  The generalized regularized long wave (GRLW) equation has been developed to
model a variety of physical phenomena such as ion-acoustic and
magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow
water and phonon packets in nonlinear crystals. This paper aims to develop and
analyze a powerful numerical scheme for the nonlinear generalized regularized
long wave (GRLW) equation by Petrov--Galerkin method in which the element shape
functions are cubic and weight functions are quadratic B-splines. The suggested
method is performed to three test problems involving propagation of the single
solitary wave, interaction of two solitary waves and evolution of solitons with
the Maxwellian initial condition. The variational formulation and semi-discrete
Galerkin scheme of the equation are firstly constituted. We estimate accuracy
of such a spatial approximation. Then Fourier stability analysis of the
linearized scheme shows that it is unconditionally stable. To verify
practicality 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 obtained
numerical results are compared with other published results and shown to be
precise and effective.
