Polynomial spectral collocation method for space fractional advection–diffusion equation

Tian, Wenyi; Deng, Weihua; Wu, Yue

doi:10.1002/num.21822

Cited by 58 publications

(33 citation statements)

References 25 publications

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“…When the solutions to (1.1) are smooth, spectral methods lead to higher-order accuracy; see, e.g., [28,45] for spectral Galerkin methods and, e.g., [16,17,26,38] for need better knowledge on the decaying rate and corresponding spectral basis induced by slow decaying weights on the half line, e.g., mapped Jacobi polynomials and generalized Laguerre functions; see, e.g., [36]. However, we do not include such a discussion here due to the limit on the length of the paper.…”

Section: Introductionmentioning

confidence: 99%

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Zhang¹,

Zeng²,

Karniadakis³

2015

SIAM J. Numer. Anal.

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We present optimal error estimates for spectral Petrov-Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov-Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates. Introduction.Numerical methods for fractional differential equations have been investigated for decades; see, e.g., [8,12,33]. However, spectral methods for these equations have a rather short history since the solutions usually have some singular lower-order derivatives. The recent investigation of spectral methods for these singular problems is motivated by the following facts. First, all numerical methods for these problems are nonlocal as spectral methods are. For example, in finite difference methods, see, e.g., [33], and in finite element methods, see, e.g., [13,18], data at almost all grids or all elements are exploited to approximate the integral operators at one grid or element. These methods are still nonlocal, even when a "fixed memory principle" is applied to reduce the use of all data; see, e.g., [15,19]. Second, spectral methods lead to high-order accuracy when the solutions are smooth; see, e.g., [7,24] for integerorder differential equations. For fractional differential equations with weakly singular kernels, solutions can be smooth even when some inputs of the equations are singular.Consider the following fractional ordinary differential equation (FODE) over the interval I = (−1, 1):

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Section: Introductionmentioning

confidence: 99%

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Zhang¹,

Zeng²,

Karniadakis³

2015

SIAM J. Numer. Anal.

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“…Another approach is to approximate the fractional derivative operators in the considered FDEs directly; see [16,27,36,38,54]. Besides finite difference methods, there exist also other numerical methods for FDEs, e.g., finite element methods [22,40], spectral methods [26,39,45], matrix methods [32,35], etc.…”

Section: Introductionmentioning

confidence: 99%

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions

Cao¹,

Zeng²,

Zhang³

et al. 2016

SIAM J. Sci. Comput.

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We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order 0 < β < 1. From the known structure of the non-smooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multi-rate fractional differential systems as well as multi-term fractional differential systems with non-smooth solutions.

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“…Using as approximation basis the set of Jacobi polynomials, pseudo-spectral discretizations of fractional derivative operators are introduced and examined. The idea is to ameliorate the methods recently proposed in [41] and [44]. To this end, a suitable techniques is suggested, where the grid used to represent the function to be differentiated, is not necessarily coincident with the collocation grid.…”

Section: Aim Of the Papermentioning

confidence: 99%

“…. , N (which is the procedure followed in [41]). The solution we suggest is to solve a simple linear system.…”

Section: Aim Of the Papermentioning

confidence: 99%

Optimal Collocation Nodes for Fractional Derivative Operators

Fatone¹,

Funaro²

2015

SIAM J. Sci. Comput.

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Spectral discretizations of fractional derivative operators are examined, where the approximation basis is related to the set of Jacobi polynomials. The pseudospectral method is implemented by assuming that the grid, used to represent the function to be differentiated, may not be coincident with the collocation grid. The new option opens the way to the analysis of alternative techniques and the search for optimal distributions of collocation nodes, based on the operator to be approximated. Once the initial representation grid has been chosen, indications for how to recover the collocation grid are provided, with the aim of enlarging the dimension of the approximation space. As a result of this process, performances are improved. Applications to fractional type advectiondiffusion equation and comparisons in terms of accuracy and efficiency are made. As shown in the analysis, special choices of the nodes can also suggest tricks to speed up computations

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Polynomial spectral collocation method for space fractional advection–diffusion equation

Cited by 58 publications

References 25 publications

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions

Optimal Collocation Nodes for Fractional Derivative Operators

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