Symplectic Finite Difference Approximations of the Nonlinear Klein-Gordon Equation

We analyze three finite difference approximations of the nonlinear Klein-Gordon equation and show that they are directly related to symplectic mappings. Two of the schemes, the Perring-Skyrme and Ablowitz-Kruskal-Ladik, are long established, and the third is a new, higher order accurate scheme. We test the schemes on traveling wave and periodic breather problems over long time intervals and compare their accuracy and computational costs with those of symplectic and nonsymplectic method-of-lines approximations and a nonsymplectic energy conserving method.