| This thesis contains two related topics, which are to design new discontinuous Galerkin (DG) schemes for two types of equations.;In the first part, we propose a DG method to solve the Hamilton-Jacobi equations. This method applies directly to the solution of the Hamilton-Jacobi equations. For the linear case, this method is equivalent to the traditional discontinuous Galerkin method for conservation laws with source terms. Thus, stability and error estimates are straightforward. For the nonlinear convex Hamiltonians, numerical experiments demonstrate that the method is stable and provides the optimal (k + 1)-th order of accuracy for smooth solutions when using piecewise k-th degree polynomials. Singularities in derivatives can also be resolved sharply if the entropy condition is not violated. Special treatment is needed for the entropy violating cases. Both one and two dimensional numerical results are provided to demonstrate the good qualities of the scheme.;In the second part, we develop a DG method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, this method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)-th order of accuracy when using piecewise k-th degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation. |
