A study on the numeri̇cal solution of the Rosenau-Kawahara equation

In the first chapter of this thesis, consisting of four chapters, after a brief literature survey about the Rosenau-Kawahara equation considered in the thesis, two model problems called Problem 1 and Problem 2, of which approximate solutions are going to be found, are introduced. In the second chapter, first of all, the Crank-Nicolson conservative implicit finite difference method, which is going to be used in this thesis, is mentioned. Then, instead of the nonlinear term in the Rosenau-Kawahara equation, using three different linearization techniques represented by Lin-I, Lin-II and Lin-III, numerical schemes of Problem 1 are obtained with the help of Crank-Nicolson conservative implicit finite difference method. In addition, the stability analysis of the numerical scheme obtained with Lin-I is examined by the von-Neumann method. Then the error norms L₂ and L∞ calculated from the schemes, the order of accuracy and the conservation constants are presented in graphs, and at the same time, wave graphs are given to show how well the problem exhibits the physical behavior with the continuity of the discrete solutions obtained. In the third chapter, the Rosenau-Kawahara equation is first split with respect to the spatial variable and converted into a coupled differential equation system called as Problem 2. Then, instead of the non-linear term in the coupled equation system, numerical schemes of Problem 2 are given using the Crank-Nicolson conservative implicit finite difference method using three different linearization techniques considered in the second chapter. In addition, the stability analysis of the numerical scheme obtained for Lin-I is examined by the von-Neumann method, as in the second chapter. Then, error norms L₂ and L∞, the order of accuracy and conservation constants obtained from the numerical schemes are presented in tables and graphs. In the fourth chapter, which is the last chapter of the thesis, a brief conclusion for future researcies is given along with the proposed schemes.