Numerical algorithm for the space-time fractional Fokker–Planck system with two internal states

Nie, Daxin; Sun, Jing; Deng, Weihua

doi:10.1007/s00211-020-01148-6

Cited by 12 publications

(7 citation statements)

References 30 publications

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“…From [55], we know that the inverse of an invertible lower triangular Toeplitz matrix is also an invertible lower triangular Toeplitz matrix. Thus, we choose the revised version of Bini's algorithm [56] to compute 22 . The computational cost and the storage requirement of (3.5) are O(M N log(M N )) and O(M N ), respectively.…”

Section: The Bilateral Preconditioning Technique and Its Implementationmentioning

confidence: 99%

“…Generally, it is difficult to obtain analytical solutions of fractional partial differential equations (FPDEs). Thus, numerous numerical methods have been proposed to solve them, see [15][16][17][18][19][20][21][22][23][24] and the references therein. Yang et al [25] proposed two novel numerical schemes to solve a time-space fractional diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

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On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

Zhao¹,

Gu³

2021

Preprint

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Time-space fractional Bloch-Torrey equations are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order scheme for this equation by employing the recently proposed L2-type formula [A. A. Alikhanov, C. Huang, Appl. Math. Comput. ( 2021) 126545]. Then, we prove the stability and the convergence of this scheme.Based on such the numerical scheme, a L2-type all-at-once system is derived. In order to solve this system in a parallel-in-time pattern, a bilateral preconditioning technique is designed according to the special structure of the system. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant for the time fractional order α ∈ (0, 0.3624). Numerical results are reported to show the efficiency of our method.

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Section: The Bilateral Preconditioning Technique and Its Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

Zhao¹,

Gu³

2021

Preprint

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“…All the convergence rates are O(h k+1 ), higher than the predicted one in Theorem 4.4. 18) with p = 0, respectively, as the exact solution and source term; from [2,19], it is known that the solution u ∈ H s+ 1 2 −ǫ (R 2 ) with ǫ > 0 arbitrary small. Here, we take T = 1, τ = T /20000, ϑ = 5, and k = 1, 2 with flux (9).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Various numerical methods, such as finite element, finite difference, and spectral method, are proposed to solve a wide variety of equations involving integral fractional Laplacian. For example, [1][2][3][4]6,19,26] use finite element method with piecewise linear polynomial to solve equations involving fractional Laplacian with homogeneous Dirichlet boundary condition; [5] discusses the regularity for fractional Poisson equation with nonhomogenous Dirichlet boundary condition and proposes a mixed finite element scheme. [14,15,17] solve the fractional Poisson equation by finite difference method and an O(h 2 ) convergence rate is obtained.…”

Section: Introductionmentioning

confidence: 99%

Local discontinuous Galerkin method for the fractional diffusion equation with integral fractional Laplacian

Nie¹,

Deng²

2021

Preprint

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“…Fractional Fokker-Planck equations have attracted many attentions in recent years and the corresponding numerical schemes are extensively proposed [1,2,6,7,10,11,13,15,16,19]. But these numerical methods are mainly constructed for the fractional Fokker-Planck equation with one diffusion operator.…”

Section: Introductionmentioning

confidence: 99%

Numerical approximations for the fractional Fokker-Planck equation with two-scale diffusion

Sun¹,

Deng²,

Nie³

2021

Preprint

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Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we first derive the fractional Fokker-Planck equation with two-scale diffusion from the Lévy process framework, and then the fully discrete scheme is built by using the L 1 scheme for time discretization and finite element method for space. With the help of the sharp regularity estimate of the solution, we optimally get the spatial and temporal error estimates. Finally, we validate the effectiveness of the provided algorithm by extensive numerical experiments.

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Numerical algorithm for the space-time fractional Fokker–Planck system with two internal states

Cited by 12 publications

References 30 publications

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

Local discontinuous Galerkin method for the fractional diffusion equation with integral fractional Laplacian

Numerical approximations for the fractional Fokker-Planck equation with two-scale diffusion

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