Discrete gradient methods have an energy conservation law

TL;DR

This paper demonstrates that discrete gradient methods for conservative PDEs inherently possess a time-discrete conservation law that preserves the same conserved quantity as the continuous system, with fluxes constructed via discrete gradients.

Contribution

It establishes a general link between discrete gradient methods and conservation laws, showing these methods preserve energy at the discrete level for various conservative PDEs.

Findings

Discrete gradient methods conserve energy in discrete time.

The discrete conservation law mirrors the continuous one in conserved density.

Fluxes are obtained by replacing derivatives with discrete gradients.

Abstract

We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved density as the continuous conservation law, while its flux is found by replacing all derivatives of the conserved density appearing in the continuous flux by discrete gradients.