This thesis consists of two parts: the first part discusses the Sparse Grid Discontinuous Galerkin (SGDG) method and its adaptive version, and their applications in Maxwell equations in nonlinear optical media [1, 2]; the second part discusses a Hamilton-Jacobi model of asynchronous data flow in parallel computers, and corresponding numerical simulation using Weighted Essentially Non-Oscillatory (WENO) method.SGDG method and its adaptive version were developed in recent years [3, 4, 5, 6, 7], to numerically solve different linear or nonlinear PDE problems, reducing degrees of freedom and computational cost. Compared to the previous works, this thesis mainly focuses on fully implicit SGDG method of nonlinear equations, which broadens the applications of the method. To achieve this goal, the existing SGDG package [8] is coupled with nonlinear solvers in PETSc [9]. Numerical simulations of several model equations and physical relevant problems are presented to demonstrate accuracy and robustness of the method.Presented in second part of the thesis are models of data flow on processors in a high performance computing framework involving computations necessitating inter-processor communications [10]. First comes an ordinary differential model, and its asymptotic limit results in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid governed by a conservation law. We derive a Hamilton-Jacobi equation associated with this conservation law for which the existence and uniqueness of solutions can be proved. High order WENO interpolation [11, 12], together with strong stability preserving (SSP) method for time discretization, is applied to simulation of the Hamilton-Jacobi model. We then present the results of numerical experiments for both discrete and continuum models; these show a qualitative agreement between the two and the effect of variations in the computing environment's processing capabilities on the progress of the modeled computation.
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