Comparaison of Exponential integrators and traditional time integration schemes for the Shallow Water equations

The time integration scheme is probably one of the most fundamental choice in the development of an ocean model. In this paper, we investigate several time integration schemes when applied to the shallow water equations. These set of equations is accurate enough when modelling a small depth ocean and is also relevant to study as it is the one solved for the barotropic (i.e. vertically averaged) component of a three dimensional ocean model. We analysed different schemes for the shallow water equations linearised around (h, 0). This simplified model give a good idea of difficulties occurring when applying a time integrator. Explicit schemes are accurate but the time step is constraint by the Courant-Friedrichs-Lewy stability condition. Implicit schemes can be unconditionally stable but not very accurate. In this article we propose a detailed comparison of such classical schemes with exponential integrators. The accuracy and the computational costs are analysed in different configurations..