Energy preserving moving mesh methods applied to the BBM equation

TL;DR

This paper develops energy-preserving moving mesh numerical methods for the BBM equation, combining partition of unity and rezoning techniques to exactly conserve a Hamiltonian approximation, with demonstrated numerical advantages.

Contribution

It introduces a novel combination of energy-preserving schemes with moving mesh methods for the BBM equation, ensuring Hamiltonian conservation.

Findings

Exact preservation of a Hamiltonian approximation in numerical schemes.

Demonstrated numerical advantages over traditional methods.

Effective application of rezoning in moving mesh context.

Abstract

Energy preserving numerical methods for a certain class of PDEs are derived, applying the partition of unity method. The methods are extended to also be applicable in combination with moving mesh methods by the rezoning approach. These energy preserving moving mesh methods are then applied to the Benjamin--Bona--Mahony equation, resulting in schemes that exactly preserve an approximation to one of the Hamiltonians of the system. Numerical experiments that demonstrate the advantages of the methods are presented.