Numerical treatment of Benjamin-Bona-Mahony-Burgers equation with fourth-order improvised B-spline collocation method

Abstract

In this work, the Benjamin-Bona-Mahony-Burgers (BBMB) equation is solved using an improvised cubic B-spline collocation technique. This equation describes the propagation of small amplitude waves in a nonlinear dispersive medium, in the modeling of unidirectional planar waves. Due to the higher smoothness and sparse nature of matrices corresponding to splines, cubic B-splines are chosen as the basis function in the collocation method. But, the optimal accuracy and order of convergence cannot be achieved using the standard B-spline collocation method. So to overcome this, improvised cubic B-splines are formed by making posteriori corrections to cubic B-spline interpolant and its higher-order derivatives. The Crank-Nicolson scheme is used to discretize the temporal domain along with the quasilinearization process to deal with the nonlinear terms. The spatial domain discretization is carried out using the improvised cubic B-spline collocation method (ICSCM). The stability analysis of the technique is performed using the von-Neumann scheme. Several test problems are solved numerically and obtained results are compared with the results available in the literature. The aim of the paper is to show that such improvised techniques which were earlier used to solve ODEs, can be applied to solve the BBMB equation also, with excellent accuracy in results.