The performance of filtered leapfrog schemes in benchmark simulations

The stabilisation of the leapfrog time-stepping scheme provided by the Robert-Asselin filter has enabled decades of atmospheric and oceanic research and weather and climate predictions. The unfortunate concomitant reduction from second-order accuracy to first-order accuracy inflicted by the filter has recently ushered in a new generation of leapfrog time filters that preserve the second-order accuracy, including the Robert-Asselin-Williams (RAW) filter and its variants. These modern filtered leapfrog schemes have previously been shown to improve numerical simulations made using both simple conceptual models and comprehensive general circulation models. However, their performance in standard benchmark experiments has not previously been assessed. Here we evaluate these filtered leapfrog schemes in four classic benchmark experiments: linear scalar advection; a nonlinear density current in the quasi-compressible equations; a nonlinear rising warm bubble in the fully compressible equations; and the linked behaviour of nonlinear twin tropical cyclones in the rotating shallow-water equations. For a given time-step size, the filtered leapfrog schemes are found to compare favourably with the third-order Runge-Kutta (RK3) scheme. They are also less computationally expensive than RK3, at roughly one-third to one-half the cost per time step. For a given computational expenditure, the filtered leapfrog schemes are found to produce smaller errors with respect to the analytical solution (where available) than RK3. Furthermore, the filtered leapfrog schemes are found to be numerically stable, even when the discretisation method splits the slow advection and diffusion modes from the fast acoustic and gravity-wave modes. Given that implementing filter upgrades requires only minimally invasive changes to an existing computer code, our results provide support for the continued use of filtered leapfrog schemes in atmosphere and ocean models.