Spectral methods have been widely applied for transient and turbulent flows. However their scope of application has been seriously limited by their inability to treat discontinuous problems and problems on nonsmooth domains. This project intends to reduce these two restrictions of spectral methods. Oscillation-free uniform high order spectral methods to simulate chemically reacting flows, especially the detonation waves and the transition from deflagration to detonation (DDT) will be designed. To ease the domain restrictions, spectral methods and the p- and the h-p version of the finite element methods will be combined. Then the combined method will be applied to general elliptic equations on nonsmooth domains and the results will be used to simulate flows in irregular channels. These methods will be implemented on high performance computers.
