Convergence and component splitting for the Crank-Nicolson--Leap-Frog integration method

A new convergence condition is derived for the Crank-Nicolson--Leap-Frog integration scheme. The convergence condition guarantees second-order temporal convergence uniformly in the spatial grid size for a wide class of implicit-explicit splittings. This is illustrated by successfully applying component splitting to first-order wave equations resulting in such second-order temporal convergence. Component splitting achieves that only on part of the space domain Crank-Nicolson needs to be used. This reduces implicit solution costs when for Leap-Frog the step size is severely limited by stability only on part of the space domain, for example due to spatial coefficients of a strongly varying magnitude or locally refined space grids.