This research will consider three topics in high order and spectral methods for the numerical solution of partial differential equations. Domain decomposition for spectral methods will be considered. Results of this portion of the work will make it easier to deal with complex gemetries. Techniques for using spectral methods for shock wave calculations will be developed. The total variation bounded spectral method will be reconstructed so as to retain spectral accuracy away from the shock. Finally, pointwise order ENO schemes will be generalized to two dimensional problems. Results of this work should find immediate application in high-speed aerodynamics and other related physical areas.
