A numerical technique for solution of the MRLW equation using quartic B-splines

Fazal-i-Haq,; Islam, Siraj Ul; Tirmizi, Ikram A.

doi:10.1016/j.apm.2010.04.012

Cited by 38 publications

(8 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For stability analysis, following Refs. [35], [47], [48], [50], [56], and [79] and the references therein we use the Von Neumann method. The nonlinear term F(u, u x ) cannot be handled by the Fourier mode method, so we must linearize it by making F u j…”

Section: The Stability Analysismentioning

confidence: 99%

“…[54] Raslan and EL-Danaf [55] used the B-spline finite element method to solve the MRLW equation. The numerical scheme based on the quartic B-spline collocation method is designed for the numerical solution of the MRLW equation by Haq et al [56] Achouri and Omrani [57] used the homotopy perturbation method to implement the MRLW equation with some initial conditions. The MRLW equation, with some initial conditions, is solved numerically via a variational iteration method by Labidi and Omrani.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation

Mohammadi

2015

Chinese Phys. B

View full text Add to dashboard Cite

The aim of the present paper is to present a numerical algorithm for the time-dependent generalized regularized long wave equation with boundary conditions. We semi-discretize the continuous problem by means of the Crank-Nicolson finite difference method in the temporal direction and exponential B-spline collocation method in the spatial direction. The method is shown to be unconditionally stable. It is shown that the method is convergent with an order of O(k 2 + h 2 ). Our scheme leads to a tri-diagonal nonlinear system. This new method has lower computational cost in comparison to the Sinc-collocation method. Finally, numerical examples demonstrate the stability and accuracy of this method.

show abstract

Section: The Stability Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation

Mohammadi

2015

Chinese Phys. B

View full text Add to dashboard Cite

show abstract

“…The GRLW equation has been studied both analytically by means of a large variety of approximation, perturbation, variational, Adomian's decomposition, etc., methods [13][14][15][16] and numerically by means of explicit and implicit procedures [17], energy-preserving (conservative) finite difference methods [18], methods of lines with Fourier-pseudospectral approximations for periodic boundary conditions [19], third-order accurate Runge-Kutta techniques [20], iterative finite difference methods [21], Galerkin, Petrov-Galerkin and cubic and quadartic B-splines techniques [22][23][24][25][26][27], meshless techniques [28], sinc-collocation procedures [29], etc.…”

Section: Introductionmentioning

confidence: 99%

Solitary waves generated by bell-shaped initial conditions in the inviscid and viscous GRLW equations

García-López

Ramos

2015

Applied Mathematical Modelling

View full text Add to dashboard Cite

a b s t r a c tThe objective of this paper is three-fold. First, the accuracy of three methods of lines that employ either piecewise analytical integration and yield exponential methods or threepoint, fourth-order accurate finite difference approximations for the spatial derivatives, is assessed by comparing the numerical solutions with the exact one of the generalized regularized long wave (GRLW) equation. Second, the three-point, fourth-order accurate, compact method is used to determine the disintegration or breakup of bell-shaped initial conditions for the GRLW and viscous GRLW equations. Third, the dynamics of the GRLW and viscous GRLW equations is studied as a function of the amplitude and width of Gaussian and triangular initial conditions. It is shown that the compact operator method yields slightly more accurate results than an exponential method that treats the source terms with fourth-order accuracy. It is also shown that the disintegration of the bell-shaped initial conditions is characterized by an initial transient, steepening of the leading front and the formation of solitary waves in the absence of viscosity. The number, speed and distance between successive solitary waves are shown to increase as the dispersion parameter is decreased in accord with a linear stability analysis. The number and speed of solitary waves are also functions of the amplitude and width of the bell-shaped initial conditions, whereas the effect of the linear advective terms is to push the solitary waves towards the downstream boundary. In the presence of viscosity, it is shown that the amplitude of the leading solitary wave that results from the disintegration of the initial conditions, decreases as a function of time and the waves exhibit a curvature which is a function of the linear and nonlinear advective and dispersion terms, and the viscosity coefficient. For large viscosities, it is shown that the initial flow adjustment results in a steep leading front whose slope decreases as time increases and waves analogous to the ones observed in the viscous Burgers' equation are formed. It is also shown that, for the GRLW equation, the number and speed of solitary waves that are formed after the break-up of the initial profile increase as the amplitude and width of the initial conditions is increased, and that Gaussian initial conditions result in faster and larger amplitude solitary waves than triangular ones. The long-term dynamics of the viscous GRLW equation is found to be nearly independent of the initial conditions.

show abstract

“…2007; Akbari and Mokhtari 2014), cubic B-spline collocation method (Khalifa et al. 2008b), quadratic B-spline collocation method (Tirmizi 2010), quadratic B-spline collocation method (Raslan 2009). …”

Section: Introductionmentioning

confidence: 99%

On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation

Pan

Zhang

2016

SpringerPlus

View full text Add to dashboard Cite

In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical “energy” of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time. The unique solvability of numerical solutions is shown. A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory. Numerical results are given to show the validity and the accuracy of the proposed method.

show abstract

A numerical technique for solution of the MRLW equation using quartic B-splines

Cited by 38 publications

References 19 publications

Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation

Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation

Solitary waves generated by bell-shaped initial conditions in the inviscid and viscous GRLW equations

On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation

Contact Info

Product

Resources

About