Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods

Khater, A. H.; Temsah, R. S.

doi:10.1016/j.camwa.2008.03.013

Cited by 79 publications

(33 citation statements)

References 28 publications

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“…Unfortunately, the weight function w(t; α) := (1 − t 1/α ) −α is a nonclassical one and no explicit formulae are known for ω (α) k and τ (α) k . But we can use the Chebyshev and Golub-Welsch algorithms to calculate the nodes and weights in (24) as discussed in 2nd Section. Let us note that the quadrature rule (24) with N = ⌈n/2⌉ becomes exact for computing the integral in (20).…”

Section: Proof By Definition We Getmentioning

confidence: 99%

“…But we can use the Chebyshev and Golub-Welsch algorithms to calculate the nodes and weights in (24) as discussed in 2nd Section. Let us note that the quadrature rule (24) with N = ⌈n/2⌉ becomes exact for computing the integral in (20). After obtaining the nodes τ (α) k and weights ω (α) k , the fractional derivative D α ⋆ L n (t; α) can be computed by…”

Section: Proof By Definition We Getmentioning

confidence: 99%

“…Spectral collocation methods are efficient and highly accurate techniques for numerical solution of nonlinear differential equations. One of their prominent features is the exponential convergence of the approximations [21][22][23][24].…”

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confidence: 99%

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Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials

Esmaeili

Shamsi

Luchko³

2011

Computers & Mathematics with Applications

150

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a b s t r a c tThis paper presents a computational technique based on the collocation method and Müntz polynomials for the solution of fractional differential equations. An appropriate representation of the solution via the Müntz polynomials reduces its numerical treatment to the solution of a system of algebraic equations. The main advantage of the present method is its superior accuracy and exponential convergence. Consequently, one can obtain good results even by using a small number of collocation points. The accuracy and performance of the proposed method are examined by means of some numerical experiments.

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Section: Proof By Definition We Getmentioning

confidence: 99%

Section: Proof By Definition We Getmentioning

confidence: 99%

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Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials

Esmaeili

Shamsi

Luchko³

2011

Computers & Mathematics with Applications

150

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“…Often, validation was only done for smooth, viscous solutions, like in Khater and Temsah [21], who used spectral integration on a Chebyshev polynomial basis. Fornberg [22] applied a Chebyshev pseudo-spectral method, implementing the boundary conditions in real space, and computed chaotic solutions at ν ≈ 4 × 10 −4 .…”

Section: Example Computations: Finite-size Effectsmentioning

confidence: 99%

On Green’s function-based time stepping for semilinear initial–boundary value problems

van Veen

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tThe Fourier spectral approach is central to the analysis of semilinear initial-boundary value problems when considered with periodic boundary conditions. It aids theoretical arguments, such as renormalisation group analysis, as well as numerical simulation through standard pseudo-spectral methods. Fixed boundary conditions, in contrast, require the use of non-uniform grids, usually generated by bases of orthogonal polynomials. On such bases, numerical differentiation is ill-conditioned and can potentially lead to a catastrophic blowup of round-off error. In this paper, we apply ideas explored by Viswanath (2013) in the context of Navier-Stokes solvers to completely eliminate numerical differentiation and linear solving from the time-stepping algorithm in favour of numerical quadrature. We propose a concrete quadrature rule and test its stability, scalability and spectral accuracy on a Kuramoto-Sivashinsky template problem over a range of five orders of magnitude of the viscosity.

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“…The fourth derivative term is the dominating term and is responsible for stabilising the equation. Several methods have been used to solve the KSe numerically and these include Chebyshev spectral collocation method [8], Quintic B-spline collocation method [9], Lattice Boltzmann method [10], meshless method of lines [11], Fourier spectral method [12] and septic B-spline collocation method [13].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Wave (Shock Profile) Propagation of the Kuramoto-Sivashinsky Equation Using an Adaptive Mesh Method

Muzadziwa¹

2018

Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

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Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods

Cited by 79 publications

References 28 publications

Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials

Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials

On Green’s function-based time stepping for semilinear initial–boundary value problems

Numerical Simulation of Wave (Shock Profile) Propagation of the Kuramoto-Sivashinsky Equation Using an Adaptive Mesh Method

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