CONVERGENCE TO STEADY-STATE SOLUTIONS OF THE EULER EQUATIONS ON UNSTRUCTURED GRIDS WITH LIMITERS

This paper addresses the practical problem of obtaining convergence to steady state solutions of the Euler equations when limiters are used in conjunction with upwind schemes on unstructured grids. The base scheme forms a gradient and limits it by imposing monotonicity conditions in the reconstruction stage. It is shown by analysis in one dimension that such an approach leads to various schemes meeting total-variation-diminishing requirements in one dimension. In multiple dimensions these schemes produce steady-state solutions that are monotone and devoid of oscillations. However, convergence stalls after a few orders of reduction in the residual, A new limiter is introduced that is particularly suited for unstructured grid applications. When reduced to one dimension, it is shown that this limiter satisfies the standard theory. With this limiter, it is shown that converged steady-state solutions can be obtained. However, the solutions are not monotone. There appears to be a conflict between achieving convergence and monotone solutions with the higher order schemes that employ limiters in the framework presented. (C) 1995 Academic Press, Inc.