Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting

TL;DR

This paper develops structure-preserving numerical integrators for dissipative systems with a thermodynamic GENERIC framework, splitting reversible and irreversible parts, and demonstrates their superior accuracy and structure preservation through numerical experiments.

Contribution

It introduces a novel splitting method that preserves the thermodynamic structure of dissipative systems, combining symplectic and explicit midpoint methods with modified energies.

Findings

Superior energy conservation and entropy production control.

Preservation of conformal symplectic structure in damped systems.

Validated through numerical experiments showing improved accuracy.

Abstract

We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible-irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g., Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.