| This dissertation presents a new type of computational method based on discontinuous Galerkin approximations, which exhibits extraordinary accuracy and robustness for a broad class of applications ranging from the analysis of convection-dominated to diffusion-dominated flow problems.; This discontinuous Galerkin technique extends the range of application of the classical discontinuous Galerkin method from first-order to first and second-order partial differential equations, including systems of nonlinear equations.; The method makes use of discontinuous hp-finite element approximations; it can deliver very high accuracy, and is robust, exhibiting only small localized oscillations near under-resolved boundary layers or interior interfaces with high gradients. The resulting space-time discretizations can be solved using explicit time-marching schemes.; The mathematical basis for this method, including a priori error estimates, numerical stability, and limitations are established in this study.; The mathematical analysis is complemented with applications to significant problems of interest in engineering. In particular, the method is used to solve the Euler equations of gas dynamics and the Navier-Stokes equations governing incompressible viscous flow. |
