A Radial Basis Function (RBF) Method for the Fully Nonlinear 1D Serre Green-Naghdi Equations

TL;DR

This paper introduces a spectral RBF-based numerical method for solving the fully nonlinear 1D Serre Green-Naghdi equations, achieving spectral accuracy with a simple MATLAB implementation.

Contribution

It presents a novel RBF spectral discretization combined with finite differences for the Serre equations, providing an efficient and accurate numerical approach.

Findings

Achieves spectral (exponential) accuracy on test cases

Provides a simple MATLAB code under 100 lines

Demonstrates effectiveness for nonlinear shallow water equations

Abstract

In this paper, we present a spectral method based on Radial Basis Functions (RBFs) for numerically solving the fully nonlinear 1D Serre Green-Naghdi equations. The approximation uses an RBF discretization in space and finite differences in time; the full discretization is obtained by the method of lines technique. For select test cases (see Bonnenton et al. [2] and Kim [11]) the approximation achieves spectral (exponential) accuracy. Complete \textsc{matlab} code of the numerical implementation is included in this paper (the logic is easy to follow, and the code is under 100 lines).