Linearly implicit exponential integrators for damped Hamiltonian PDEs

TL;DR

This paper introduces structure-preserving linearly implicit exponential integrators for damped Hamiltonian PDEs, which are computationally efficient and maintain key invariants, demonstrated through applications to several classical equations.

Contribution

The paper develops novel linearly implicit exponential integrators for damped Hamiltonian PDEs, including an exponential Kahan's method, with proven preservation of invariants and improved computational efficiency.

Findings

Integrators effectively preserve dissipation rates and Hamiltonian invariants.

Numerical experiments demonstrate the methods' efficiency and accuracy.

Applications to damped Burger's, KdV, and nonlinear Schrödinger equations show practical utility.

Abstract

Structure-preserving linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping. Linearly implicit integrators are derived by polarizing the polynomial terms of the Hamiltonian function and portioning out the nonlinearly of consecutive time steps. They require only a solution of one linear system at each time step. Therefore they are computationally more advantageous than implicit integrators. We also construct an exponential version of the well-known one-step Kahan's method by polarizing the quadratic vector field. These integrators are applied to one-dimensional damped Burger's, Korteweg-de-Vries, and nonlinear Schr{\"o}dinger equations. Preservation of the dissipation rate of linear and quadratic conformal invariants and the Hamiltonian is illustrated by numerical experiments.