Study of generalized regularized long wave equation via septic Hermite collocation method with Crank–Nicolson and SSP-RK43 schemes to capture the various solitons

Abstract

The paper aims to develop a powerful numerical technique for the generalized regularized long wave equation. The proposed technique is based on the orthogonal collocation on finite element method and septic Hermite interpolation polynomials as basis functions called septic Hermite collocation method (SHCM). These basis functions are C3 continuous, ensuring the continuity of dependent variable and its first three derivatives throughout the solution range. Two schemes viz one is implicit Crank–Nicolson scheme and other is Runge–Kutta method of four stages and third-order (SSP-RK43) are used for time discretization and SHCM is used for space discretization. The stability analysis of SHCM with Crank–Nicolson is performed and the scheme is found to be unconditionally stable. The stability of SHCM with SSP-RK43 by CFL conditions is also studied and the method is found to be conditionally stable. The proposed methods are applied to three problems involving single solitary waves, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. The formation and behavior of soliton waves are studied numerically. To demonstrate the robustness of proposed technique, error norms and the invariants of motion, i.e., mass, momentum, and energy are calculated. The results are in good agreement with the exact ones and found to be better than the numerical solutions available in the literature. The order of convergence along spatial and temporal directions has been calculated numerically. It is observed that the order of convergence of the explicit SSP-RK43 is higher than the implicit Crank–Nicolson and SHCM with SSP-RK43 provides better accuracy than the SHCM with Crank–Nicolson. The SHCM is capable of displaying solitary wave collision. The proposed technique is simple, fast, efficient, and easy to implement. It does not require unnecessary integration or calculation of weight functions as the case of other numerical methods for instance Galerkin methods, spectral methods, B-spline finite element methods etc.