Conservative and dissipative discretisations of multi-conservative ODEs and GENERIC systems

TL;DR

This paper introduces structure-preserving discretisation methods for conservative and GENERIC systems, ensuring invariants and dissipation are maintained at the discrete level, leading to more accurate and reliable numerical simulations.

Contribution

It presents an arbitrary-order conservative discretisation for ODEs and a GENERIC-compatible scheme that conserves energy and dissipates entropy, using auxiliary variables to replicate continuous proofs discretely.

Findings

Numerical examples demonstrate improved accuracy in Kepler and Kovalevskaya problems.

The schemes effectively preserve invariants and dissipation in complex systems.

Enhanced stability and fidelity in long-term simulations of thermodynamic systems.

Abstract

Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and dissipation laws on discretisation in time can yield vastly better approximations for the same computational effort, compared to schemes that are not structure-preserving. In this work we present two novel contributions: (i) an arbitrary-order time discretisation for general conservative ordinary differential equations that conserves all known invariants and (ii) an energy-conserving and entropy-dissipating scheme for both ordinary and partial differential equations written in the GENERIC format, a superset of Poisson and gradient-descent systems. In both cases the underlying strategy is the same: the systematic introduction of auxiliary variables, allowing for the replication at the discrete level of the proofs of conservation or dissipation. We illustrate the advantages of our approximations with numerical examples of the Kepler and Kovalevskaya problems, a combustion engine model, and the Benjamin-Bona-Mahony equation.