High-order multi-symplectic schemes for the nonlinear Klein-Gordon equation

In this paper, we construct multi-symplectic schemes with any order of accuracy for the nonlinear Klein-Gordon equation by concatenating the symplectic schemes for ODEs. Some existing schemes, such as the Preissman scheme and the Leap-frog scheme, and new multi-symplectic schemes are constructed. We also show that the composition method, which plays a crucial role in finding the high-order symplectic integrators for the ODEs, can also be applied to construct high-order multi-symplectic schemes for PDEs. Extension of the concatenating method to more than one space dimension is also discussed. Numerical experiments are presented to show the order and the efficiency of the constructed multi-symplectic schemes.