High-Accuracy Finite Element Modeling of the Rosenau–Hyman Equation

The present study focuses on the numerical solution of the Rosenau–Hyman (R–H) equation, also known as the generalized Korteweg–de Vries equation, which describes the dynamics of shallow water waves and pattern formation in liquid drops. To this end, a collocation finite element method based on septic B-spline approximation is proposed and applied to the R–H equation for different parameter values of the test problem. In addition, a von Neumann stability analysis is performed, demonstrating that the proposed scheme is unconditionally stable. The efficiency and reliability of the method are illustrated by solving a test problem and computing the L2 and L∞ error norms. The numerical results are found to be in very good agreement with the corresponding analytical solutions, indicating that the proposed B-spline collocation algorithm is both accurate and robust. To further demonstrate the effectiveness of the method in solving nonlinear equations, the results are presented graphically as well as in tabular form. The close consistency between analytical and numerical results suggests that the proposed approach is a powerful and attractive tool for investigating characteristic features of nonlinear phenomena in various fields of science.