A new numerical application of the generalized Rosenau-RLW equation

. This study implemented a collocation nite element method based on septic B-splines as a tool to obtain the numerical solutions of the nonlinear generalized RosenauRLW equation. One of the advantages of this method is that when the bases are chosen at a high degree, better numerical solutions are obtained. E ectiveness of the method is demonstrated by solving the equation with various initial and boundary conditions. Further, in order to detect the performance of the method, L2 and L1 error norms and two lowest invariants IM and IE were computed. The obtained numerical results were compared with some of those in the literature for similar parameters. This comparison clearly shows that the obtained results are better than and in good conformity with some of the earlier results. Stability analysis demonstrates that the proposed algorithm, based on a Crank Nicolson approximation in time, is unconditionally stable

A numerical study using finite element method for generalized RosenauKawahara-RLW equation

In this paper, we are going to obtain the soliton solution of the generalized RosenauKawahara-RLW equation that describes the dynamics of shallow water waves in oceans and rivers. We confirm that our new algorithm is energy-preserved and unconditionally stable. In order to determine the performance of our numerical algorithm, we have computed the error norms L2 and L∞. Convergence of full discrete scheme is firstly studied. Numerical experiments are implemented to validate the energy conservation and effectiveness for longtime simulation. The obtained numerical results have been compared with a study in the literature for similar parameters. This comparison clearly shows that our results are much better than the other results

Solitons and shock waves solutions for the rosenau-kdv-RLW equation

In this article, a space time numerical scheme has been proposed to approximate solutions of the nonlinear Rosenau-Korteweg-de Vries-Regularized Long Wave (Rosenau-KdV-RLW) equation which represents the dynamics of shallow water waves. The scheme is based on a septic B-spline finite element method for the spatial approximation followed by a method of lines for the temporal integration. The proposed scheme has been illustarated with two test problems involving single solitary and shock waves. To demonstrate the competency of the present numerical algorithm the error norms L2 , L and two lowest invariants MI and E I have been calculated. Linear stability analysis of the scheme has been studied using von-Neumann theory. The illustrated results confirm that the method is efficient and preserves desired accuracy

Solitary-wave solutions of the GRLW equation using septic B-spline collocation method

In this work, solitary-wave solutions of the generalized regularized long wave (GRLW) equation are obtained by using septic B-spline collocation method with two different lin- earization techniques. To demonstrate the accuracy and efficiency of the numerical scheme, three test problems are studied by calculating the error norms L 2 and L ∞ and the invari- ants I 1 , I 2 and I 3 . A linear stability analysis based on the von Neumann method of the numerical scheme is also investigated. Consequently, our findings indicate that our numer- ical scheme is preferable to some recent numerical schemes

Numerical approximation to a solution of the modified regularized long wave equation using quintic B splines

In this work, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by the method based on collocation of quintic B-splines over the finite elements. A linear stability analysis shows that the numerical scheme based on Von Neumann approximation theory is unconditionally stable. Test problems including the solitary wave motion, the interaction of two and three solitary waves and the Maxwellian initial condition are solved to validate the proposed method by calculating error norms L2 and L∞ that are found to be marginally accurate and efficient. The three invariants of the motion have been calculated to determine the conservation properties of the scheme. The obtained results are compared with other earlier result

Petrov Galerkin finite element method for solving the MRLW equation

In this article, a Petrov-Galerkin method, in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified regularized long wave (MRLW) equation. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and L2, L∞ error norms. The computed results show that the present scheme is a successful numerical technique for solving the MRLW equation. A linear stability analysis based on the Fourier method is also investigate

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion-acoustic and magnetohydro dynamic waves in plasma,nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop andanalyze a powerful numerical scheme for the nonlinear GRLWequation by Petrov–Galerkin method in which the elementshape functions are cubic and weight functions are quadratic B-splines. The proposed method is implemented to three ref-erence problems involving propagation of the single solitarywave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational for-mulation and semi-discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of thelinearized scheme we show that the scheme is uncondition-ally stable. To verify practicality and robustness of the new scheme error norms L2, L∞ and three invariants I1, I2,and I3 are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective

A numerical solution of the modified regularized long wave (MRLW) equation using quartic B-splines

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by subdomain finite element method using quartic B-spline functions. Solitary wave motion, interaction of two and three solitary waves and the development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the proposed method are tested by calculating the numerical conserved laws and error norms L₂ and L∞. The obtained results show that the method is an effective numerical scheme to solve the MRLW equation. In addition, a linear stability analysis of the scheme is found to be unconditionally stable.Publisher's Versio