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The generalized regularized long wave (GRLW) equation has been developed tomodel a variety of physical phenomena such as ion-acoustic andmagnetohydrodynamic waves in plasma, nonlinear transverse waves in shallowwater and phonon packets in nonlinear crystals. This paper aims to develop andanalyze a powerful numerical scheme for the nonlinear generalized regularizedlong wave (GRLW) equation by Petrov--Galerkin method in which the element shapefunctions are cubic and weight functions are quadratic B-splines. The suggestedmethod is performed to three test problems involving propagation of the singlesolitary wave, interaction of two solitary waves and evolution of solitons withthe Maxwellian initial condition. The variational formulation and semi-discreteGalerkin scheme of the equation are firstly constituted. We estimate accuracyof such a spatial approximation. Then Fourier stability analysis of thelinearized scheme shows that it is unconditionally stable. To verifypracticality and robustness of the new scheme error norms $L_{2}$, $L_{\infty}$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated. The obtainednumerical results are compared with other published results and shown to beprecise and effective.

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