A Novel Improved Collocation Approach to Solve Good Boussinesq Equation Describing Propagation of Shallow Water Waves

Abstract

The present work is about obtaining a high-order accurate numerical approach to approximate the solution of the good Boussinesq equation (GBeq). In present approach, the quintic B-spline collocation procedure equipped with new approximations for the second-order and the fourth-order spatial derivatives is employed to discretize the spatial variables and Crank–Nicolson scheme is used to obtain temporal integration of the GBeq. The proposed approach achieves sixth-order accuracy and second-order accuracy in spatial and temporal directions, respectively. By von-Neumann stability analysis, the unconditionally stability of the suggested approach is proved. The efficiency and applicability of the computational approach is verified by examining the sample problems including motion of single solitary, interaction of two solitons and birth of solitons. The $L_{\infty }$ error norm is computed and compared with the existing studies in the literature. The comparisons demonstrate that the suggested approach is superior to some existing techniques in terms of accuracy. Also, the rate of convergence and invariant constant are numerically computed and seen to match with their theoretical values.