Stability Analysis of an Odd--Even-Line Hopscotch Method for Three-Dimensional Advection--Diffusion Problems

Abstract

A linear stability analysis is given for an odd--even-line hopscotch (OELH) method, which has been developed for integrating three-space dimensional, shallow water transport problems. Sufficient and necessary conditions are derived for strict von Neumann stability for the case of the general, constant coefficient, linear advection--diffusion model problem. The analysis is based on an equivalence with an associated scheme which is composed of the leapfrog, the Du Fort--Frankel, and the Crank--Nicolson schemes. The results appear to be rather intricate. For example, the resulting expressions for critical stepsizes reveal that the presence of horizontal diffusion generally leads to a smaller value, in spite of the fact that we have unconditional stability for pure diffusion problems. It is pointed out that this is due to the Du Fort--Frankel deficiency. On the other hand, it is also shown, by a numerical experiment, that in practice it is sufficient to obey the weaker Courant--Friedrichs--Lewy (CFL) condition associated with the case of pure horizontal advection, unless a huge number of integration steps are to be taken.