On the multisymplecticity of partitioned Runge-Kutta and splitting methods

Authors:
Brett N. Ryland;Robert I. Mclachlan;Jason Frank
Affiliations:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand;Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand;Center for Mathematics and Computer Science (CWI), Amsterdam, The Netherlands
Venue:
International Journal of Computer Mathematics - Splitting Methods for Differential Equations
Year:
2007

Citing 9
Cited 9

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On the multisymplecticity of partitioned Runge-Kutta and splitting methods

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Abstract

Although Runge-Kutta and partitioned Runge-Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrödinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on the (second order) box scheme, and construct well-defined, explicit integrators, of various orders, with local discrete multisymplectic conservation laws, based on partitioned Runge-Kutta methods. We also show that two popular explicit splitting methods are multisymplectic.