Two Multigrid Time Stepping Schemes which Preserve Monotonicity and TVD

Antony Jameson

Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305

Justin W.L. Wan (speaker)

Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Abstract

In this talk, we present two efficient multigrid time stepping schemes for scalar linear and nonlinear wave equations based on a upwind biased interpolation and restriction, and a nonstandard coarse grid update formula. Furthermore, we prove that these schemes preserve monotonicity and are total variation diminishing (TVD). Thus, no numerical oscillation is introduced, resulting in fast wave propagation to the steady state.

Multigrid for solving elliptic partial differential equations (PDEs) has been proven, both numerically and theoretically, to be a successful and powerful techniques. Efficient multigrid methods have also been proposed for solving non-elliptic equations, in particular, Euler and Navier Stokes equations. One approach is to accelerate the evolution of a hyperbolic system to a steady state on multiple grids by taking larger time steps on coarse grids without violating the stability condition. Thus, the low frequency disturbances are rapidly expelled through the outer boundary whereas the high frequency errors are locally damped. It can be proved that an M-level multigrid cycle consisting of one smoothing step on each coarse grid results in an effective time step of the sum of the time steps of all the coarser grids.

While the above approach has achieved great success, for instance, in Euler calculations, the understanding of the numerical property is relatively limited and analyses are scarce. In this paper, two efficient multigrid time stepping schemes proposed by Jameson is studied and extended for the steady state solution of scalar linear and nonlinear wave equations. One scheme is multiplicative in nature and the other additive. Thus the former is usually more effective but the latter is more parallel. The upwind biased interpolation and restriction are defined based on the characteristic directions. As opposed to elliptic multigrid, they are not transpose of each other. For nonlinear wave equations, the characteristic directions change at every grid points and in every time steps. To define the appropriate upwind interpolation and restriction, we solve a local Riemann problem which determines locally the characteristic direction. Up to the authors' knowledge, Riemann solutions have not been used in the context of multigrid interpolation/restriction.

In addition, we present the numerical analysis of these schemes; the primary focus is on the monotonicity preserving and total variation diminishing properties. Both concepts are fundamental in designing discretization schemes for conservation laws, but nevertheless, have never been used to analyze multigrid methods in the literature. We prove that both the two-level multiplicative and additive schemes preserve monotonicity and are TVD; and the same holds for the multilevel additive scheme. Finally, numerical results for solving the linear wave equation and the nonlinear Burgers' equation in one and two dimensions are presented to demonstrate the effectiveness of the proposed schemes and verify that no oscillation occurs during the multigrid time stepping.