| This dissertation consists of two parts.; In Part I, we study formally high order accurate discontinuous Galerkin methods on general arbitrary grid for multi-dimensional hyperbolic systems of conservation laws. We extend the notion of E-flux from scalar to system, and found that after flux splitting upwind flux is a Riemann solver free E-flux for systems. Therefore we are able to show that the discontinuous Galerkin methods satisfy a cell entropy inequality for square entropy (in semidiscrete sense) if the multi-dimensional systems are symmetric. Similar result was obtained for scalar equations in multidimensions. We also developed a 2nd order finite difference version of the discontinuous Galerkin methods. Numerical experiments have been performed with excellent results.; In Part II, the viscous Moore-Greitzer equation modeling the airflow through the compression system in turbomachines, such as a jet engine, are derived using homogenization of the Navier-Stokes equations. The homogenization makes rig orous a separation of scales arguments, based on the different spatial scales in the engine and the different time scales in the flow. The size of the rotor-stator pair of blades provides a small parameter, which is the size of the local cell. The stator and rotor blades in the compressor produce a very turbulent flow on a fast time scale. The leading order equation, for the fast-time and local scale, describes this turbulent flow. On a much larger spatial scale and a slower time scale, there exist modulations of the flow including instabilities called surge and stall. The third order equation, in the small parameter, describes these global flow modulations, when averaged over the small (local) spatial scales and fast time scale. The viscous Moore-Greitzer equation is obtained when small intertial and eddy-viscosity terms are dropped from the slow-time, spatially global equations. A system of spatially local pressure equations and spatially global, slow-time equations for the velocity are also derived that can be solved numerically without any approximations. |
