Multi-stage schemes for the Euler and Navier-Stokes equations with optimal smoothing

Abstract

there was a local preconditioning matrix that removes the spread among the characteristic speeds as much as The effect of a recently derived local preconditioning possible. It achieves what can be shown to be the 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 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.