Sixth order Hermite collocation method for analysis of MRLW equation

Abstract

The septic Hermite collocation method (SHCM) is proposed for solving the modified regularized long wave equations (MRLW), which may be utilized to simulate wide range of problems of applied sciences. SHCM is a hybrid of the orthogonal collocation and the finite element methods with septic Hermite interpolating polynomials as the basis function. The SHCM is used for the space discretization and Crank–Nicolson (CN) scheme for the time discretization. The stability analysis of the proposed method is carried out using Fourier series analysis and the SHCM is shown to be unconditionally stable. The proposed method is found to be six order convergent in space direction and second order convergence in time direction. To demonstrate the applicability of proposed algorithm, several important forms of the MRLW equation are solved. Three motion invariants: mass, energy, and momentum have been computed numerically and proved to match their exact values. The behavior of solitary waves, as well as wave undulations, have been graphically depicted. The SHCM is capable of displaying solitary waves collision. Also, error norms are computed at various time levels and compared with literature results. The outcome is found to be superior than the previous values.