Hyperbolic relaxation technique for solving the dispersive Serre-Green-Naghdi Equations with topography

TL;DR

This paper introduces a hyperbolic relaxation method for solving the dispersive Serre-Green-Naghdi equations with topography, validated through analytical solutions and experimental comparisons.

Contribution

It extends a relaxation technique to fully non-linear equations with topography effects, providing a new numerical approach for these complex fluid dynamics problems.

Findings

The relaxation method accurately approximates the Serre-Green-Naghdi equations.

Analytical solutions verify the correctness of the relaxed model.

Numerical results agree well with experimental data.

Abstract

The objective of this paper is to propose a hyperbolic relaxation technique for the dispersive Serre-Green-Naghdi equations (also known as the fully non-linear Boussinesq equations) with full topography effects introduced in Green, A.E. and Naghdi, P.M. (J. Fluid Mech., 78, 237-246, 1976) and Seabra-Santos el al (J. Fluid Mec.h, 176, 117-134, 1997). This is done by revisiting a similar relaxation technique introduced in Guermond el al (J. Comput. Phys., 399, 108917, 2019) with partial topography effects. We also derive a family of analytical solutions for the one-dimensional dispersive Serre-Green-Naghdi equations that are used to verify the correctness the proposed relaxed model. The method is then numerically illustrated and validated by comparison with experimental results.