Numerical solution of multiscale problems in atmospheric modeling

Explicit time integration methods are characterized by a small numerical effort per time step. In the application to multiscale problems in atmospheric modeling, this benefit is often more than compensated by stability problems and stepsize restrictions resulting from stiff chemical reaction terms and from a locally varying Courant–Friedrichs–Lewy (CFL) condition for the advection terms. In the present paper, we address this problem by a rather general splitting technique that may be applied recursively. This technique allows the combination of implicit and explicit methods (IMEX splitting) as well as the local adaptation of the time stepsize to the meshwidth of non-uniform space grids in an explicit multirate discretization of the advection terms. Using a formal representation as partitioned Runge–Kutta method, convergence of order p ⩽ 3 may be shown if some additional order conditions are satisfied. In a series of numerical tests, the convergence results are verified and the consequences of different splitting strategies like flux splitting and cell splitting are analysed.