Evaluation of time integrators for the shallow water equations

The efficiency of low-order and high-order time integrators and the efficiency of implicit methods versus explicit methods for the shallow water equations were compared. The high-order Runge Kutta 4 method is more efficient than low order methods like Leap-Frog and the stabilized Runge-Kutta method, because the extra F evaluations are compensated by a large stability region and fourth-order accuracy. The fourth-order accuracy is essential to remain accurate with respect to second-order methods, when the largest possible timestep is used. When for accuracy reasons the timestep is inside the stability region, the Runge-Kutta 4 method is more efficient, otherwise the implicit method is more efficient. This is especially the case if the solutions are almost stationary. Conclusions are tentative because the influence of the error due to space discretization, which may partly offset the shown error, is not included.