Two novel classes of energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions

We develop two novel classes of energy-preserving algorithms for the sine-Gordon (SG) equation subject to Neumann boundary conditions. The cosine pseudo-spectral method is first employed for spatial discretization under two different meshes to obtain two structure-preserving semi-discrete schemes, which are recast into a finite-dimensional Hamiltonian system and thus admit an energy conservation law. Then we combine a prediction-correction Crank-Nicolson method with an energy projection technique to arrive at fully discrete energy-preserving schemes. Alternatively, we introduce a supplementary variable to transform the SG model into a relaxation system, which is named the supplementary variable method (SVM). Furthermore, we apply 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 are presented to demonstrate the accuracy and efficiency of the proposed schemes. Detailed numerical comparisons among these methods themselves and other reported ones are carried out as well.