zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An O(Nlog 2 N) alternating-direction finite difference method for two-dimensional fractional diffusion equations. (English) Zbl 1229.65165

Summary: Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of O(N 3 ) per time step and memory of O(N 2 ) for where N is the number of grid points.

In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of O(Nlog 2 N) per time step and memory of O(N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination.

Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5 h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.

MSC:
65M06Finite difference methods (IVP of PDE)
35K20Second order parabolic equations, initial boundary value problems
35R11Fractional partial differential equations
65F10Iterative methods for linear systems
65T50Discrete and fast Fourier transforms (numerical methods)
References:
[1]Ammar, G. S.; Gragg, W. B.: The generalized Schur algorithm for the superfast solution of Toeplitz systems. Rational approximation and its applications in mathematics and physics, Lect. notes math. 1237, 315-330 (1987) · Zbl 0614.65023
[2]Ammar, G. S.; Gragg, W. B.: Superfast solution of real positive definite Toeplitz systems, SIAM J. Matrix anal. Appl. 9, 61-76 (1988) · Zbl 0658.65022 · doi:10.1137/0609005
[3]Benson, D.; Wheatcraft, S. W.; Meerschaert, M. M.: Application of a fractional advection – dispersion equation, Water resour. Res. 36, 1403-1413 (2000)
[4]Benson, D.; Wheatcraft, S. W.; Meerschaert, M. M.: The fractional-order governing equation of Lévy motion, Water resour. Res. 36, 1413-1423 (2000)
[5]Böttcher, A.; Silbermann, B.: Introduction to large truncated Toeplitz matrices, (1999)
[6]Carreras, B. A.; Lynch, V. E.; Zaslavsky, G. M.: Anomalous diffusion and exit time distribution of particle travers in plasma turbulence models, Phys. plasma 8, 5096-5103 (2001)
[7]Chan, R. H.; Ng, M. K.: Conjugate gradient methods for Toeplitz systems, SIAM rev. 38, 427-482 (1996) · Zbl 0863.65013 · doi:10.1137/S0036144594276474
[8]Cui, M.: Compact finite difference method for the fractional diffusion equation, J. comput. Phys 228, 7792-7804 (2009) · Zbl 1179.65107 · doi:10.1016/j.jcp.2009.07.021
[9]Davis, P. J.: Circulant matrices, (1979)
[10]Ervin, V. J.; Heuer, N.; Roop, J. P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. anal 45, 572-591 (2007) · Zbl 1141.65089 · doi:10.1137/050642757
[11]Ervin, V. J.; Roop, J. P.: Variational formulation for the stationary fractional advection dispersion equation, Numer. methods partial differ. Equat. 22, 558-576 (2005) · Zbl 1095.65118 · doi:10.1002/num.20112
[12]Ervin, V. J.; Roop, J. P.: Variational solution of fractional advection dispersion equations on bounded domains in rd, Numer. methods partial differ. Equat. 23, 256-281 (2007) · Zbl 1117.65169 · doi:10.1002/num.20169
[13]Golub, G. H.; Van Loan, C. F.: Matrix computations, (1996)
[14]Gray, R. M.: Toeplitz and circulant matrices: a review, Found. trends commun. Inform. theory 2, No. 3, 155-239 (2006)
[15]Hilfer, R.: Applications of fractional calculus in physics, (2000)
[16]Kirchner, J. W.; Feng, X.; Neal, C.: Fractal stream chemistry and its implications for containant transport in catchments, Nature 403, 524-526 (2000)
[17]Klafter, J.; Sokolov, I. M.: Anomalous diffusion spreads its wings, Phys. world 29 – 32, No. August (2005)
[18]Langlands, T. A. M.; Henry, B. I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. comput. Phys 205, 719-736 (2005) · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[19]Li, X.; Xu, C.: The existence and uniqueness of the week solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. comput. Phys 8, 1016-1051 (2010)
[20]Lin, R.; Liu, F.; Anh, V.; Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. math. Comput. 212, 435-445 (2009) · Zbl 1171.65101 · doi:10.1016/j.amc.2009.02.047
[21]Magin, R. L.: Fractional calculus in bioengineering, (2006)
[22]Meerschaert, M. M.; Benson, D. A.; Baeumer, B.: Multidimensional advection and fractional dispersion, Phys. rev. E 59, 5026-5028 (1999)
