Stability Analysis of an Odd–even-line Hopscotch Method for Three-dimensional Advection–diffusion Problems∗

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.