Exponential Integrators Preserving Local Conservation Laws of PDEs with Time-Dependent Damping/Driving Forces

TL;DR

This paper develops second-order structure-preserving numerical methods for PDEs with time-dependent damping or driving forces, ensuring local conservation laws are maintained, and demonstrates their effectiveness through applications to Schrödinger and Camassa-Holm equations.

Contribution

The paper introduces a novel class of second-order discretizations that preserve local conservation laws in PDEs with time-dependent damping/driving terms, extending structure-preserving algorithms.

Findings

Methods successfully preserve local conservation laws.

Numerical experiments show improved stability and accuracy.

Approach outperforms existing schemes in key tests.

Abstract

Structure-preserving algorithms for solving conservative PDEs with added linear dissipation are generalized to systems with time-dependent damping/driving terms. This study is motivated by several PDE models of physical phenomena, such as Korteweg-de Vries, Klein-Gordon, Schr\"{o}dinger, and Camassa-Holm equations, all with damping/driving terms and time-dependent coefficients. Since key features of the PDEs under consideration are described by local conservation laws, which are independent of the boundary conditions, the proposed (second-order in time) discretizations are developed with the intent of preserving those local conservation laws. The methods are respectively applied to a damped-driven nonlinear Schr\"{o}dinger equation and a damped Camassa-Holm equation. Numerical experiments illustrate the structure-preserving properties of the methods, as well as favorable results over other competitive schemes.