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An SDG Galerkin structure‐preserving scheme for the Klein‐Gordon‐Schrödinger equation
Author(s) -
Wang Jialing,
Wang Yushun
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6342
Subject(s) - mathematics , discretization , partial differential equation , ordinary differential equation , mathematical analysis , bounded function , norm (philosophy) , hamiltonian (control theory) , klein–gordon equation , discontinuous galerkin method , schrödinger equation , order of accuracy , galerkin method , finite element method , differential equation , method of characteristics , mathematical optimization , physics , nonlinear system , quantum mechanics , political science , law , thermodynamics
In this paper, we use the Galerkin weak form to construct a structure‐preserving scheme for Klein‐Gordon‐Schrödinger equation and analyze its conservative and convergent properties. We first discretize the underlying equation in space direction via a selected finite element method, and the Hamiltonian partial differential equation can be casted into Hamiltonian ordinary differential equations based on the weak form of the system afterwards. Then, the resulted ordinary differential equations are solved by the symmetric discrete gradient method, which yields a charge‐preserving and energy‐preserving scheme. Moreover, the numerical solution of the proposed scheme is proved to be bounded in the discreteL ∞norm and convergent with the convergence order of O ( h 2 + τ 2 ) in the discreteL 2norm without any grid ratio restrictions, where h and τ are space and time step, respectively. Numerical experiments conducted last to verify the theoretical analysis.