General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs
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In this paper we propose and investigate a general approach to constructing local energy-preserving algorithms which can be of arbitrarily high order in time for solving Hamiltonian PDEs. This approach is based on the temporal discretization using continuous Runge-Kutta-type methods, and the spatial discretization using pseudospectral methods or Gauss–Legendre collocation methods. The local energy conservation law of our new schemes is analyzed in detail. The effectiveness of the novel local energy-preserving integrators is demonstrated by coupled nonlinear Schrödinger equations and 2D nonlinear Schrödinger equations with external fields. Our new schemes are compared with some classical multi-symplectic and symplectic schemes in numerical experiments. The numerical results show the remarkable long-term behaviour of our new schemes.
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