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

Abstract

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. IntroductionThe subject of this paper is the odd-even hopscotch (OEH) method for the numerical integration in time of time-dependent partial differential equations (PDEs) (Gordon [2], Gourlay [3][4][5] •• , qa) the (constant) velocity, and f.> 0 a diffusion parameter. When combined with simple central differences the OEH method shows an equivalence to the combined leapfrog-Du Fort-Frankel method. Via this equivalence we derive the explicit expression of the critical time step for von Neumann stability for problem (1.1). This expression is easily found by applying a useful theorem due to Hindmarsh, Gresho and Griffiths [9]. This theorem plays an important role in their stability analysis of the forward Euler-central difference scheme. The interesting thing on the critical time step is that it is independent of f. whereas it is smaller than the critical time step for zero diffusion, but only in the multi-dimensional case. We wish to remark that this pathological behaviour of the leapfrog-Du Fort-Frankel method has been observed earlier (see [13] and the references therein). However, to the best of our knowledge, the explicit expression of the critical time step is new.An immediate consequence of this pathological behaviour is that adding artificial diffusion to the OEH central difference scheme may render the process unstable. This observation is in clear contrast to the common practice which teaches us that introducing artificial diffusion has a stabilizing effect. We note that this remark does not contradict the findings of Gourlay and Morris [6] in their investigation of the OEH scheme for nonlinear shock calculations as they restrict their attention to the one-dimensional case.The second aspect of the OEH method considered in this paper is the drawback we refer to as the Du Fort-Frankel (DFF) accuracy deficiency. The consequence of the DFF deficiency is that convergence takes place for a smaller set o...