Local preconditioning of the Euler and Navier-Stokes equations.

Abstract

A convergence acceleration technique for the Euler and Navier-Stokes equations is presented, based on local preconditioning of these systems of equations. The key to the success of local Euler preconditioning is equalizing the characteristic wave speeds of the Euler equations as much as possible. By equalizing the wave speeds, the efficiency of wave propagation is improved, resulting in convergence acceleration of the time-marching scheme and other benefits such as clustering of numerical eigenvalues and accuracy improvement in the low-speed limit. A large family of Euler preconditioners is studied with regard to various design criteria formulated for enhancing the performance of Euler time-marching schemes in the areas of efficiency, accuracy, and robustness. A major result is the derivation of a class of preconditioners that can successfully compute stagnating flow, which has previously been problematic for preconditioners developed before this work. The Navier-Stokes preconditioning also accelerates the convergence and clusters eigenvalues. It is developed based on a Fourier analysis of the discretized equations and a dispersion analysis of the differential equations. The principle of the Navier-Stokes preconditioning is to remove the dependence of the physical time scales on both the Mach number and the cell-Reynolds number. The discrete Fourier analysis suggests a hybrid Navier-Stokes preconditioner, combining an optimal Euler preconditioner with the Jacobi block for the discretized viscous/conductive terms. The dispersion analysis produces an analytical form of the preconditioner, which can equalize, in absolute value, the complex wave speeds of the Navier-Stokes equations; these include both effects of viscous damping and wave propagation. The Jacobi and analytical techniques can also be combined in a single preconditioner, for greater robustness and efficiency. Numerical tests confirm that both types of preconditioners can increase the efficiency of the convergence.