CFD codes. A time step based on point-Jacobi results in the distributions shown in Figs. l b and 2b. Figures l c and 2c are obtained by a rescaled point-Jacobi time step, whose eigenvalue distribution has a maximum extent of one along the Table 1: Constants in Time Step for 1-D Advection/Diffusion Residual Operator (Figs. 1, 2) negative real axis. This choice of time step may be written as, where diag(6,) and diag(6,,) are the diagonal Jacobian elements of the difference operators defined in Eqs. 3