A cubic B-spline Galerkin approach for the numerical simulation of the GEW equation

Abstract

The generalized equal width (GEW) wave equation is solved numerically by using lumped Galerkin approach with cubic B-spline functions. The proposed numerical scheme is tested by applying two test problems including single solitary wave and interaction of two solitary waves. In order to determine the performance of the algorithm, the error norms L 2 and L∞ and the invariants I 1 , I 2 and I 3 are calculated. For the linear stability analysis of the numerical algorithm, von Neumann approach is used. As a result, the obtained findings show that the presented numerical scheme is preferable to some recent numerical methods.

Numerical solutions of the generalized equal width wave equation using the Petrov–Galerkin method

“…The three invariants in this case are tabulated in Table (7) . It is clear that the quantities are satisfactorily constant and very closed with the methods [8,31,32] during the computer run. Fig.…”

“…The error aberration varies from −8×10 −2 to 1 × 10 −2 and the maximum errors happen around the central position of the solitary wave. For our second experiment, we take the parameters p = 3, c = 0.3, h = 0.1, ∆t = 0.2, ε = 3, µ = 1, x 0 = 30 with interval [0, 80] to coincide with that of previous papers [8,31,32]. Thus the solitary wave has amplitude 1.0 and the computations are carried out for times up to t = 20.…”

“…In Table(4) the performance of the our new method is compared with other methods [8,31,32] at t = 20. It is observed that errors of the method [8,31,32] are considerably larger than those obtained with the present scheme. The motion of solitary wave using our scheme is graphed at time t = 0, 10, 20 in Fig.(3).…”

“…The equation is solved numerically by a meshless method based on a global collocation with standard types of radial basis functions (RBFs) by [14]. Quintic B-spline collocation method with two different linearization techniques and a lumped Galerkin method based on B-spline functions were employed to obtain the numerical solutions of the GEW equation by Karakoc and Zeybek,[8,31] respectively. Roshan [32], applied Petrov-Galerkin method using the linear hat function and quadratic B-spline functions as test and trial function respectively for the GEW equation.…”

A Quartic Subdomain Finite Element Method for the Modified KdV Equation

“…Gardner et al [5] showed the existence and uniqueness of solutions of the KdV equation. Many researches have used various numerical methods including finite difference method [6,7], finite element method [8][9][10][11][12][13][14][15][16][17][18], pseudospectral method [3] and heat balance integral method [19] to solve the equation. MKdV equation is a special case of the generalized Korteweg de-Vries (GKdV) equation having the form…”