Efficient energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions

TL;DR

This paper introduces two new energy-preserving numerical methods for solving the sine-Gordon equation with Neumann boundary conditions, combining spectral spatial discretization with structure-preserving time integration.

Contribution

The paper develops two novel classes of fully discrete energy-preserving algorithms using spectral methods and relaxation systems for the sine-Gordon equation.

Findings

Methods demonstrate high accuracy and efficiency

Numerical results confirm energy preservation and stability

Algorithms outperform existing schemes in benchmarks

Abstract

We present two novel classes of fully discrete energy-preserving algorithms for the sine-Gordon equation subject to Neumann boundary conditions. The cosine pseudo-spectral method is first used to develop structure-preserving spatial discretizations under two different meshes, which result two finite-dimensional Hamiltonian ODE systems. Then we combine the prediction-correction Crank-Nicolson scheme with the projection approach to arrive at fully discrete energy-preserving methods. Alternatively, we introduce a supplementary variable to transform the initial model into a relaxation system, which allows us to construct structure-preserving algorithms more easily. We then discretize the relaxation system directly by using the cosine pseudo-spectral method in space and the prediction-correction Crank-Nicolson scheme in time to derive a new class of energy-preserving schemes. The proposed methods can be solved effectively by the discrete Cosine transform. Some benchmark examples and numerical comparisons are presented to demonstrate the accuracy, efficiency and superiority of the proposed schemes.