A finite difference solution of the regularized long-wave equation

Mathematical Problems in Engineering (Impact Factor: 0.76). 01/2006; DOI: 10.1155/MPE/2006/85743
Source: DOAJ


A linearized implicit finite difference method to obtain numerical solution of the one-dimensional regularized long-wave (RLW) equation is presented. The performance and the accuracy of the method are illustrated by solving three test examples of the problem: a single solitary wave, two positive solitary waves interaction, and an undular bore. The obtained results are presented and compared with earlier work.

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    • "Because of having limited analytical solutions, numerical analysis of the RLW equation has an importance in its study. Various techniques have been developed to obtain the numerical solution of this nonlinear partial differential equation, some of which are finite difference methods [19] [3] [8] [11] [15], finite element methods [2] [20] [17] [22] [24] [25] [12] [16] [13] [14] [23] and spectral methods [6] [21] [5]. The paper is outlined as follows. "
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    ABSTRACT: In this paper, the exponential B-spline functions are used for the numerical solution of the RLW equation. Three numerical examples related to propagation of single solitary wave, interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method. Obtained results are compared with some early studies.
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    • "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. "
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    ABSTRACT: Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa–Holm equation, the BBM equation and the hyperelastic-rod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity.
    Journal of Computational and Applied Mathematics 02/2011; 235(8-235):1925-1940. DOI:10.1016/j.cam.2010.09.015 · 1.27 Impact Factor
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    ABSTRACT: In this work, our main purpose is to develop of a sufficiently robust, accurate and efficient numerical scheme for the solution of the regularized long wave (RLW) equation, an important partial differential equation with quadratic nonlinearity, describing a large number of physical phenomena. The crucial idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semi-discretization and the backward difference formula for the time discretization. Furthermore, a suitable linearization preserves a linear algebraic problem at each time level. We present error analysis of the proposed scheme for the case of nonsymmetric discretization of the dispersive term. The appended numerical experiments confirm theoretical results and investigate the conservative properties of the RLW equation related to mass, momentum and energy. Both procedures illustrate the potency of the scheme consequently. PACS Codes: 02.70.Dh, 02.60 Cb, 02.60.Lj, 03.65.Pm, 02.30.-f. MSC: 65M60, 65M15, 65M12, 65L06, 35Q53.
    Boundary Value Problems 01/2013; 2013(1). DOI:10.1186/1687-2770-2013-116 · 1.01 Impact Factor
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