Explicit high-order structure-preserving approach for sine-Gordon equation with Neumann boundary conditions

We first derive high-order ‘symmetric image’ formulas to effectively address Neumann boundary conditions for the sine-Gordon equation. Then we propose a high-order energy-preserving approach by integrating the scalar auxiliary variable method with a fourth-order compact discretization and these derived formulas. The approach is highly efficient due to its explicit solver. Numerical experiments are carried out to demonstrate the solution accuracy, exact energy preservation, and waveform evolution.