On the Numerical Solution of Conservation Laws by Minimizing Residuals

Abstract

The numerical solution of conservation laws by minimizing the residuals of an overdetermined set of discrete equations is studied. Previous research has shown that for certain formulations, minimizing the residuals in the L1 norm will yield solutions that resolve discontinuities that are very sharp and correctly placed. In this study, we analyze a previously proposed method that numerically solves the 2D advection equation with discontinuous data. The method is able to resolve the discontinuity over one mesh cell, without generating spurious oscillations. However, we have found that incorrect solutions are generated for some data. This had led us to formulate and prove two theorems concerning these results. We also provide an analysis of the solution procedure, along with suggestions for developing future schemes that are more applicable to a wide range of problems.