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
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