Linearly implicit exponential integrators for damped Hamiltonian PDEs

TL;DR

This work addresses numerically integrating damped Hamiltonian PDEs while exactly preserving linear conformal invariants and accurately dissipating higher-order invariants. It introduces linearly implicit exponential integrators built from polarization of the Hamiltonian and discrete-gradient concepts, yielding schemes that require only a single linear solve per time step. A two-step polarized discrete-gradient framework and exponential Kahan-type variants are developed and applied to 1D damped Burger's, KdV, and NLS equations, with results showing exact or near-exact invariant preservation and favorable long-time behavior. The findings demonstrate computational efficiency over fully implicit exponential methods, suggesting these linearly implicit exponential integrators as practical tools for dissipative PDEs and potential extensions to higher dimensions and time-varying damping.

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{ö}dinger equations. Preservation of the dissipation rate of linear and quadratic conformal invariants and the Hamiltonian is illustrated by numerical experiments.

Figures (11)

• Figure 1: Solutions of the damped Burger's equation with EK.
• Figure 2: Damped Burger's equation (EK): mass (left), exact Hamiltonian (middle) and modified Hamiltonian (right).
• Figure 3: Damped Burger's equation (EK): residuals of mass (left), Hamiltonian (middle) and modified Hamiltonian (right).
• Figure 4: Damped Burger's equation (CIMP): residuals of mass (left), Hamiltonian (right).
• Figure 5: Solutions of the damped Burger's equation with Kahan's method \ref{['Two_step_expkahan']}.