The research planned is related to the fast and accurate computation of non-homogeneous/nonlinear Cauchy-Riemann equations in the complex plane; the modular code development on serial and parallel computers, and its applications to solving nonlinear problems, in particular problems in compressible fluid flows. The theory of non-homogeneous Cauchy-Riemann equations developed here provides a unified framework for extending the complex variable theory to solving large classes of problems involving non-analytic functions. The effect of non-homogeneous terms is shown to alter the analytic solution by an additive non-analytic term which is a double integral. The fast and accurate method of evaluating this integral provides a firm ground to successful application of the theory to computing the solution at a superior computational complexity. The robust code to be developed on these ideas will be used to solve large classes of applied problems including (i) rising bubbles in a compressible fluid; and (ii) cavitating compressible flows past curved obstacles. Computational Fluid Dynamics is an area of research with implication for a wide variety of scientific and engineering applications in atmospheric, physical and engineering sciences.
