Third-order Finite Volume/Finite Element Solution of the Fully Nonlinear Weakly Dispersive Serre Equations

TL;DR

This paper introduces a third-order hybrid finite volume/finite element scheme for solving the fully nonlinear, weakly dispersive Serre equations, effectively handling higher-order dispersive terms with improved accuracy and stability.

Contribution

It reformulates the Serre equations to eliminate mixed derivatives, enabling a stable, accurate hybrid numerical scheme combining finite volume and finite element methods.

Findings

The scheme achieves third-order accuracy.

It accurately models discontinuous flows and dam-break problems.

The method is simple to implement and stable across various test cases.

Abstract

The nonlinear weakly dispersive Serre equations contain higher-order dispersive terms. This includes a mixed derivative flux term which is difficult to handle numerically. The mix spatial and temporal derivative dispersive term is replaced by a combination of temporal and spatial terms. The Serre equations are re-written so that the system of equations contain homogeneous derivative terms only. The reformulated Serre equations involve the water depth and a new quantity as the conserved variables which are evolved using the finite volume method. The remaining primitive variable, the velocity is obtained by solving a second-order elliptic equation using the finite element method. To avoid the introduction of numerical dispersion that may dominate the physical dispersion, the hybrid scheme has third-order accuracy. Using analytical solutions, laboratory flume data and by simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including discontinuous flows.