On superconvergent discontinuous Galerkin methods for Schrödinger equations and sparse grid central discontinuous Galerkin method

"In this thesis, we design and analyze a discontinuous Galerkin (DG) method for one-dimensional Schrodinger equations under a general class of numerical fluxes, and another efficient DG method for high-dimensional hyperbolic equations.In the first DG method, we develop an ultra-weak discontinuous Galerkin (UWDG) method to solve the one-dimensional nonlinear Schrödinger equation. Stability conditions and error estimates are derived for the scheme with a general class of numerical fluxes. The error estimates are based on detailed analysis of the projection operator associated with each individual flux choice. Depending on the parameters, we find out that in some cases, the projection can be defined element-wise, facilitating analysis. In most cases, the projection is global, and its analysis depends on the resulting 2 x 2 block-circulant matrix structures. For a large class of parameter choices, optimal a priori L2 error estimates can be obtained. Numerical examples are provided verifying theoretical results.In addition to the stability and error analysis, we analyze the superconvergence properties of the UWDG method for one-dimensional linear Schrodinger equation with various choices of flux parameters. Depending on the flux choices and if the polynomial degree k is even or odd, we prove 2k or (2k - 1)-th order superconvergence rate for cell averages and numerical flux of the function, as well as (2k - 1) or (2k - 2)-th order for numerical flux of the derivative. In addition, we prove superconvergence of (k + 2) or (k + 3)-th order of the UWDG solution towards a special projection. At a class of special points, the function values and the first and second order derivatives of the UWDG solution are superconvergent with order k + 2, k + 1, k, respectively. The proof relies on the correction function techniques initiated in Cao, et. al. (2014), and applied to Cao, et. al. (2017) for direct DG (DDG) methods for diffusion problems. By negative norm estimates, we apply the post processing technique and show that the accuracy of our scheme can be enhanced to order 2k. Theoretical results are verified by numerical experiments.In the second DG method, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the dual mesh is introduced and analyzed, resulting in a sparse finite element space that can be used for non-periodic problems. Theoretical results, such as L2 stability and error estimates are obtained for scalar problems. CFL conditions are studied numerically comparing discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods. Numerical results including scalar linear equations, acoustic and elastic waves are provided."--Pages ii-iii.