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Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. (English) Zbl 0946.65132
The author considers partial differential equations of the form \[ M\partial_tx+ K\partial_xz= \nabla_z S(z), \] where \(z\in\mathbb{R}^d\) and \(M,K\in \mathbb{R}^{d\times d}\) are skew-symmetric matrices, and shows that Gauss-Legendre collocation in space and time leads to numerical methods that preserve symplectic conservation laws. He suggests several semi-explicit symplectic methods that use explicit or linearly implicit discretizations in time.

MSC:
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
35L15 Initial value problems for second-order hyperbolic equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
35L45 Initial value problems for first-order hyperbolic systems
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