Matching multistage schemes to viscous flow.

Abstract

A method to accelerate convergence to steady state by explicit time-marching schemes for the compressible Navier-Stokes equations is presented. The combination of cell-Reynolds-number-based multistage time stepping and local preconditioning makes solving steady-state viscous flow problems competitive with the convergence rates typically associated with implicit methods, without the associated memory penalty. Initially, various methods are investigated to extend the range of multistage schemes to diffusion-dominated cases. It is determined that the Chebyshev polynomials are well suited to serve as amplification factors for these schemes; however, creating a method that can bridge the continuum from convection-dominated to diffusion-dominated regimes proves troublesome, until the Manteuffel family of polynomials is uncovered. This transformation provides a smooth transition between the two extremes; and armed with this information, sets of multistage coefficients are created for a given spatial discretization as a function of cell Reynolds number according to various design criteria. As part of this process, a precise definition for the numerical time step is hammered out, something which up to this time, has been set via algebraic arguments only. Next are numerical tests of these sets of variable multistage coefficients. To isolate the effects of the variable multistage coefficients, the test case chosen is very simple: circular advection-diffusion. The numerical results support the analytical analysis by demonstrating an order of magnitude improvement in convergence rate for single-grid relaxation and a factor of three for multigrid relaxation. Building upon the success of the scalar case, preconditioning is applied to make the Navier-Stokes system of equations behave more nearly as a single scalar equation. Then, by applying the variable multistage coefficient scheme to a typical boundary-layer flow problem, the results affirm the benefits of local preconditioning for low-Mach-number flows and the variable coefficient scheme provides a factor of five savings over the fixed-coefficient scheme.