Fourth order compact scheme for navier-stokes and convection-diffusion equations

Abstract

The present dissertation entitled “FOURTH ORDER COMPACT SCHEME FOR NAVIER-STOKES AND CONVECTION-DIFFUSION EQUATIONS” comprises the study of fourth order compact finite difference schemes under the supervision of Dr. Vivek Sangwan, Assistant Professor, School of Mathematics and Computer Applications, Thapar University, Patiala. Differential equations arise in the mathematical modelling of many physical, chemical and biological phenomena and many more areas of science and engineering such as fluid dynamics, electromagnetism, material science, astrophysics, economy etc. Therefore, it becomes a necessity to develop methods for solving differential equations. There are two principle approaches for finding the solutions of differential equations. One is the asymptotic approach and the other one is numerical approach. In this regard, numerical approaches have many benefits over asymptotic approaches as these techniques are much easier to use as compared to asymptotic approaches. Also these require the problem solver very less information about the problem to be solved. In numerical approaches, one finds a solution of a differential equation numerically. The aim of this work is to study the higher order compact finite difference schemes for solving the differential equations. These schemes enjoys the benefits of higher order accuracy, high stability, easy to implement and many more. Also, these schemes uses a compact 9−point stencil which has the much added advantage in terms of less boundary conditions requirement while solving the problem numerically and retaining higher order accuracy. The schemes provide very powerful tool for solving very complicated differential equations like convection-dominated problems and Navier-Stokes equations with high Reynolds numbers.