Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method

TL;DR

This paper introduces a systematic discretization method for Hamiltonian and dissipative PDEs that preserves energy or its monotonic decrease, ensuring accurate numerical simulations of physical systems.

Contribution

The paper develops the 'Average Vector Field' method, a novel approach that exactly preserves energy in Hamiltonian PDEs and the correct dissipation in dissipative PDEs.

Findings

Successfully applied to various PDEs including sine-Gordon and Korteweg-de Vries.

Preserves energy exactly in Hamiltonian systems.

Ensures correct energy dissipation in dissipative systems.

Abstract

We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrodinger, (linear) time-dependent Schrodinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.