High-order Structure-preserving Methods for Damped Hamiltonian System

TL;DR

This paper introduces high-order structure-preserving numerical methods for damped Hamiltonian systems that accurately maintain energy dissipation properties, validated through numerical experiments on nonlinear PDEs.

Contribution

The paper develops a novel combination of exponential integrators and energy-preserving collocation methods for damped Hamiltonian systems, achieving high-order accuracy while preserving energy dissipation.

Findings

Methods effectively preserve energy dissipation ratio.

Numerical experiments confirm high-order accuracy.

Applicable to nonlinear PDEs like Burgers and KdV equations.

Abstract

We present a novel methodology for constructing arbitrarily high-order structure-preserving methods tailored for damped Hamiltonian systems. This method combines the idea of exponential integrator and energy-preserving collocation methods, effectively preserving the energy dissipation ratio introduced by the damping terms. We demonstrate the conservative properties of these methods and confirm their order of accuracy through numerical experiments involving the damped Burger's equation and Korteweg-de-Vries equation.