A moving collocation method for solving time dependent partial differential equations

A new moving mesh method is introduced for solving time dependent partial differential equations (PDEs) in divergence form. The method uses a cell averaging cubic Hermite collocation discretization for the physical PDEs and a three point finite difference discretization for the PDE which determines the moving mesh. Numerical results are presented for a selection of difficult bench-mark problems, including Burgers' equation and Sod's shocktube problem. They indicate third order convergence for the method, slower than the traditional (fourth order) cubic Hermite collocation on a fixed mesh but much faster than the first order of the commonly used moving finite difference methods. Numerical experiments also show that, in comparison with finite differences and fixed mesh collocation, moving collocation produces more accurate results for small and moderate numbers of mesh points.