The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations

Abstract

Abstract Errors caused by discrete time stepping may be an important component of total model error in contemporary atmospheric and oceanic simulations. To reduce time-stepping errors in leapfrog integrations, the Robert–Asselin–Williams (RAW) filter was proposed by the author as a simple improvement to the widely used Robert–Asselin (RA) filter. The present paper examines the behavior of the RAW filter in semi-implicit integrations. First, in a linear theoretical analysis, the stability and accuracy are interrogated by deriving analytic expressions for the amplitude errors and phase errors. Then, power-series expansions are used to interpret the leading-order errors for small time steps and hence to identify optimal values of the filter parameters. Finally, the RAW filter is tested in a realistic nonlinear setting, by applying it to semi-implicit integrations of the elastic pendulum equations. The results suggest that replacing the RA filter with the RAW filter could reduce time-stepping errors in semi-implicit integrations.