Arbitrarily high-order energy-conserving methods for Poisson problems

TL;DR

This paper develops high-order energy-conserving numerical methods for Poisson problems, extending HBVMs to effectively preserve energy and Casimirs, with implementation details and numerical validation.

Contribution

It introduces a generalized class of energy-conserving methods for Poisson problems, enhancing HBVMs to better conserve invariants like Casimirs.

Findings

Methods effectively conserve energy and Casimirs.

Numerical tests confirm theoretical properties.

Implementation details facilitate practical use.

Abstract

In this paper we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some numerical tests are reported, in order to assess the theoretical findings.