A New Highly Accurate Numerical Scheme for Benjamin-Bona-Mahony-Burgers Equation Describing Small Amplitude Long Wave Propagation

In this article, a new highly accurate numerical scheme is proposed and used for solving the initial-boundary value problem of the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation. The BBM-Burgers equation is fully discretized by the Crank-Nicolson type method using the first-order forward finite difference approximation for the derivative in time and the standard second-order central difference approximations for all spatial derivatives. The nonlinear term appearing in the implicit scheme is firstly linearized in terms of a new dependent variable by utilizing the well known Taylor series expansion and then the resulting tri-diagonal linear algebraic equation system is solved by a direct solver method. To test the accuracy and efficiency of the scheme, three experimental test problems are taken into consideration of which the two have analytical solutions and the other one has not an analytical one. The computed results are compared with those of some studies in the literature for the same values of parameters. It is shown that the obtained results from the present method, which is stable and easy-to-use, get closer and closer to the exact solutions when the step sizes refine. This fact is also an other evidence of the accuracy and reliability of the method. Moreover, a low level data storage requirement and easy-to-implement algorithm of the present method can be considered among its notable advantages over other numerical methods. In addition, the unconditionally stability of the present scheme is shown by the von Neumann method.