| In this thesis we develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple spatial dimensions. They include KdV type equations involving third derivatives, bi-harmonic type equations involving fourth derivatives and partial differential equations involving fifth derivatives.; Theoretically we construct correct interface numerical fluxes for all the high order PDEs listed above, and we prove L2 stability for a wide class of nonlinear problems. For the KdV type equations, we prove a cell entropy inequality for the square entropy for a class of nonlinear PDEs of this type both in one and multiple spatial dimensions, and give an error estimate for the linear cases in the one dimensional case. We also prove a cell entropy inequality for the square entropy for PDEs with fifth derivatives. The LDG methods we constructed for the high order PDEs are especially suitable for problems which are “convection dominate”, i.e. those with small coefficients to the higher derivative terms.; Numerically we present several examples to illustrate the capability of these methods. For the KdV type equations, we give examples to illustrate the high order accuracy of the method for both one dimensional and two dimensional, both linear and nonlinear problems. We also present other examples, including soliton propagations, zero-dispersion limit of conservation laws and applications to nonlinear dispersive equations. For bi-harmonic type equations and partial differential equations with fifth derivatives, we give preliminary numerical examples to show the correctness of these methods. We also present new results on a post-processing technique, originally designed for methods with good negative-order error estimates. Numerical experiments show that this technique works as well for the new higher derivative cases, effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.; The methods have the usual advantage of local discontinuous Galerkin methods, namely they are extremely local and efficient for parallel implementations and easy for h-p adaptivity. |
