Discontinuous Galerkin methods are widely used in many practical fields. In this thesis, we focus on a new class of discontinuous Galerkin methods for second-order wave equations. This thesis is constructed by three main parts. In the first part, we study the convergence properties of the energy-based discontinuous Galerkin proposed in  for wave equations. We improve the existing suboptimal error estimates to an optimal convergence rate in the energy norm. In the second part, we generalize the energy-based discontinuous Galerkin method proposed in  to the advective wave equation and semilinear wave equation in second-order form. Energy-conserving or energy-dissipating methods follow from simple, mesh-independent choices of the interelement fluxes. Error estimates in the energy norm are established. In the third part, we focus on establishing methods to overcome the computa- tional stiffness from the high-order piecewise polynomial approximations in the energy-based discontinuous Galerkin methods and reduce the computational cost of the inversion of the stiffness matrix.
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Zhang, Lu, "A New Class of Discontinuous Galerkin Methods for Wave Equations in Second-Order Form" (2020). Mathematics Theses and Dissertations. 5.