A Second Order Finite Difference Approximation for the Fractional Diffusion Equation

Nasir, H. M.; Gunawardana, B.L.K.; Abeyrathna, H. M. N. P.

doi:10.7763/ijapm.2013.v3.212

Cited by 27 publications

(32 citation statements)

References 11 publications

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“…Generally speaking, if c −1 = c 1 it is hard/impossible to get a high order scheme for (1) with K 1 K 2 = 0. However, for second order approximations, as do in [20], we can firstly plug (12) into the two-sided problem (1), and then expand ∂u(x+βh,t) ∂t in Taylor's series w.r.t x, similar to the way of getting (23), (24) and (28). That is,…”

Section: Derivation Of the General Numerical Schemementioning

confidence: 99%

“…In [20,22], it is found that the Grünwald approximation has a superconvergent point. Based on this fact, in this subsection we derive a series of second order quasi-compact approximations for the combined left Riemann-Liouville fractional derivatives.…”

Section: Some Of the Second Order Approximationsmentioning

confidence: 99%

“…, then the quasi-compact approximations corresponding to (20), (21), and (28) share the following form…”

Section: Performing the Inverse Fourier Transform On (35) Leads Tomentioning

confidence: 99%

“…For the right Riemann-Liouville fractional derivative, the same results can be obtained if δ α h,p f (x), p = −1, 0, 1 is substituted by σ α h,p f (x), p = −1, 0, 1, and f (x − ·) by f (x + ·). By realigning the equi-spaced grid points, Nasir et al in [20] refer to that δ α h,1 f (x) can approximate −∞ D α x f (x + βh) with second order accuracy, where…”

Section: Some Of the Second Order Approximationsmentioning

confidence: 99%

“…The superconvergence of the Grünwald discretizaton to the Riemann-Liouville derivative at some special point is introduced in Sec. 8.2 of [22] and a further discussion is given in [20]. Using the superconvergence of the Grünwald discretization, in this paper we develop a series of quasi-compact second order, third order, and fourth order schemes for space fractional diffusion equation:…”

Section: Introductionmentioning

confidence: 99%

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A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Zhao

Deng

2014

Numer. Methods Partial Differential Eq.

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Based on the superconvergent approximation at some point (depending on the fractional order α, but not belonging to the mesh points) for Grünwald discretization to fractional derivative, we develop a series of high order quasi-compact schemes for space fractional diffusion equations. Because of the quasi-compactness of the derived schemes, no points beyond the domain are used for all the high order schemes including second order, third order, fourth order, and even higher order schemes; moreover, the algebraic equations for all the high order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the nonhomogeneous boundary conditions are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders.

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Section: Derivation Of the General Numerical Schemementioning

confidence: 99%

Section: Some Of the Second Order Approximationsmentioning

confidence: 99%

“…, then the quasi-compact approximations corresponding to (20), (21), and (28) share the following form…”

Section: Performing the Inverse Fourier Transform On (35) Leads Tomentioning

confidence: 99%

Section: Some Of the Second Order Approximationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Zhao

Deng

2014

Numer. Methods Partial Differential Eq.

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A fast second‐order difference scheme for the space–time fractional equation

Wang

Sun

2019

Numerical Methods Partial

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In this paper, a fast second-order accurate difference scheme is proposed for solving the space-time fractional equation. The temporal Caputo derivative is approximated by  L2-1 formula which employs the sum-of-exponential approximation to the kernel function appeared in Caputo derivative. The second-order linear spline approximation is applied to the spatial Riemann-Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes. KEYWORDS convergence, fast method, finite difference scheme, space-time fractional equation, stability 1 Numer Methods Partial Differential Eq. 2019;35:1326-1342. wileyonlinelibrary.com/journal/num

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High order schemes for the tempered fractional diffusion equations

Deng

2015

Adv Comput Math

107

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Lévy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering Lévy measure of Lévy flights leads to the tempered stable Lévy processes which combine both the α-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable Lévy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Grünwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Grünwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.Mathematics Subject Classification (2010): 26A33, 65M12, 65M06

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A Second Order Finite Difference Approximation for the Fractional Diffusion Equation

Cited by 27 publications

References 11 publications

A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

A fast second‐order difference scheme for the space–time fractional equation

High order schemes for the tempered fractional diffusion equations

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