[23]Meerschaert, M. M.; Benson, D. A.; Baeumer, B.: Operator Lévy motion and multiscaling anomalous diffusion, Phys. rev. E 63, 02112 (2001)
[24]Meerschaert, M. M.; Scheffler, H. P.: Nonparametric methods for heavy tailed vector data: A survey with applications from finance and hydro logy, Recent advances and trends in nonparametric statistics, 265-279 (2003)
[25]Meerschaert, M. M.; Scheffler, H. P.; Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation, J. comput. Phys. 211, 249-261 (2006) · Zbl 1085.65080 · doi:10.1016/j.jcp.2005.05.017
[26]Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations, J. comput. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[27]Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. Math. 56, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[28]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. rep. 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[29]Metzler, R.; Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. phys. A math. Gen. 37, R161-R208 (2004) · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01
[30]Metzler, R.; Klafter, J.; Sokolov, I.: Anomalous transport in external fields: continuous time random walks and fractional diffusion equations extended, Phys. rev. E 58, 1621-1633 (1998)
[31]Murio, D. A.: Implicit finite difference approximation for time fractional diffusion equations, Comput. math. Appl. 56, 1138-1145 (2008) · Zbl 1155.65372 · doi:10.1016/j.camwa.2008.02.015
[32]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[33]Podlubny, I.: Fractional differential equations, (1999)
[34]Raberto, M.; Scalas, E.; Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study, Physica 314, 749-755 (2002) · Zbl 1001.91033 · doi:10.1016/S0378-4371(02)01048-8
[35]Reeves, D. M.; Benson, D. A.; Meerschaert, M. M.; Scheffler, H. P.: Transport of conservative solutes in simulated fractured media: 2. Ensemble solute transport and the correspondence to operator-stable limit distributions, Water resour. Res. 44, W05410 (2008)
[36]Roop, J. P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2, J. comput. Appl. math 193, 243-268 (2006) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[37]Sabatelli, L.; Keating, S.; Dudley, J.; Richmond, P.: Waiting time distributions in financial markets, Eur. phys. J. B 27, 273-275 (2002)
[38]Scher, H.; Montroll, E. W.: Anomalous transit-time dispersion in amorphous solids, Phys. rev. B 12, 2455-2477 (1975)
[39]Shlesinger, M. F.; West, B. J.; Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence, Phys. rev. Lett. 58, 1100-1103 (1987)
[40]Sokolov, I. M.; Klafter, J.; Blumen, A.: Fractional kinetics, Physics today, (November) 55, 28-53 (2002)
[41]Sousa, E.: Finite difference approximates for a fractional advection diffusion problem, J. comput. Phys. 228, 4038-4054 (2009) · Zbl 1169.65126 · doi:10.1016/j.jcp.2009.02.011
[42]Su, L.; Wang, W.; Yang, Z.: Finite difference approximations for the fractional advection diffusion equation, Phys. lett. A 373, 4405-4408 (2009)
[43]Varga, R. S.: Matrix iterative analysis, (2000)
[44]Wang, H.; Wang, K.; Sircar, T.: A direct O(Nlog2n) finite difference method for fractional diffusion equations, J. comput. Phys. 229, 8095-8104 (2010) · Zbl 1198.65176 · doi:10.1016/j.jcp.2010.07.011
[45]Zaslavsky, G. M.; Stevens, D.; Weitzner, H.: Self-similar transport in incomplete chaos, Phys. rev. E 48, 1683-1694 (1993)
[46]Zhang, Y.: A finite difference method for fractional partial differential equation, Appl. math. Comput. 215, 524-529 (2009) · Zbl 1177.65198 · doi:10.1016/j.amc.2009.05.018
[47]Zhang, Y.; Benson, D. A.; Meerschaert, M. M.; Labolle, E. M.; Scheffler, H. P.: Random walk approximation of fractional-order multiscaling anomalous diffusion, Phys. rev. E 74, No. 2, 026706 (2006)
[48]Zhang, Y.; Benson, D. A.; Meerschaert, M. M.; Scheffler, H. P.: On using random walks to solve the space-fractional advection-dispersion equations, J. stat. Phys. 367, No. 1, 89-110 (2006) · Zbl 1092.82038 · doi:10.1007/s10955-006-9042-x