A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system

TL;DR

This paper introduces a modified Galerkin finite element method tailored for the fully nonlinear Serre-Green-Naghdi system, enabling the use of low-order elements while accurately handling third-order derivatives.

Contribution

It presents a novel numerical approach that simplifies finite element implementation for complex shallow water equations without sacrificing accuracy.

Findings

The method effectively captures wave dynamics in shallow water.

It conserves key physical quantities during simulations.

Validated against laboratory experiments and theoretical results.

Abstract

A new modified Galerkin / Finite Element Method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low-order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results.