Optimal Collocation Nodes for Fractional Derivative Operators
Abstract
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 advection-diffusion 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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References
1.
R. Askey and J. Fitch, Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl., 26 (1969), pp. 411--437.
2.
E. Barkai, R. Metzler, and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), pp. 132–-138.
3.
D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), pp. 1403-–1412.
4.
A. S. Chaves, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, 239 (1998), pp. 13--16.
5.
W. H. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), pp. 204--226.
6.
W. H. Deng and C. Li, Finite difference methods and their physical constraints for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 27 (2011), pp. 1561--1583.
7.
K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), pp. 229–-248.
8.
K. Diethelm, N. J. Ford, and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), pp. 31–-52.
9.
V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations, 22 (2006), pp. 558--576.
10.
V. J. Ervin and J. P. Roop, Variational formulation for the fractional advection dispersion equations on bounded domains in $R^{d}$, Numer. Methods Partial Differential Equations, 23 (2007), pp. 256--281.
11.
V. J. Ervin, N. Heuer, and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2007), pp. 572--591.
12.
L. Fatone, D. Funaro, and V. Scannavini, Finite-differences preconditioners for superconsistent pseudospectral approximations, ESAIM Math. Model. Numer. Anal., 41 (2007), pp. 1021--1039.
13.
D. Funaro, Polynomial Approximation of Differential Equations, Lecture Notes in Phys. 8, Springer-Verlag, New York, 1992.
14.
D. Funaro, Spectral Elements for Transport-Dominated Equations, Lect. Notes Comput. Sci. Eng. 1, Springer-Verlag, New York, 1997.
15.
D. Funaro, Superconsistent discretizations, J. Sci. Comput., 17 (2002), pp. 67--79.
16.
D. Funaro, Superconsistent discretizations of integral type equations, Appl. Numer. Math., 48 (2004), pp. 1--11.
17.
R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dynam., 29 (2002), pp. 129–-143.
18.
B. Henry and S. Wearne, Fractional reaction-diffusion, Phys. A, 276 (2000), pp. 448-–455.
19.
J. S. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge, UK, 2007.
20.
M. M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 2535–-2542.
21.
M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method, Int. J. Pure Appl. Math., 74 (2012), pp. 287–-297.
22.
T. Langlands and B. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), pp. 719–-736.
23.
X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108–-2131.
24.
X. Li and C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), pp. 1016–-1051.
25.
Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533–-1552.
26.
F. Liu, M. M. Meerschaert, R. J. McGough, P. Zhuang, and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal., 16 (2013), pp. 9–-25.
27.
Q. Liu, F. Liu, I. Turner, and V. Anh, Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method, J. Comput. Phys., 222 (2007), pp. 57--70.
28.
C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), pp. 704–-719.
29.
R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, CT, 2006.
30.
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
31.
M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65--77.
32.
M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), pp. 80--90.
33.
R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1--77.
34.
R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), pp. R161--R208.
35.
A. Pablo, F. Quirós, A. Rodriguez, and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), pp. 1378--1409.
36.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego CA, 1999.
37.
J. Shen, T. Tang, and L.-L. Wang, Spectral Methods, Algorithms, Analysis and Applications, Springer Ser. Comput. Math. 41, Springer, Heidelberg, 2011.
38.
N. Sugimoto, Burgers equation with a fractional derivative: Hereditary effects on nonlinear acoustic waves, J. Fluid Mech., 225 (1991), pp. 631-–653.
39.
N. Sugimoto and T. Kakutani, Generalized Burgers equation for nonlinear viscoelastic waves, Wave Motion, 7 (1985), pp. 447-–458.
40.
Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), pp. 193–-209.
41.
G. Szegö, Orthogonal Polynomials, AMS, New York, 1939.
42.
W. Y. Tian, W. Deng, and Y. Wu, Polynomial spectral collocation method for space fractional advection-diffusion equation, Numer. Methods Partial Differential Equations, 30 (2014), pp. 514--535.
43.
L.-L. Wang, X. Zhao, and Z. Zhang, Superconvergence of Jacobi-Gauss-type spectral interpolation, J. Sci. Comput., 59 (2014), pp. 667--687.
44.
G. M. Zaslavsky, Chaos, fractional kinetic, and anomalous transport, Phys. Rep., 371 (2002), pp. 461--580.
45.
M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximations, J. Comput. Phys., 47 (2013), pp. 2108–-2131.
46.
M. Zayernouri and G. E. Karniadakis, Exponentially accurate spectral and spectral element methods for fractional ODEs, J. Comput. Phys., 257 (2014), pp. 460–-480.
47.
M. Zayernouri and G. E. Karniadakis, Fractional spectral collocation method, SIAM J. Sci. Comput., 36 (2014), pp. A40–-A62.
48.
H. Zhang, F. Liu, and V. Anh, Garlerkin finite element approximations of symmetric space-fractional partial differential equations, Appl. Math. Comput., 217 (2010), pp. 2534--2545.
49.
Z. Zhang, Superconvergence points of polynomial spectral interpolation, SIAM J. Numer. Anal., 50 (2012), pp. 2966--2985.
Information & Authors
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Published In
SIAM Journal on Scientific Computing
Pages: A1504 - A1524
DOI: 10.1137/140993697
ISSN (online): 1095-7197
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© 2015, Society for Industrial and Applied Mathematics.
History
Submitted: 30 October 2014
Accepted: 21 April 2015
Published online: 18 June 2015
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