A well-balanced spectral volume scheme with the wetting–drying property for the shallow-water equations

Abstract

The shallow-water equations are widely used to model surface water bodies, such as lakes, rivers and the swash zone in coastal flows. Physically congruent solutions are characterized by non-negative water depth, and many numerical methods may fail to preserve this property at the discrete level when moving wet–dry transitions are present in the physical domain. In this paper, we present a spectral-volume method for the approximate solution of the one-dimensional shallow-water equations, which is third-order accurate in wet regions, far from discontinuities, and which is well balanced for water at rest states: the stability of the solution is ensured if reconstruction and limitation of variables preserves non-negativity of the depth and a suitable constraint for the time step length is satisfied. A number of numerical experiments are reported, showing the promising capabilities of the model to solve problems with non-trivial topographies and friction.