Moving Mesh Methods with Upwinding Schemes for Time-Dependent PDEs

tion, which is often called a reaction–convection–diffusion equation, is representative of many problems which are It is well known that moving mesh and upwinding schemes are two kinds of techniques for tracking the shock or steep wave front solved by moving mesh methods in one dimension. in the solution of PDEs. It is expected that their combination should Much work has been devoted to the construction of produce more robust methods. Several upwinding schemes are shock capturing schemes on uniform grids, such as Goduconsidered for non-uniform meshes. A self-adaptive moving mesh nov, MUSCL, PPM, FCT, ENO, and PHM (see Salari and method is also described. Numerical examples are given to illustrate Steinber [21] and its references, Shu and Osher [22], and that in some cases, especially for hyperbolic conservative laws with nonconvex flux, the upwinding schemes improve the results of the Marquina [16] for more information). Most of these moving mesh methods. Comparing the results of several upwinding schemes possess the total variation diminishing (TVD) feaschemes, we find the local piecewise hyperbolic method (PHM) is ture which is necessary for high order oscillation-free very efficient and accurate when combined with a moving mesh schemes. The conventional scheme on a uniform grid needs strategy. Q 1997 Academic Press enough points to capture the shock or steep wave front, which often puts too many points in the area where the solution is very flat and smooth. Moving mesh methods,