A numerical method for diffusion–convection equation using high-order difference schemes

Golbabai, A.; Arabshahi, M.M.

doi:10.1016/j.cpc.2010.03.008

Cited by 10 publications

(9 citation statements)

References 27 publications

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“…In this subsection we present some of the numerical experiments we made to test our method for the pseudo-parabolic equation (31). Taken a finite domain Ω = {(x, y, t) : 0 < x < 1, 0 < y < 1, 0 ≤ t ≤ T }, we consider Eqs.…”

Section: Results For Pseudo-parabolic Problemmentioning

confidence: 99%

“…while the function f that appears in (31) is given by f (x, y, t) = (2 + α + 2β) exp(2t)(cos(x) + sin(y)).…”

Section: Results For Pseudo-parabolic Problemmentioning

confidence: 99%

“…In the literature, excellent summaries have been provided in [39,5,22,23,25,53]. Among the numerical methods used in simulation to solve convection-diffusion problems, we can mention different approximating schemes such as the finite difference (FD) method [31,51], the finite volume (FV) scheme [20,18], the finite element (FE) technique [19,63], and the meshfree radial basis function (RBF) method [1,15,17].…”

Section: Introductionmentioning

confidence: 99%

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A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Garmanjani

Cavoretto

Esmaeilbeigi

2018

Computers & Mathematics with Applications

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Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations.

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Section: Results For Pseudo-parabolic Problemmentioning

confidence: 99%

“…while the function f that appears in (31) is given by f (x, y, t) = (2 + α + 2β) exp(2t)(cos(x) + sin(y)).…”

Section: Results For Pseudo-parabolic Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Garmanjani

Cavoretto

Esmaeilbeigi

2018

Computers & Mathematics with Applications

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“…Convection-diffusion equations are widely used in fluid mechanics, oil reservoir exploration, chemical engineering and many other fields (Morton 1996;Wang et al 2000;Wang and Li 2007;Lin et al 2009;Gao et al 2019;Tian 2019;Huang et al 2019). Numerical solutions of convection-diffusion equations often use finite difference methods (Chou and Shu 2007;Golbabai and Arabshahi 2010;Sun and Li 2014), finite-volume methods (FVM) (Liang and Zhao 2006;Gao et al 2019;Tian 2019;Huang et al 2019;Angermann and Wang 2019) and finite element methods (FEM) (Chana et al 2014;Cheichan et al 2019;Zhang and Chen 2019). Due to the merit of local conservation and suitability for complicated domains, FVMs are widely used in engineering computations.…”

Section: Introductionmentioning

confidence: 99%

A high-order modified finite-volume method on Cartesian grids for nonlinear convection–diffusion problems

Wang

2020

Comp. Appl. Math.

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Recently, Buchmüller and Helzel proposed a modified dimension-by-dimension finitevolume (FV) WENO method on Cartesian grids for multidimensional nonlinear conservation laws which can retain the full order of accuracy of the underlying one-dimensional (1D) reconstruction. In this work, we extend this method to multidimensional convection-diffusion equations. The 1D sixth-order central reconstruction of the conserved quantity is utilized for discretizing the diffusion terms in which the diffusion coefficients may be nonlinear functions of the conserved quantity. Using high-order accurate conversions between edge-averaged values and edge center values of any sufficiently smooth quantity, high-order accurate convective and viscous numerical fluxes at cell interfaces are computed. The present modified FV method uses fourth-order accurate conversions for the diffusive fluxes. Numerical examples show that the present method achieves fourth-order accuracy for multidimensional smooth problems, and is suitable for the numerical simulation of viscous shocked flows.

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“…Recently, various powerful mathematical methods such as the homotopy perturbation method, variational iteration method, Adomian decomposition method and others [1][2][3] have been proposed to obtain approximate solutions in partial differential equations (PDEs). The 2-D parabolic differential equations appeared in many scientific fields of engineering and science such as neutron diffusion, heat transfer and fluid flow problems.…”

Section: Introductionmentioning

confidence: 99%

On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation

Golbabai

Molavi-Arabshahi²

2011

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In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered

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A numerical method for diffusion–convection equation using high-order difference schemes

Cited by 10 publications

References 27 publications

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

A high-order modified finite-volume method on Cartesian grids for nonlinear convection–diffusion problems

On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation

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