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Invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation. (English) Zbl 0790.65101

A formalism to derive second-order invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation is presented. Three algorithms are proposed which conserve in discrete form either the total energy or the momentum.

A geometric interpretation of the algorithms is given and their stability and accuracy are investigated. An efficient solution procedure is discussed for the computer implementation of the proposed algorithms and several numerical examples are presented, which include collisions of solitary waves to demonstrate the conservation and robustness properties of those algorithms.

MSC:
65Z05Applications of numerical analysis to physics
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
81Q05Closed and approximate solutions to quantum-mechanical equations
35Q53KdV-like (Korteweg-de Vries) equations