High Order Accuracy Methods for PDEs with Shocks and Uncertainties

Abstract : The aim of this project is the construction of efficient high fidelity schemes for systems of PDEs that contain shock waves. We advocate here the Discontinuous collocation method, i.e. multi-domain spectral method with stable and conservative penalty interface conditions. In this work, the previously developed methodology is generalized to inhomogeneous grid to simulate the two dimensional supersonic injector-cavity system. Non-physical modes generated at the domain interfaces due to the spatial grid inhomogeneity is minimized using the new weighted multi-domain spectral penalty method. The method yields accurate and stable solutions of the injector-cavity system which agree well with experiments qualitatively. Through the direct numerical simulation of the injector-cavity system using the weighted method, the geometric effect of the cavity wall on the pressure fluctuation is investigated. It is shown that the recessed slanted cavity attenuates pressure fluctuations inside cavity enabling the cavity to act potentially as a stable flameholder for scramjet engine.