A kernel-based meshless conservative Galerkin method for solving Hamiltonian wave equations

TL;DR

This paper introduces a meshless Galerkin method using radial basis functions for Hamiltonian wave equations, ensuring energy conservation through projection operators and energy-preserving discretization schemes.

Contribution

The paper presents a novel meshless Galerkin approach with energy conservation for Hamiltonian wave equations, including a new projection-based formulation and energy-preserving discretization.

Findings

The method conserves global energy exactly.

Numerical examples demonstrate high accuracy and energy preservation.

The approach provides a complete error analysis of the discretization.

Abstract

We propose a meshless conservative Galerkin method for solving Hamiltonian wave equations. We first discretize the equation in space using radial basis functions in a Galerkin-type formulation. Differ from the traditional RBF Galerkin method that directly uses nonlinear functions in its weak form, our method employs appropriate projection operators in the construction of the Galerkin equation, which will be shown to conserve global energies. Moreover, we provide a complete error analysis to the proposed discretization. We further derive the fully discretized solution by a second order average vector field scheme. We prove that the fully discretized solution preserved the discretized energy exactly. Finally, we provide some numerical examples to demonstrate the accuracy and the energy conservation.