Geometric numerical integrators for Hunter-Saxton-like equations

TL;DR

This paper introduces new geometric numerical integrators for Hunter--Saxton-like equations using multi-symplectic and Hamiltonian structures, with numerical experiments showing their effectiveness.

Contribution

It develops novel multi-symplectic and Hamiltonian-preserving discretizations for Hunter--Saxton equations, including boundary condition treatment, not previously clarified.

Findings

Numerical experiments show the proposed integrators perform well.

New formulations enable better boundary condition handling.

Explicit Euler and variational derivative methods are effectively applied.

Abstract

We present novel geometric numerical integrators for Hunter--Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter--Saxton equation, the modified Hunter--Saxton equation, and the two-component Hunter--Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.