SAV Based Conservative Finite Element Method For Several Schr(?)dinger Equations

The Schr(?)dinger equation is a fundamental equation of quantum mechanics,and it is usually difficult to obtain an accurate analytical solution.Therefore,developing efficient numerical methods to solve equations to obtain high-precision numerical solutions is a research hotspot.Since the Schr(?)dinger equation has the characteristics of nonlinearity,high oscillation and conservation.These characteristics bring difficulties and challenges to high-precision numerical simulation.Designing an efficient algorithm that preserves the conservation properties of the Schr(?)dinger equation is a challenging research topic.We expect to build a high-precision numerical simulation method with good approximation and stability,and the final numerical solution can satisfy the conservation properties of some physical quantities of the original problem model(such as mass conservation,energy conservation,etc.),so that in long-term numerical simulation can accurately display physical phenomena.In this paper,we propose,analyze,and numerically verify conservative finite element methods for the nonlinear Schrodinger equation,Schrodinger-Poisson equation,and the magnetic field-constrained Schrodinger equation.By introducing scalar auxiliary variables(SAV),these kinds of Schr(?)dinger equations are rewritten into a new equivalent format,and the energy is transformed into a quadratic form.We first used the standard continuous finite element method for spatial discretization,and proved that the semi-discrete finite element scheme can preserve the conservation of mass and energy laws.The Crank-Nicolson method and the relaxed Runge-Kutta method are used for time discretization,respectively.The fully discrete scheme proposed by the Crank-Nicolson method is implicit,and the nonlinear term is processed by Newton iteration.This numerical scheme can maintain mass and energy conservation.Besides,the fully discrete scheme obtained by applying the relaxed Runge-Kutta method is explicit,the relaxation Runge-Kutta scheme can maintain mass or energy conservation according to the appropriate relaxation coefficient.We give extensive numerical examples to demonstrate the accuracy of the proposed method,as well as the conservation of mass and energy in long-term simulations.