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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.

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.