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An energy‐preserving C rank– N icolson G alerkin method for H amiltonian partial differential equations
Author(s) -
Li Haochen,
Wang Yushun,
Sheng Qin
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22062
Subject(s) - mathematics , discretization , partial differential equation , hamiltonian (control theory) , crank–nicolson method , finite element method , mathematical analysis , ordinary differential equation , norm (philosophy) , galerkin method , discontinuous galerkin method , differential equation , mathematical optimization , physics , thermodynamics , political science , law
A semidiscretization based method for solving Hamiltonian partial differential equations is proposed in this article. Our key idea consists of two approaches. First, the underlying equation is discretized in space via a selected finite element method and the Hamiltonian PDE can thus be casted to Hamiltonian ODEs based on the weak formulation of the system. Second, the resulting ordinary differential system is solved by an energy‐preserving integrator. The relay leads to a fully discretized and energy‐preserved scheme. This strategy is fully realized for solving a nonlinear Schrödinger equation through a combination of the Galerkin discretization in space and a Crank–Nicolson scheme in time. The order of convergence of our new method is O ( τ 2 + h 2 ) if the discrete L 2 ‐norm is employed. An error estimate is acquired and analyzed without grid ratio restrictions. Numerical examples are given to further illustrate the conservation and convergence of the energy‐preserving scheme constructed.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1485–1504, 2016