Multigrid Methods for Steady Euler Equations Based on Multi-stage Jacobi Relaxation

Abstract

First order accurate upwind methods of flux-difference type applied to steady Euler equations generate a set of discrete equations of positive type. This set can be solved by any classic relaxation method in multigrid form. The set of discrete equations generated by a higher order accurate form does not have this property and cannot be solved in the same way. The common approach is then to use defect correction [1, 2, 3]. In this procedure the multigrid method is applied to the first order accurate form and constitutes an inner iteration for a higher order correction only made on the finest grid. The defect correction proves to work well in many applications. The speed of convergence is however largely determined by the outer iteration and sometimes is found to be rather dissapointing, especially when the first order and the higher order solutions differ significantly. It can be expected that if the higher order approximation also could be used in the multigrid itself a better performance could be obtained. A second difficulty is that often convergence cannot be obtained unless a suitable initial flow field is specified, i.e. there is a risk of choosing an initial approximation which is out of the attraction region of the iterative method.