Conservative Discontinuous Galerkin Methods For Nonlinear Schr(?)dinger Equations

This article focuses on the nonlinear Schr¨odinger equations in the DDG and LDG two different schemes. And we use Crank-Nicolson and Strang-splitting methods to discrete time for solving the problem.The first part introduces the origin and status of the Schr¨odinger equations. First, we give the background of the equation and its importance in quantum mechanics from the emotional point of view. And then describing some of the conclusions in finite difference methods, spectral methods, split methods and finite element methods applying on the Schr¨odinger equations.The DDG and LDG methods of semi-discrete and fully discrete schemes are described in the second and third parts of this article, also, mass and energy conservation properties are theoretically proved following the schemes. First of all, we give the DDG and LDG schemes of the equation(2.1) in 1D and 2D space, along discrete time with Crank-Nicolson methods to obtain fully discrete. On the same time, the proof of the mass and energy conservations of the DDG semi-discrete scheme can be seen in the chapter. What special attention we should give to ”energy” is that it is not the traditional energy format, see definition in Theorem(2.4.1). The third part proves the mass and energy conservation in LDG semi-discrete format.In the fourth section, we present special treatments to the nonlinear terms of the nonlinear Schr¨odinger equations. Strang-splitting methods translating the equation(2.1) into a ODE equation(4.1) and a linear Schr¨odinger equation(4.2).It is convenient for solving nonlinear Schr¨odinger equations, and we can put more attention to explore nature properties in the article.We emphasis the fifth part because it gives numerical processing in the 1D and2 D dimensions, verifying most of the properties discussed above.Numerical results can tell to us that the convergence of the original problem can reach to 3-order.Also, in the computational efficiency, DDG are better than LDG both in 1D and2 D cases. When verifying the mass property, DDG and LDG both can conserve mass in 1D and 2D space.