The linearized Crank-Nicolson scheme for the solution of the Kuramoto-Sivashinsky equation

This master's thesis aims to find approximate solutions of a one-dimensional nonlinear Kuramoto-Sivashinsky (KS) type partial differential equation with given initial-boundary conditions using the Richtmyer linearized Crank-Nicolson type finite difference method. The thesis consists of five chapters. In Chapter 1, a brief introduction to the KS equation which is considered as the model problem followed by a review of existing studies in the literature regarding its solutions. In Chapter 2, the classical finite difference methods with some necessary definitions and concepts to be used in the thesis are briefly presented. In Chapter 3, the Richtmyer linearized Crank-Nicolson type finite difference scheme for the model problem given by the KS equation is derived. Additionally, in this chapter, the unconditional stability of the scheme was demonstrated using the von-Neumann method and its convergence order together with the local truncation error was calculated. In Chapter 4, three test problems, two of which possess exact solutıons, in order to verify the accuracy and reliability of the presented scheme. The results obtained from the scheme are compared both internally and with the results reported by other researchers in the literature. In this context, the L2, L∞ and GRE error norms, as well asCPU (s) processor times, are calculated. Furthermore, some graphs are provided to demonstrate the continuity of the approximate results and to show that they exhibit the correct physical behavior of the problem. In the final chapter which is Chapter 5, the numerical results obtained are generally evaluated, and a brief suggestion for future studies is made in light of the findings reached throughout the thesis.