A MOVING COLLOCATION METHOD FOR SOLVING TIME-DEPENDENT PARTIAL-DIFFERENTIAL EQUATIONS

A new moving mesh method is introduced for solving time dependent part ial differential equations (PDEs) in divergence form. The method uses a cell averaging cubic Hermite collocation discretization for the phys ical PDEs and a three point finite difference discretization for the P DE which determines the moving mesh. Numerical results are presented f or a selection of difficult bench-mark problems, including Burgers' eq uation and Sod's shocktube problem. They indicate third order converge nce for the method, slower than the traditional (fourth order) cubic H ermite collocation on a fixed mesh but much faster than the first orde r of the commonly used moving finite difference methods. Numerical exp eriments also show that, in comparison with finite differences and fix ed mesh collocation, moving collocation produces more accurate results for small and moderate numbers of mesh points.