High‐order compact difference scheme for the regularized long wave equation

Abstract

SUMMARYA numerical simulation of the regularized long wave (RLW) equation is obtained using a high-order compact difference method, based on the fourth-order compact difference scheme in space and the fourthorder Runge-Kutta method in time integration. The method is tested on the problems of propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves, the development of an undular bore and the solitary waves induced by boundary motion. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. L 2 and L ∞ error norms are used to measure differences between the exact and numerical solutions. The results obtained by proposed method are compared with those of other recently published methods and shown to be more accurate and efficient.

A numerical solution of the RLW equation by Galerkin method using quartic B‐splines

“…A number of the waves generated are caused by the forcing time when the forcing time is kept fixed with varying amplitudes; amplitudes of the induced waves are recorded in Table X. It is observed from the table that amplitudes of the generated waves are governed mainly by the amplitudes of the boundary forcing and higher forcing amplitude has also produced more waves, which confirms the observation of earlier published articles [5,8].…”

“…U n j and its first derivative are calculated from Equation (5). Conservation quantities, L 2 error norm…”

Geometric finite difference schemes for the generalized hyperelastic-rod wave equation

“…Using the summation by part rule (26), it can be checked that the discrete operators D 1 and D 2 are antisymmetric for the discrete scalar product (22). The discrete gradients (28), (30) and (34) …”

“…For the BBM equation, conservative finite difference schemes were proposed in [25] with a convergence and stability analysis. We also refer to [26,27]. As far as the hyperelastic-rod wave equation, the authors are only aware of the numerical scheme given in [28] which is based on a Galerkin approximation and preserves a discretization of the energy.…”

Solving the generalized regularized long wave equation on the basis of a reproducing kernel space

“…Moreover numerical techniques such as finite difference methods [6][7][8][9][10][11], a spectral method [12], finite element methods based on the least square principle [13][14][15], finite element methods based on Galerkin and collocation principles [16][17][18][19][20][21][22][23][24], the Petrov-Galerkin method [25], the radial basis function collocation method [26,27], the Sinc-collocation method [28], the collocation method with quintic B-splines [29] and the cubic B-spline finite element method [30,31] have been devised for finding numerical solutions of special kinds of GRLW equations.…”