Using Optimal Time Step Selection to Boost the Accuracy of FD Schemes for Variable-Coefficient PDEs
International Conference of Applied and Engineering Mathematics, World Congress on Engineering 2009, London, U.K., July 3, 2009
We extend the technique of optimal time step (OTS) selection for finite difference (FD) schemes of time dependent PDEs to PDEs where the leading-order spatial derivative term has a spatially varying coefficient. The basic approach involves identifying a transformation of the domain that eliminates the spatial dependence of the coefficient for the leading-order term. This change of variables is then used to define an optimal computational grid for the FD scheme on the original domain. By using both the optimal grid and OTS selection, we are able to boost the order of accuracy above what would be expected from a formal analysis of the FD scheme. We illustrate the utility of our method by applying it to variable-coefficient wave and diffusion equations. In addition, we demonstrate the viability of OTS selection for the two-step Kreiss-Petersson-Ystrom discretization of the wave equation.