The multiplier method to construct conservative finite difference schemes for ordinary and partial differential equations

TL;DR

This paper introduces the multiplier method for constructing conservative finite difference schemes that preserve conservation laws in ordinary and partial differential equations, applicable without requiring Hamiltonian structures.

Contribution

The paper proposes a novel multiplier-based discretization approach that ensures conservation laws are exactly preserved in finite difference schemes, regardless of the system's variational structure.

Findings

Discrete densities are exactly conserved.

Method is consistent for any order when the inverse multiplier exists.

Applicable to dissipative problems and scalar ODEs with singular multipliers.

Abstract

We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on discretizing conservation law multipliers and their associated density and flux functions. We show that the proposed discretization is consistent for any order of accuracy when the discrete multiplier has a multiplicative inverse. Moreover, we show that by construction, discrete densities can be exactly conserved. In particular, the multiplier method does not require the system to possess a Hamiltonian or variational structure. Examples, including dissipative problems, are given to illustrate the method. In the case when the inverse of the discrete multiplier becomes singular, consistency of the method is also established for scalar ODEs provided the discrete multiplier and density are zero-compatible.