Energy-preserving $H^1$-Galerkin schemes for the Hunter--Saxton equation

TL;DR

This paper introduces energy-preserving Galerkin schemes for the Hunter--Saxton equation, ensuring Hamiltonian conservation and efficient implementation, validated through numerical experiments.

Contribution

It presents two novel weak forms and Galerkin discretizations that preserve the Hamiltonian for the Hunter--Saxton equation.

Findings

Schemes successfully conserve the Hamiltonian.

Methods are computationally efficient with cheap $H^1$ elements.

Numerical experiments confirm effectiveness.

Abstract

We consider the numerical integration of the Hunter--Saxton equation, which models the propagation of weakly nonlinear orientation waves. For the equation, we present two weak forms and their Galerkin discretizations. The Galerkin schemes preserve the Hamiltonian of the equation and can be implemented with cheap $H^1$ elements. Numerical experiments confirm the effectiveness of the schemes.