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Discontinuous Galerkin methods for Hamiltonian ODEs and PDEs. (English) Zbl 1380.65288

Summary: In this article, we present a unified framework of discontinuous Galerkin (DG) discretizations for Hamiltonian ODEs and PDEs. We show that with appropriate numerical fluxes the numerical algorithms deduced from DG discretizations can be combined with the symplectic methods in time to derive the multi-symplectic PRK schemes. The resulting numerical discretizations are applied to the linear and nonlinear Schrödinger equations. Some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
35L65 Hyperbolic conservation laws
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