Conformal variational discretisation of infinite dimensional Hamiltonian systems with gradient flow dissipation

TL;DR

This paper introduces a mixed variational discretisation method for infinite-dimensional Hamiltonian systems with gradient flow dissipation, preserving geometric structure and physical properties in numerical approximations.

Contribution

It proposes a novel variational approach that maintains the separation of conservative and dissipative parts, ensuring physical fidelity in numerical solutions.

Findings

Conservation laws are preserved in numerical discretisation.

The method achieves a priori convergence estimates.

Numerical tests confirm theoretical properties on Korteweg-de Vries and Navier-Stokes equations.

Abstract

Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a nonconservative part, in the form of gradient flow dissipation. Traditional numerical approximations of this class of problem typically fail to retain the separation into conservative and nonconservative parts hence leading to unphysical solutions. In this work we propose a mixed variational method that gives a semi-discrete problem with the same geometric structure as the infinite-dimensional problem. As a consequence the conservation laws and the dissipative terms are retained. A priori convergence estimates on the solution are established. Numerical tests of the Korteweg-de Vries equation and of the two-dimensional Navier-Stokes equations on the torus and on the sphere are presented to corroborate the theoretical findings.