Exponentially fitted methods with a local energy conservation law

TL;DR

This paper introduces an exponentially fitted Discrete Variational Derivative method that conserves charge and energy for oscillatory Hamiltonian PDEs, demonstrated on the nonlinear Schrödinger equation.

Contribution

It presents a novel exponentially fitted scheme that preserves key physical invariants for complex Hamiltonian PDEs, improving computational efficiency and accuracy.

Findings

Conserves charge and energy discretely

Performs well on oscillatory breather wave problem

Outperforms existing conservative schemes

Abstract

A new exponentially fitted version of the Discrete Variational Derivative method for the efficient solution of oscillatory complex Hamiltonian Partial Differential Equations is proposed. When applied to the nonlinear Schroedinger equation, the new scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem whose solution is an oscillatory breather wave.