Corrigendum to “A finite difference method with non-uniform timesteps for fractional diffusion equations” [Comput. Phys. Comm. 183 (12) (2012) 2594–2600]

Yuste, S. B.; Quintana-Murillo, J.

doi:10.1016/j.cpc.2013.11.002

Cited by 4 publications

(3 citation statements)

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“…First, we choose the time mesh through nonuniform time stepsizes. Next, Module 1 computes U K M,N by the scheme (6) or (9). From Module 2, one can get the MSD of stochastic process by expectation formula.…”

Section: Localization Diffusion: Numerical Resultsmentioning

confidence: 99%

“…Growth matrices C 1 (t k+1 ) and C 2 (t k+1 ) correspond to two equations in (9). By solving (6) or (9), one can get all u k m,n , the approximations of the exact solution.…”

Section: Numerical Schemes With Nonuniform Time Stepsizesmentioning

confidence: 99%

“…In order to obtain MSD, one must know the probability density function (PDF) of stochastic processes, which can be obtained by solving the corresponding Fokker-Planck equations. Studying the numerical methods of solving diffusion equations is a very active field, and many effective schemes have been developed [9,10,11,12,13,14,15,16]. In this paper, we mainly focus on the numerical schemes for the newly developed models [3], i.e., the two-dimensional Fokker-Planck equations governing the PDF of the tempered fractional Brownian motion (tfBm), and reveal the mechanism of the motion of the particles.…”

Section: Introductionmentioning

confidence: 99%

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Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion

Liu

Deng

2019

Numer Algor

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In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the singularity in the time direction. When 0 < H < 0.5, a change of variables ∂ t 2H = 2Ht 2H−1 ∂t avoids the singularity of numerical computation at t = 0, which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For 0.5 < H < 1, the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. By numerically solving the corresponding Fokker-Planck equation, we obtain the mean squared displacement of stochastic processes, which conforms to the characteristics of the tempered fractional Brownian motion.

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Section: Localization Diffusion: Numerical Resultsmentioning

confidence: 99%

“…Growth matrices C 1 (t k+1 ) and C 2 (t k+1 ) correspond to two equations in (9). By solving (6) or (9), one can get all u k m,n , the approximations of the exact solution.…”

Section: Numerical Schemes With Nonuniform Time Stepsizesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion

Liu

Deng

2019

Numer Algor

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Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations

Yuste

Quintana-Murillo

2015

Numer Algor

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The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors.

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FEM for time-fractional diffusion equations, novel optimal error analyses

Mustapha

2018

Math. Comp.

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A semidiscrete Galerkin finite element method applied to timefractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with respect to both the convergence order of the approximate solution and the regularity of the initial data. By using novel energy arguments, for each fixed time t, optimal error bounds in the spatial L 2 -and H 1 -norms are derived for both cases: smooth and nonsmooth initial data.

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Corrigendum to “A finite difference method with non-uniform timesteps for fractional diffusion equations” [Comput. Phys. Comm. 183 (12) (2012) 2594–2600]

Cited by 4 publications

References 0 publications

Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion

Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion

Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations

FEM for time-fractional diffusion equations, novel optimal error analyses

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