Symplectic integration of PDEs using Clebsch variables

TL;DR

This paper introduces a symplectic integration method for PDEs like Burgers' equation by lifting them to a collective Hamiltonian system using Clebsch variables, leading to improved conservation properties in numerical simulations.

Contribution

It presents a novel approach to numerically integrate Lie-Poisson PDEs by transforming them into symplectic systems via Clebsch variables, enabling the use of symplectic integrators.

Findings

Excellent conservation properties demonstrated in numerical examples

Increased phase-space dimension is offset by symplectic integration advantages

Method applicable to various PDEs with Lie-Poisson structure

Abstract

Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations,...) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.