Solution of 1-dimensional Benjamin-Bona-Mahony-Burgers equationwith a discrete approximation method

Since the subject of this thesis is the calculation of the approximate numerical solutions of the 1-dimensional non-linear Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation, which explains the propagation of small amplitude long waves in a nonlinear dispersion medium, depending on the given initial and boundary conditions, in Chapter 1, the importance of the equation is briefly discussed and studies available in the literature are mentioned. In Chapter 2 forming the basis of this study, a numerical scheme is presented to obtain some calculated results together with the approximate numerical solutions depending on the appropriate initial and boundary conditions of the BBM-Burgers equation, which is given in 1-dimensional non-linear general form, which is considered as a model problem. For this, the BBM-Burgers equation is first completely discretized using the Crank-Nicolson type method using the first order forward finite difference approach for time derivatives and the usual second order central difference approach for all spatial derivatives, and thus a nonlinear implicit scheme is obtained. Then, since the solution of the implicit scheme with a direct method is aimed, a Rubin-Graves type linearization approach is used instead of the nonlinear term in the scheme, and the initial and boundary value problem is reduced to a system of linear algebraic equations consisting of tri-diagonal matrices and which can be solved directly by a direct solution method. In Chapter 3, three experimental test problems are given, two with exact solutions and one without an exact solution, to check the accuracy and effectiveness of the scheme presented in the previous section. In Chapter 4 the results (point values of the solution, the error norms L2 and L∞, the accuracy orders of the scheme, and CPU times) calculated from the scheme presented in Chapter 2 are compared with the results of the studies available in the literature for the same parameter values seen in the problem. From the approximate numerical results obtained, it is seen that as the spatial and temporal step lengths are decreased, the presented scheme gave good enough results approaching analytical solutions and is in acceptable agreement with those of other researchers. In addition, the stability of the easy-to-use scheme is examined with the von-Neumann technique and it is shown that the scheme is unconditionally stable. In the last chapter, Chapter 5, a brief conclusion and further work are mentioned.