A posteriori variational multiscale methods for the 1D convection-diffusion equations

Abstract

The present work is a continuation of a paper presented by the two first authors in the proceedings of the “Computational Science for the 21st century” conference held in Tours in 1997 honouring the 60th birthday of Roland Glowinski. It is devoted to the solution of 1D convection-diffusion equations in dominant convection regime situations. In that paper, an “a posteriori” VMS filtering technique was introduced. We present an extension of this technique to nonlinear convection-diffusion equations (a traffic model), providing an efficient method for the resolution of shocks from just the Galerkin solution at targeted times. We also present a residual-based “a posteriori” VMS filtering, that provides quite accurate stable solutions, can be extended to multi-dimensional problems, and can be applied locally. We finally present some numerical tests exhibiting the high accuracy of the obtained solutions.