Solution of Evolutionary Partial Differential Equations Using Adaptive Finite Differences with Pseudospectral Post-processing

A coordinate transformation approach is described that enables pseudospectral methods to be applied efficiently to unsteady differential problems with steep solutions. The work is an extension of a method presented by Mulholland, Huang, and Sloan for the adaptive pseudospectral solution of steady problems. A coarse grid is generated by a moving mesh finite difference method that is based on equidistribution, and this grid is used to construct a time-dependent coordinate transformation. A sequence of spatial transformations may be generated at discrete points in time, or a single transformation may be generated as a continuous function of space and time. The differential problem is transformed by the coordinate transformation and then solved using a method that combines pseudospectral discretisation in space with a suitable integrator in time. Numerical results are presented for unsteady problems in one space dimension.