An unconditionally stable scheme for the shallow water equations

A finite-difference scheme for solving the linear shallow water equations in a bounded domain is described. Its time step is not restricted by a Courant-Friedrichs-Levy (CFL) condition. The scheme, known as Israeli-Naik-Cane (INC). is the offspring of semi-Lagrangian (SL) schemes and the Cane-Patton (CP) algorithm. In common with the latter it treats the shallow water equations implicitly in v and with attention to wave propagation in x. Unlike CP, it uses an SL-like approach to the zonal variations, which allows the scheme to apply to the full primitive equations. The great advantage, even in problems where quasigeostrophic dynamics are appropriate in the interior, is that the INC scheme accommodates complete boundary conditions.