Cubic B-Spline Finite Element Method for Solving Whitham–Broer–Kaup Shallow Water Model

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Abstract
This paper aims to develop an efficient and accurate numerical method for solving the third-order nonlinear Whitham–Broer–Kaup (WBK) shallow water model. To achieve this objective, we applied a cubic B-spline-based finite element method, which offers higher-order continuity and precise geometric representation. The study of this model is important due to its wide applications in fluid dynamics, plasma physics, and nonlinear optics. We established the existence and uniqueness of the numerical solutions and performed a convergence analysis for the semi-discrete Galerkin scheme. The system of ordinary differential equations produced by the application of this method is solved by the Crank–Nicolson method. To validate the current approach, three test problems are analyzed. The errors are measured by using the [ ] and [ ] norms. Numerical results demonstrate that the method provides highly accurate approximations, with close agreement between numerical and exact solutions, confirming the efficiency, stability, and reliability of the cubic B-spline finite element method for modeling complex nonlinear wave phenomena. In addition, we compared the present method with several published methods. The results show that the proposed method provides a better solution in terms of absolute error and computational efficiency.

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