Numerical scheme for the generalised Serre–Green–Naghdi model

Abstract

A numerical scheme for solving the recently derived generalised Serre–Green–Naghdi equations which produce a family of equations modelling waves in shallow water with varying dispersion relationships, is described. The numerical scheme extends schemes applied to the classical Serre–Green–Naghdi equations written in conservation law form and is the first validated scheme for all admitted dispersion relationships. The consistency of the described numerical scheme for all members of the family of equations is in contrast to other numerical schemes which require specialised modifications for individual members of the family of equations. A typical second-order implementation of this numerical scheme is then demonstrated and validated using known analytic solutions; the travelling solitary wave and the dam-break problem. The numerical method is further validated for general members of the family of equations using forced solutions. Select members of this family are then used to simulate the evolution of rectangular waves of depression for which there are experimental results. The numerical method is shown to be conservative, robust and second-order accurate for the entire family of equations. The validated numerical solutions support a classification of the family of equations based on their linear dispersive properties which includes advancing dispersive wave trains in contrast to classical trailing dispersive wave trains.