A symplectic approximation with nonlinear stability and convergence analysis for efficiently solving semi-linear Klein–Gordon equations

It is noted that the geometric integration for nonlinear Hamiltonian PDEs has led to the development of numerical schemes which systematically incorporate qualitative features of the underlying problem into their structure. The symplectic approximation to nonlinear Hamiltonian PDEs should be useful when studying the geometric integration. However, it is also an important aspect to analyse the nonlinear stability and convergence when a fully discrete symplectic scheme is designed for nonlinear Hamiltonian PDEs. In this paper, we develop a symplectic approximation for efficiently solving semi-linear Klein–Gordon equations, which can be formulated as an abstract Hamiltonian system of second-order ordinary differential equation (ODE). To this end, we first analyse an extended Runge–Kutta–Nyström-type (RKN-type) approximation based on the operator-variation-of-constants formula (also known as the Duhamel Principle) for the abstract Hamiltonian system of second-order ODE. We then present the symplectic conditions for the approximation, and derive some practical symplectic approximation schemes for semi-linear Klein–Gordon equations. The most important is that we commence the nonlinear stability and convergence analysis for the symplectic approximation to semi-linear Klein–Gordon equations. The results of various numerical experiments, including Klein–Gordon equations in the nonrelativistic limit regime where the solution is highly oscillating in time, demonstrate the remarkable advantage and efficiency of the symplectic approximation schemes in comparison with existing numerical schemes in the literature.