Unconditionally stable space–time discontinuous residual distribution for shallow-water flows

This article describes a discontinuous implementation of residual distribution for shallow-water flows. The emphasis is put on the space–time implementation of residual distribution for the time-dependent system of equations with discontinuity in time only. This lifts the time-step restriction that even implicit continuous residual distribution schemes invariably suffer from, and thus leads to an unconditionally stable discretisation. The distributions are the space–time variants of the upwind distributions for the steady-state system of equations and are designed to satisfy the most important properties of the original mathematical equations: positivity, linearity preservation, conservation and hydrostatic balance. The purpose of the several numerical examples presented in this article is twofold. First, to show that the discontinuous numerical discretisation does indeed exhibit all the desired properties when applied to the shallow-water equations. Second, to investigate how much the time step can be increased without adversely affecting the accuracy of the scheme and whether this translates into gains in computational efficiency. Comparison to other existing residual distribution schemes is also provided to demonstrate the improved performance of the scheme.