Understanding the treatment of waves in atmospheric models. Part 1: The shortest resolved waves of the 1D linearized shallow-water equations

摘要

This article provides an intercomparison of the dispersive and diffusive properties of several standard numerical methods applied to the 1D linearized shallow-water equations without the Coriolis term, including upwind and central finite-volume, spectral finite-volume, discontinuous Galerkin, spectral element, and staggered finite-volume. All methods are studied up to tenth-order accuracy, where possible. A consistent framework is developed which allows for direct intercomparison of the ability of these methods to capture the behaviour of linear gravity waves. The Courant-Friedrichs-Lewy (CFL) condition is also computed, which is important for gauging the stability of these methods, and leads to a measure of approximate equal error cost. The goal of this work is threefold: first, to determine the shortest wavelength which can be considered `resolved' for a particular method; second, to determine the effect of increasing the order of accuracy on the ability of a method to capture wave-like motion; and third, to determine which numerical methods offer the best treatment of wave-like motion.