Time and spatial discretization-based B-spline and finite difference approaches with bi-numerical computation on general Rosenau-regularized long-wave equations

In this work, two numerical methods are proposed to obtain the approximate solution of the general Rosenau regularized long wave equation. The first method incorporates a linearization technique, while the second employs Butcher's fifth-order Runge-Kutta method. Both approaches utilize septic B-splines as basis functions. In the second method, the approximate solutions are obtained without applying any transformation or linearization technique to handle the nonlinearity. Instead, Butcher's fifth-order Runge-Kutta method is directly implemented for the numerical treatment of the resulting system of first-order differential equations, without employing a finite difference scheme at each time level. The linear stability and convergence analysis for the linearized septic B-spline scheme is investigated. The error norms and invariants, like mass and energy, are calculated for two test problems to enhance the efficiency and accuracy of the present method and also compared to the existing results.