A new approach for numerical solutions of Rosenau-Burgers equation

This master thesis consists of five chapters. In the first chapter of the thesis, after giving a brief information about the physical properties of the Rosenau-Burgers equation discussed in the thesis, the studies to be done in the thesis are briefly mentioned. In the second chapter of the thesis, some basic concepts such as stability, consistency and convergence along with standart finite difference and conservative finite difference approaches are presented. In this section, after giving a brief overview of stability analysis, the von Neumann Fourier series method is explained. In the third chapter, a detailed literature review about the Rosenau-Burgers equation is made and information about the current studies is presented and the model problems to be considered in the thesis are given together with the initial and boundary conditions. In the fourth chapter, a linearization technique is used instead of the non-linear term in the Rosenau-Burgers equation, in which standart and conservative finite difference approaches are applied, respectively. The numerical schemes obtained as a result of these applications are applied to the model problems with initial and boundary conditions given in the third chapter. Then, numerical diagrams are obtained by using the same linearization technique and conservative finite difference approaches instead of the nonlinear term in the space split Rosenau-Burgers equation obtained by using the V=U_{xx} transformation. These numerical schemes are applied to the model problems with initial and boundary conditions in the third chapter, and approximate results are calculated. Finally, in this section, the error norms calculated from the approximate solution by applying the conservative finite difference schemes corresponding to the standart, conservative and space split equations to the model problems are presented in tables and graphs. In addition, the numerical results obtained with each scheme are compared with the results given in some existing studies in the literature. In the last chapter of the thesis, which is the fifth chapter, the numerical results calculated from all of the applied methods are compared in the third chapter for the model problems with initial and boundary conditions, and a brief evaluation has been made about the approaches.