A local energy conservation law is proposed for the Klein–Gordon–Schr ¨odinger equations, which is held in any local time–space region. The local property is independent of the boundary condition and more essential than the global energy conservation law. To develop a numerical method preserving the intrinsic properties as much as possible, we propose a local energy-preserving(LEP) scheme for the equations. The merit of the proposed scheme is that the local energy conservation law can hold exactly in any time–space region. With the periodic boundary conditions, the scheme also possesses the discrete change and global energy conservation laws. A nonlinear analysis shows that the LEP scheme converges to the exact solutions with order O(τ2+ h2). The theoretical properties are verified by numerical experiments.
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机译:Inertio-gravity Poincaré waves and the quantum relativistic Klein-Gordon equation, near-inertial waves and the non-relativistic Schr?dinger equation
机译:On perturbation of a functional with the mountain pass geometry Applications to the nonlinear Schr?dinger–Poisson equations and the nonlinear Klein–Gordon–Maxwell equations