Local preconditioning of the Euler equations.

Abstract

A local preconditioning matrix for the multi-dimensional Euler equations is derived that reduces the spread of the characteristic speeds from a factor (M + 1)/min(M,$\vert$M $-$ 1$\vert$) to a factor 1/$\sqrt{1 - \min(M\sp2,M\sp{-2})}$, where M is the Mach number. It is shown that the latter value is the lowest attainable. Numerical experiments with this preconditioning, applied to a single grid, explicit upwind discretization of the two-dimensional Euler equations, show that it significantly increases the rate of convergence to a steady solution, as predicted theoretically. The numerical implementation for the first-order Roe scheme requires a slight modification of the dissipation term in the flux formula for the sake of stability. This modification not only avoids the high-frequency instability but also prevents loss of accuracy for almost incompressible flow. Other benefits expected from the use of the new preconditioning relate to its ability to make the system of the Euler equations behave more as a scalar equation. One specific application discussed is the development of explicit marching schemes that effectively damp most high-frequency Fourier modes, as desired in multi-grid relaxation. Smoothing properties of Euler schemes with multi-grid relaxation have traditionally been analyzed on the basis of a scalar convection equation and are known to depend strongly on the Courant number employed. The effect of the preconditioning matrix on an Euler scheme is to equalize the Courant numbers associated with the different waves, so that optimal smoothing can be achieved simultaneously for all underlying characteristic convection schemes. Some convergence acceleration technique (e.q. artificial compressibility) are simply attempts at what we have done. Others (e.g. GMRES) are based on totally different concepts, and would benefit from a clustering of the eigenvalues, so that they would work well in conjunction with preconditioning.