Preconditioned iterative methods for space-time fractional advection-diffusion equations

Zhao, Zhi‐Ping; Jin, Xiaoqing; Lin, Matthew M.

doi:10.1016/j.jcp.2016.05.021

Cited by 27 publications

(28 citation statements)

References 35 publications

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“…These motivated us to study (1.1)–(1.2) in numerical way. A lot of researches have proposed different efficient numerical methods to study time fractional differential equations, in which the time fractional derivatives are usually modelled in Caputo's sense, interested readers can refer to and the references therein. There are many favorable approximations for the Caputo fractional derivative, here we review several researches which provide some recent progress related to our work.…”

Section: Introductionmentioning

confidence: 99%

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

Lyu

Vong

2018

Numerical Methods Partial

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In this work, we study finite difference scheme for coupled time fractional Klein‐Gordon‐Schrödinger (KGS) equation. We proposed a linearized finite difference scheme to solve the coupled system, in which the fractional derivatives are approximated by some recently established discretization formulas. These formulas approximate the solution with second‐order accuracy at points different form the grid points in time direction. Taking advantage of this property, our proposed linearized scheme evaluates the nonlinear terms on the previous time level. As a result, iterative method is dispensable. The coupled terms in the scheme bring difficulties in analysis. By carefully studying these effects, we proved that the proposed scheme is unconditionally convergent and stable in discrete L 2 norm with energy method. Numerical results are included to justify the theoretical statements.

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Section: Introductionmentioning

confidence: 99%

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

Lyu

Vong

2018

Numerical Methods Partial

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“…It was discovered that the stiffness matrices of the numerical discretizations of constant-order sFDEs on a uniform partition typically possess a Toeplitz-like structure [32], which reduces the memory requirement from O(N 2 ) to O(N ) and computational complexity from O(N 3 ) to O(N log N ) per Krylov subspace iteration via the discrete fast Fourier transform. Furthermore, different preconditioners were employed to futher improve the computational efficiency and even convergence behavior [1,5,14,19,20,21,25,31,34].…”

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confidence: 99%

A fast method for variable-order space-fractional diffusion equations

Jia

Zheng

et al. 2020

Numer Algor

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We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order spacefractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical approximation does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O(kN log 2 N ) memory and O(kN log 3 N ) computational complexity with N and k being the numbers of unknowns and the approximants, respectively. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.Keywords Variable-order space-fractional diffusion equation · Collocation method · Divide-and-conquer algorithm · Toeplitz matrix Mathematics Subject Classification (2010) 65F05 · 65M70 · 65R20 IntroductionField tests showed that space-fractional diffusion equations (sFDEs) provide more accurate descriptions of challenging phenomena of superdiffusive transport and long range interaction, which occur in solute transport in heteroge-

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“…[2,3]). The order (n − 1 < < n) left and right Riemann-Liouville (R-L) fractional derivatives of the function g(t) on [0, T] are defined as i left R-L fractional derivative: 0 t g(t) s) −n+1 ds;…”

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confidence: 99%

A compact finite difference scheme for the fourth‐order time‐fractional integro‐differential equation with a weakly singular kernel

Qiu

Guo

2019

Numerical Methods Partial

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In this paper, a compact finite difference scheme is constructed and investigated for the fourth-order time-fractional integro-differential equation with a weakly singular kernel. In the temporal direction, the Caputo derivative term is treated by means of L 1 discrete formula and the Riemann-Liouville fractional integral term is discretized by the second-order convolution quadrature rule. A fully discrete compact difference scheme is constructed with the space discretization by the fourth-order compact approximation. The stability and convergence are obtained by the discrete energy method, the Cholesky decomposition and the reduced-order method. Numerical experiments are presented to verify the theoretical analysis. KEYWORDSCholesky decomposition, compact difference scheme, fourth-order time-fractional integro-differential equation, second-order convolution quadrature rule, stability and convergence 1 Numer Methods Partial Differential Eq. 2020;36:439-458. wileyonlinelibrary.com/journal/num

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Preconditioned iterative methods for space-time fractional advection-diffusion equations

Cited by 27 publications

References 35 publications

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

A fast method for variable-order space-fractional diffusion equations

A compact finite difference scheme for the fourth‐order time‐fractional integro‐differential equation with a weakly singular kernel

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