On the multisymplecticity of partitioned Runge-Kutta and splitting methods
International Journal of Computer Mathematics , Volume 84 - Issue 6 p. 847- 869
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.
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|Taylor & Francis|
|International Journal of Computer Mathematics|
|Adaptive Multisymplectic Box Schemes for Hamiltonian Wave Equations|
Ryland, B.N, McLachlan, R.I, & Frank, J.E. (2007). On the multisymplecticity of partitioned Runge-Kutta and splitting methods. International Journal of Computer Mathematics, 84(6), 847–869. doi:10.1080/00207160701458633