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

Numerical investigations of shallow water waves via generalized equal width (GEW) equation

In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element approximation have been studied respectively. In addition to that a powerful Fourier series analysis has been performed and indicated that our method is unconditionally stable. Finally, proficiency and practicality of the method have been demonstrated by illustrating it on two important problems of the GEW equation including propagation of single solitons and collision of double solitary waves. The performance of the numerical algorithm has been demonstrated for the motion of single soliton by computing L∞ and L2 norms and for the other problem computing three invariant quantities I1, I2 and I3. The presented numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid

Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation

This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modi ed Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation nite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms L2, L1 and three invariants I1; I2 and I3 are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation

Galerkin finite element solution for benjamin-bona-mahony-burgers equation with cubic b-splines

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony– Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic Bspline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms L2, L∞ and three invariants I1, I2 and I3 are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation