Development Of Discontinuous Galerkin Methods For Nonlinear Problems And Time Discretization Methods

In this thesis we develop local discontinuous Galerkin(LDG)methods for the fourth-order fully nonlinear Cahn-Hilliard(CH)type equations and Allen-Cahn/Cahn-Hilliard(AC/CH)system.The energy stability of the LDG meth-ods is proved for any orders of accuracy on arbitrary triangulations in any space dimension for the general nonlinear case.The LDC discretization results in high order accuracy,nonlinear stable and suitable for hp-adaptation scheme.Numeri-cal examples for the Cahn-Hilliard equations and the Allen-Cahn/Cahn-Hilliard system in one and two dimensions are presented and the numerical results illus-trate the accuracy and capability of the methods.These results indicate that the LDC method is a good tool for solving such nonlinear equations in mathematical physics.We also develop a discontinuous Galerkin method on triangular meshes to solve the reactive dynamic user equilibrium model for pedestrian flows.The pedestrian density in this model is governed by the conservation law in which the flow flux is implicitly dependent on the density through the EikonaI equation.To solve the Eikonal equation efficiently at each time level,we use the fast sweeping method.Two numerical examples are then used to demonstrate the effectiveness of the algorithm.This algorithm efficiently solves the pedestrian flows problem on arbitrary geometry domain.We explore three efficient time discretization techniques for the LDG meth-ods to solve partial differential equations(PDEs)with higher order spatial deriv-atives.The main difficulty is the stiffness of the LDC spatial discretization op-erator,which would require a unreasonably small time step for an explicit local time stepping method.We focus our discussion on the semi-implicit spectral deferred correction(SDC)method,and study its stability and accuracy when coupled with the LDG spatial discretization.We also discuss two other time discretization techniques,namely the additive Runge-Kutta(ARK)method and the exponential time differencing(ETD)method,coupled with the LDG spatial discretization.A comparison is made among these three time discretization tech-niques,to conclude that all three methods are efficient when coupled with the LDG spatial discretization for solving PDEs containing higher order spatial deriv-atives.In particular,the SDC method has the advantage of easy implementation for arbitrary order of accuracy,and the ARK method has the smallest CPU cost in our implementation.