Time Integration of the Shallow Water Equations in Spherical Geometry

The shallow water equations in spherical geometry provide a prototype for developing and testing numerical algorithms for atmospheric circulation models. In a previous paper we have studied a spatial discretization of these equations based on an Osher-type finite-volume method on stereographic and latitude–longitude grids. The current paper is a companion devoted to time integration. Our main aim is to discuss and demonstrate a third-order, A-stable, Runge–Kutta–Rosenbrock method. To reduce the costs related to the linear algebra operations, this linearly implicit method is combined with approximate matrix factorization. Its efficiency is demonstrated by comparison with a classical, third-order explicit, Runge–Kutta method. For that purpose we use a known test set from literature. The comparison shows that the Rosenbrock method is by far superior.