Multi-domain spectral penalty method for hyperbolic systems: Theory and applications

In this dissertation we study the multi-domain spectral penalty method for hyperbolic systems. We first describe the theoretical aspects of the penalty method and explain how the method can be extended to the multi-domain numerical method by using the linear hyperbolic systems. This penalty method is given in a general form for the conservation law and the issue of instability at the interface is discussed. Based on this study we apply the multi-domain spectral penalty method to the real problems. We first solve the gravitational collapse model of a massless scalar field from the Numerical Relativity. We also apply the method to the supersonic recessed cavity flame-holder simulations from the Computational Fluid Dynamics. Through these applications we verify the validity of the method and obtain the new results that are not available in low order methods.