Finite difference approximations for 2-dimensional nonlinear coupled burgers' equation

This master thesis consists of six chapters. In the introductory chapter of the thesis, preliminary information was given about the purpose of this study after briefly mentioning the structure of the 2-dimensional coupled Burgers' equation to be considered in this thesis. In the second chapter, explicit, implicit and Crank-Nicolson classical finite difference methods to be used in the thesis are explained and then the difference schemes obtained by applying these methods to the heat transfer equation are given as a sample application. In the third chapter, the literature search of the 2-dimensional coupled Burgers' equation is described in detail, then three model problems with different initial and boundary conditions are presented. In addition, L₂ and L_{∞} error norms are used in this section to measure the accuracy and consistency of numerical schemas. In the fourth chapter, the schematics obtained by applying three model probing methods of explicit, implicit and Crank-Nicolson classical finite difference methods are given and then numerical solutions of problems are obtained with these schemes. The numerical results obtained were presented in tabular form, comparing with the available full solutions and other results in the literature. In addition, theL₂ and L_{∞} error norms calculated for model Problem 1 and Problem 3 having exact solutions are shown in the tables. The fifth chapter forms the main part of this thesis. In this section, a linearization technique of the Rubin-Graves rgl type instead of the nonlinear UU_{x}, VU_{y},UV_{x} and VV_{y} terms in the 2-Dimensional Coupled Burgers' Numerical solutions of the model problems were obtained by using the finite difference schemes. The numerical solutions obtained were compared with the existing complete solutions and with the other results in the literature. At the same time, L₂ and L_{∞} error norms were calculated. In addition, for Problem 1 and Problem 3 both numerical and complete solutions are shown graphically, whereas for Problem 2 only numerical results are plotted. Lastly, in the sixth chapter, explicit and implicit and Crank-Nicolson methods in the fourth section of the thesis for Exact Problems 1 and 3, and L₂ and L_{∞} error norms calculated by applying the Rubin-Graves type linearization technique in the fifth section are compared within themselves.