There are many new developments in scientific computing, in its application to fluid flows and wave phenomena, which warrant their consolidation in a single source, covering some of the key developments. I have been convinced by many students and peers that there is a definitive need for a single source book which deals with topics covered here. I would like to acknowledge their inspiration. My main motivation in writing this manuscript is to communicate something new and powerful as opposed to conventional derivatives of products churned out by existing schools of thought.

However, this book also provides general introduction to computational fluid dynamics (CFD), using well tested classical methods of solving partial differential equations (PDEs) for the sake of completeness. These are to be found in Chaps. 1 to 9 and 13, but re-interpreted using the spectral analysis method introduced in Chaps. 4, 8 and 10. This provides an unity of approach in understanding numerical methods for parabolic, elliptic and hyperbolic PDEs. The spectral analysis tool has been refined in recent years by the author's group, with which disparate methods can be easily compared.