Mass Preserving Discontinuous Galerkin Methods For Schr(?)dinger Equations

In this thesis, we study the mass preserving discontinuous Galerkin methods for Schrodinger equations.In Chapter3, we first choose the numerical flux for one-dimensional and two-dimensional linear Schrodinger equations. Then, for the semi-discrete schemes and the fully-discrete schemes, we analyze the conservation of the mass. For space discretization, we use the DDG method. For time discretization, we use the Crank-Nicolson method. Numerical experiments show that, for polynomial elements of degree k, it have the optimal (k+1)th order of accuracy. After a long time calculate, it also get the optimal order of convergence. And, for different time and subdivision, the mass is also preserved.In Chapter4, we discuss the nonlinear Schrodinger equation with the potential function of two forms. We also choose the numerical flux and analyze the mass conservation of the semi-discrete schemes and the fully-discrete schemes. Different to linear Schrodinger equation, we use DDG method for space discretization, but we use Strang splitting method for time discretization. Numerical experiments show that, for polynomial elements of degree k, it also have the optimal (k+1)th order of accuracy. And after a long time calculate, it also get the optimal order of convergence. And, more remarkable, in the2D case, the penalty term β0[u]/h indeed plays a special role of stabilization for polynomials of even degree. In contrast, optimal accuracy for polynomial elements of odd degree can be achieved for β0=0. At last, the numerical tests show that the mass conservation is done.In Chapter5, we estimate the error of the semi-discrete schemes of the linear Schrodinger equation. First, we define a global projection. The main idea of introduce the global projection is because interface conditions dictated by the choice of numerical fluxes. The global projection we choose, it not only satisfy the interface conditions but also can make the trouble terms at the cell interfaces are eliminated or controlled. Next, we study the existence of the global projection. In a word, for any θ∈[0,1], PU is uniquely defined. At last, we get the error estimate of the semi-discrete schemes of the linear Schrodinger equation. For the error analysis of two dimensional case, if continue to adopt the method of one dimension, has some trouble cannot be eliminated or under control. We need to consider another solution, this also is continuing to work.Compare to traditional Schrodinger equation of conserving format, our method not only has the mass conservation, high precision and stability for a long period of time, and need not to transform equation, greatly shorten the calculation time.