Symplectic Finite Difference Approximations of the Nonlinear Klein-Gordon Equation

We analyse 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 travelling 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. Keywords: Symplectic approximations, Klein--Gordon equation, finite--differences. AMS(MOS) subject index: 65M99,65M05,65M20,35Q20. Running title: Symplectic finite difference approximations. 1 Introduction The nonlinear Klein--Gordon equation, u tt \Gamma \Deltau + g(u) = 0 ; (1) is used to model many different nonlinear phenomena, including the propagation of dislocations in crystals, the behaviour of elementary particles..