A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

Chen, S.; Liu, F.; Jiang, Xiaoyun; Turner, Ian; Anh, Vo

doi:10.1016/j.amc.2014.08.031

Cited by 42 publications

(29 citation statements)

References 33 publications

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“…The difference scheme was proven to be convergent on analysing the spectral radius of the coefficient matrix of the discretised linear system, and here we show that the scheme is uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A similar result for two-dimensional (2D) FDE has been obtained in the Riesz sense [11,24], and for two-sided FDE in the 1D case [13].…”

Section: Introductionsupporting

confidence: 53%

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Lei

Chen

Zhang

2016

East Asian J. Appl. Math.

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High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant preconditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N log N ) operation cost and O(N ) memory requirement. Illustrative numerical examples are also presented.

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Section: Introductionsupporting

confidence: 53%

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Lei

Chen

Zhang

2016

East Asian J. Appl. Math.

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“…2 Remark 4.1. We conclude from Theorems 3 and 4 that the stiffness matrix A for the finite volume scheme (5), which is discretized on a locally refined composite mesh, can be stored in O (N) of memories and Aw can be evaluated in O (N log N) operations for any vector w ∈ R N .…”

Section: Theorem 4 the Submatrices A Lr And A Rl Can Be Approximatmentioning

confidence: 98%

“…We perform numerical experiments to investigate the performance of the fast finite volume method (5). In the numerical comparisons we measure the discrete maximum norm of the global truncation error…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Extensive research has been conducted in the development of numerical methods for fractional differential equations [5,7,9,10,[12][13][14]18,[26][27][28]. Because of the nonlocal nature of fractional differential operators, numerical methods for space-fractional differential equations usually generate full stiffness matrices and were traditionally solved via Gaussian elimination.…”

Section: Introductionmentioning

confidence: 99%

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A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh

Jia

Wang

2015

Journal of Computational Physics

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Numerical methods for fractional differential equations generate full stiffness matrices, which were traditionally solved via Gaussian type direct solvers that require O (N 3 ) of computational work and O (N 2 ) of memory to store where N is the number of spatial grid points in the discretization. We develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite volume schemes defined on a locally refined composite mesh for fractional differential equations to resolve boundary layers of the solutions. Numerical results are presented to show the utility of the method.

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“…17,18 Liu et al 9 discussed the Fitzhugh-Nagumo mono domain model with the space fractional equation. Chen et al 19 studied two-sided space-fractional diffusion model with variable diffusivity coefficients in a fast semi-implicit difference method. For recent references in this field, see Refs.…”

Section: Introductionmentioning

confidence: 99%

Fractional boundary layer flow and radiation heat transfer of MHD viscoelastic fluid over an unsteady stretching surface

2015

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This paper presents an investigation for magnetohydrodynamic (MHD) viscoelastic fluid boundary layer flow and radiation heat transfer over an unsteady stretching sheet in presence of heat source. Time dependent fractional derivative is first introduced in formulating the boundary layer equations. Numerical solutions are obtained by using the finite difference scheme and L1-algorithm approximation. Results indicate that the proposed model describes a basic delaying times framework for viscoelastic flow and radiation heat transfer. The effects of involved parameters on velocity and temperature fields are shown graphically and analyzed in detail.

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A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

Cited by 42 publications

References 33 publications

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh

Fractional boundary layer flow and radiation heat transfer of MHD viscoelastic fluid over an unsteady stretching surface

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