This thesis is devoted to efficient numerical methods and their implementations for two classes of wave equations. The first class is linear wave equations in very high frequency regime, for which one has to use some asymptotic approach to address the computational challenges. We focus on the use of the Gaussian beam superposition to compute the semi--classical limit of the Schr"{o}dinger equation. The second class is dispersive wave equations arising in modeling water waves. For the Whitham equation, so-called the Burgers--Poisson equation, we design, analyze, and implement local discontinuous Galerkin methods to compute the energy conservative solutions with high-order of accuracy.

Our Gaussian beam (GB) approach is based on the domain-propagation GB superposition algorithm introduced by Liu and Ralston [Multiscale Model. Simul., 8(2), 2010, 622--644]. We construct an efficient numerical realization of the domain propagation-based Gaussian beam superposition for solving the Schr"odinger equation. The method consists of several significant steps: a semi-Lagrangian tracking of the Hamiltonian trajectory using the level set representation, a fast search algorithm for the effective indices associated with the non-trivial grid points that contribute to the approximation, an accurate approximation of the delta function evaluated on the Hamiltonian manifold, as well as efficient computation of Gaussian beam components over the effective grid points. Numerical examples in one and two dimensions demonstrate the efficiency and accuracy of the proposed algorithms.

For the Burgers--Poisson equation, we design, analyze and test a class of local discontinuous Galerkin methods. This model, proposed by Whitham [Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The $L^2$-stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method.