Abstract
Structure-preserving algorithms, or multi-symplectic methods for partial differential equations, though less developed, have been considered as important as those for ordinary differential equations. In Chap. 9, the idea of ERKN methods for oscillatory ordinary differential equations is extended to the integration of oscillatory partial differential equations. Multi-symplectic discretizations of the Hamiltonian wave equations and the corresponding discrete conservation laws are investigated. The discretization by two symplectic ERKN methods in time and space, or by a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method in space, leads to a multi-symplectic integrator. Two explicit multi-symplectic extended leap-frog integrators are derived. The numerical stability and dispersive properties of the integrators are analyzed. The two integrators are applied to the linear wave equation and the sine-Gordon equation.
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Wu, X., You, X., Wang, B. (2013). Extended Leap-Frog Methods for Hamiltonian Wave Equations. In: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35338-3_9
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DOI: https://doi.org/10.1007/978-3-642-35338-3_9
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