Nonlinear operator integration factor splitting for the shallow water equations

The purpose of this paper is to explore an alternative to the traditional interpolation based semi-Lagrangian time integrators employed in atmospheric models. A novel aspect of the present study is that operator splitting is applied to a purely hyperbolic problem rather than the incompressible Navier-Stokes equations. The underlying theory of operator integration factor splitting is reviewed and the equivalence with semi-Lagrangian schemes is established. A nonlinear variant of integration factor splitting is proposed where the advection operator is expressed in terms of the relative vorticity and kinetic energy. To preserve stability, a fourth order Runge-Kutta scheme is applied for sub-stepping. An analysis of splitting errors reveals that OIFS is compatible with the order conditions for linear multi-step methods. The new scheme is implemented in a spectral element shallow water model using an implicit second order backward differentiation formula for Coriolis and gravity wave terms. Numerical results for standard test problems demonstrate that much larger time steps are possible.