Numerical Simulation of Compressible Flow in Complex Domain

This thesis deals with the theoretical analysis and practical applications of the hp discontinuous Galerkin method for convection-diffusion problems. At first, we apply this method to a scalar non-stationary convection-diffusion equation with nonlinear convective as well as diffusive terms which represents a model problem for the solution of the system of the compressible Navier-Stokes equations describing a motion of viscous compressible fluids. The resulting, fully discrete schemes, where the time discretization is done by backward difference formula, including symmetric, non-symmetric and incomplete variants of stabilizations are theoretically analyzed. Under additional assumptions on mesh, regularity of exact solution and nonlinear convective and diffusive terms, we derive hp a priori error estimates for the DG-BDF method of the second order in time, which are optimal in the L2(H1)-seminorm and suboptimal in the L(L2)- norm. Further, we extend the concept of the discontinuous Galerkin method to the compressible Navier-Stokes equations, incorporating a semi-implicit discretization with respect to time which leads to the solution of a sequence of linear algebraic systems. We develop an efficient strategy for the solution of these acquired systems. It is based on a simple adaptive technique for the choice..