FINITE-DIFFERENCE CALCULUS INVARIANT STRUCTURE OF A CLASS OF ALGORITHMS FOR THE NONLINEAR KLEIN-GORDON EQUATION

In a previous work, the authors have presented a formalism for derivin g systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quant ities. This formalism is herein cast in the context of exact finite di fference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus: in the same sense as that of the cor responding differential equation being derivable from its associated e nergy function (a conserved quantity). A clear ramification of this re sult is that the derived algorithms preserve certain discrete invarian t quantities, which are the consistent counterpart of the invariant qu antities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the cont ext of finite difference calculus. Some numerical experiments are pres ented to illustrate the conservation property of the proposed algorith ms.