A numerical scheme for the approximate solution of the Newell-Whitehead-segel equation

This master's thesis which aims to find approximate numerical solutions of one-dimensional nonlinear Newell-Whitehead-Segel (NWS) type partial differential equations under specific initial and boundary conditions by means of Richtmyer linearized Crank-Nicolson type finite differences method is planned in five chapters. In Chapter 1, firstly the NWS equation to be considered as the model problem, is briefly introduced. A literature review regarding the solutions of the equation is then presented. In Chapter 2, the fundamental principles of the finite differences method to be used in this thesis are given along with some necessary definitions and concepts. In Chapter 3, the Crank–Nicolson-type finite difference scheme with Richtmyer linearization was derived for the model problem given by the NWS equation in Chapter 1. Furthermore, the order of convergence of the scheme obtained in this chapter was computed, and the scheme was shown to be unconditionally stable by means of von-Neumann stability analysis. In Chapter 4, three example problems with exact solutions given by the NWS equation have been briefly introduced. The scheme presented in Chapter 3 is successfully is applied to each problem, and some computed numerical results are compared among themselves as well as with the results by other researches in the literatures using tables. Additionally, some graphs are provided to illustrate the continuity of the computed numerical results and to show that each considered problem exhibits its correct physical behaviors. In Chapter 5, which is the final chapter of the thesis, the theoretical contributions and numerical findings of the thesis are summarized along with a brief suggestion for future research.