On the odd-even hopscotch scheme for the numerical integration of time-dependent partial differential equations

This paper is devoted to the odd-even hopscotch scheme for the numerical integration of time-dependent partial differential equations. Attention is focussed on two aspects. Firstly, via the equivalence to the combined leapfrog-Du Fort–Frankel method we derive the explicit expression of the critical time step for von Neumann stability for a class of multi-dimensional convection-diffusion equations. This expression can be derived directly by applying a useful stability theorem due to Hindmarsh, Gresho and Griffiths [9]. The interesting thing on the critical time step is that it is independent of the diffusion parameter and yet smaller than the critical time step for zero diffusion, but only in the multi-dimensional case. This curious phenomenon does not occur for the one-dimensional problem. Secondly, we consider the drawback of the Du Fort–Frankel accuracy deficiency of the hopscotch scheme. To overcome this deficiency we discuss global Richardson extrapolation in time. This simple device can always be used without reducing feasibility. Numerical examples are given to illustrate the outcome of the extrapolation.