A fast accurate approximation method with multigrid solver for two-dimensional fractional sub-diffusion equation

Lin, Xue-lei; Lu, Xiwen; Ng, Michael K.; Sun, Hai-Wei

doi:10.1016/j.jcp.2016.07.031

Cited by 13 publications

(17 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This observation is analogous to the standard parabolic case [1] and backward Euler convolution quadrature for the model (1) [14]. Iterative solvers have been employed in a number of numerical studies of the subdiffusion model (1) and found some empirical success (see, e.g., [6,16] for multigrid methods). However, the error analysis of such iterative schemes is still largely missing, and the numerical experiments focused mostly on smooth solutions,…”

Section: Introductionmentioning

confidence: 85%

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Jin

2019

Commun. Appl. Math. Comput.

View full text Add to dashboard Cite

In this work, we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order (0, 1) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss-Seidel smoother to approximate the solution efficiently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.

show abstract

Section: Introductionmentioning

confidence: 85%

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Jin

2019

Commun. Appl. Math. Comput.

View full text Add to dashboard Cite

show abstract

“…This article defines and implements a new multigrid method for the two-dimensional fractional sub-diffusion equation in the C/C++ programming language [13]. Their method is compared against existing methods to establish its speed and accuracy.…”

Section: (A) Description Of Chosen Articlesmentioning

confidence: 99%

“…We present the completion percentage over time for each article in this study. The solid line is for article [18], the dotted line is for [14], the long dashed line is for [17], the short dashed line is for [16], the long dash dotted line is for [13], the short dash dotted line is for [15] and the long dash double dotted line is for [12]. For each figure, the x-axis is time measured in hours; the y-axis is the completion measured in percentage.…”

Section: (I) Vignette 1: Missing Dataset Prevents Reproductionmentioning

confidence: 99%

“…We build on our own previous work that evaluates the potential for reproducing published results in 306 computational physics articles [10]. From this work, we examine seven articles to understand barriers to reproducibility more deeply [12][13][14][15][16][17][18]. These seven articles cover a relatively diverse set of topics and use a variety of computational platforms.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Learning from reproducing computational results: introducing three principles and theReproduction Package

Krafczyk

Shi

Bhaskar

et al. 2021

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

We carry out efforts to reproduce computational results for seven published articles and identify barriers to computational reproducibility. We then derive three principles to guide the practice and dissemination of reproducible computational research: (i) Provide transparency regarding how computational results are produced; (ii) When writing and releasing research software, aim for ease of (re-)executability; (iii) Make any code upon which the results rely as deterministic as possible. We then exemplify these three principles with 12 specific guidelines for their implementation in practice. We illustrate the three principles of reproducible research with a series of vignettes from our experimental reproducibility work. We define a novel Reproduction Package , a formalism that specifies a structured way to share computational research artifacts that implements the guidelines generated from our reproduction efforts to allow others to build, reproduce and extend computational science. We make our reproduction efforts in this paper publicly available as exemplar Reproduction Packages . This article is part of the theme issue ‘Reliability and reproducibility in computational science: implementing verification, validation and uncertainty quantification in silico ’.

show abstract

“…Hence, numerical methods turn to be important approaches to solve TFDEs and FDEs. Numerical methods for solving FDEs have been extensively studied in [8][9][10][11][12][13][14][15][16] in recent decades. For tempered fractional derivatives, if the same difference approximation for fractional derivatives is directly adopted, the stability of the resulted finite difference scheme can only be guaranteed when the spatial step size is small enough, see for example [6].…”

Section: Introductionmentioning

confidence: 99%

Fast solution algorithms for exponentially tempered fractional diffusion equations

Lei

Fan

Chen

2018

Numerical Methods Partial

View full text Add to dashboard Cite

In this article, a fast‐iterative method and a fast‐direct method is proposed for solving one‐dimensional and two‐dimensional tempered fractional diffusion equations with constant coefficients. The proposed iterative method is accelerated by circulant preconditioning which is shown to converge superlinearly while the proposed direct method is based on circulant and skew‐circulant representation for Toeplitz matrix inversion. In one‐dimensional case, the operation cost of the proposed methods are both shown to be O ( N log N ) with O ( N ) memory requirement in each time step, where N is the number of spatial nodes. With the alternating direction implicit method, it is proven that the proposed fast solution algorithms can be extended to handle two‐dimensional tempered fractional diffusion equations with O ( N 2 log N ) operation cost and O ( N 2 ) memory requirement in each time step, where the number of spatial nodes in x ‐direction and y ‐direction both equal to N . Numerical examples are provided to illustrate the effectiveness and efficiency of the proposed methods.

show abstract

A fast accurate approximation method with multigrid solver for two-dimensional fractional sub-diffusion equation

Cited by 13 publications

References 19 publications

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Learning from reproducing computational results: introducing three principles and theReproduction Package

Fast solution algorithms for exponentially tempered fractional diffusion equations

Contact Info

Product

Resources

About