A Fully Implicit Domain Decomposition Algorithm for Shallow Water Equations on the Cubed-Sphere

Popular approaches for solving the shallow water equations (SWE) for climate modeling are explicit and semi-implicit methods, both have certain constraints on the time step size. In this paper, we propose and study a fully implicit method which imposes no limit on the time step size, but requires the solution of a large sparse nonlinear system of equations at every time step. The focus of the paper is a parallel, fully coupled, Newton-Krylov-RAS algorithm with a Jacobian matrix explicitly calculated on a weakly non-matching cubed-sphere mesh. Here RAS is a restricted additive Schwarz method. We show numerically that with such a preconditioned nonlinearly implicit method the time step size is no longer constrained by the CFL condition and report superlinear speedup of the algorithm on machines with thousands of processors, and for problems with smooth and non-smooth solutions.