INVARIANT-CONSERVING FINITE-DIFFERENCE ALGORITHMS FOR THE NONLINEAR KLEIN-GORDON EQUATION

A formalism for systematically deriving second order accurate finite d ifference algorithms which conserve certain invariant quantities in th e original nonlinear PDEs is presented. Three algorithms are derived f or the nonlinear Klein-Gordon equation (NLKGE) based on the proposed f ormalism. The local conservation laws of the NLKGE form the basic star ting point in our derivation, which hinges essentially on the commutat ivity of certain finite difference operators. Such commutativity in th e discrete approximations allows a preservation of the derivation prop erties of the continuous counter-parts at the PDE level. With appropri ate boundary conditions, the proposed algorithms preserve in the discr ete sense either the total system energy or the system's linear moment um. Several variants of the present algorithms and their relation to p reviously proposed algorithms are discussed. An analysis of the accura cy and stability is conducted to compare the different variants of the proposed algorithms. The preservation of energy of the present algori thms for the NLKGE can also be viewed as providing a method of stabili zation for conditionally stable algorithms for the linear wave equatio n. The computer implementation of the proposed algorithms, with the tr eatment of the boundary conditions, is presented in detail. Numerical examples are given concerning soliton collisions in the sine-Gordon eq uation, the double sine-Gordon equation, and the phi+/-4 ('phi-four') equation. The numerical results demonstrate that the present algorithm s can preserve accurately (up to 10 decimal digits) the total system e nergy for a very coarse grid. Reliable algorithms for Josephson juncti on models, which contain dissipation, damping mechanisms and driving b ias current, are obtained as direct by-products of the proposed invari ant-conserving algorithms for the NLKGE. Even though presented mainly for the 1-D case, the proposed algorithms are generalizable to the 2-D and 3-D cases, and to the case of complex-valued NLKGE.