Abstract
Lévy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering Lévy measure of Lévy flights leads to the tempered stable Lévy processes which combine both the α-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable Lévy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Grünwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Grünwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baeumera, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233(10), 2438–2448 (2010)
Buschman, R.G.: Decomposition of an integral operator by use of Mikusinski calculus. SIAM J. Math. Anal. 3(1), 83–85 (1972)
Carmi, S., Turgeman, L., Barkai, E.: On distributions of functionals of anomalous diffusion paths. J. Stat. Phys. 141(6), 1071–1092 (2010)
Cartea, Á., del-Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A 374(2), 749–763 (2007)
Cartea, Á., del-Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76, 041105 (2007)
del-Castillo-Negrete, D.: Truncation effects in superdiffusive front propagation with Lévy flights. Phys. Rev. E 79, 031120 (2009)
Çelik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231(4), 1743–1750 (2012)
Chechkin, A.V., Gonchar, V.Yu., Klafter, J., Metzler, R.: Natural cutoff in Lévy flights caused by dissipative nonlinearity. Phys. Rev. E 72, 010101 (2005)
Deng, W.H., Chen, M.H., Barkai, E.: Numerical algorithms for the forward and backward fractional Feynman-Kac equations. J. Sci. Comput. 62(3), 718–746 (2015)
Fornberg, B.: Calculation of weights in finite difference formulas. SIAM Rev. 40(3), 685–691 (1998)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)
Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, New York (1974)
Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)
Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London (1993)
Srivastava, H.M., Buschman, R.G.: Convolution Integral Equations with Special Function Kernels. Wiley, New York (1977)
Lynch, V.E., Carreras, B.A., del-Castillo-Negrete, D., Ferreira-Mejias, K.M., Hicks, H.R.: Numerical methods for the solution of partial differential equations of fractional order. J. Comput. Phys. 192, 406–421 (2003)
Liu, F., Ahn, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166(2), 209–219 (2004)
Marom, O., Momoniat, E.: A comparison of numerical solutions of fractional diffusion models in finance. Nonl. Anal.: R.W.A. 10(6), 3435–3442 (2009)
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 37(31), R161–R208 (2004)
Mantegna, R.N., Stanley, H.E.: Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys. Rev. Lett. 73(22), 2946–2949 (1994)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)
Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L17403 (2009)
Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics, vol. 43. Walter de Gruyter, Berlin/Boston (2012)
Nasir, H.M., Gunawardana, B.L.K., Abeyrathna, H.M.N.P.: A second order finite difference approximation for the fractional siffusion equation. Inter. J. Appl. Phys. Math. 3, 237–243 (2013)
Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, 1–12 (2006)
Pang, H.-K., Sun, H.W.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231(2), 693–703 (2012)
Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, Berlin (2007)
Sokolov, I.M., Chechkin, A.V., Klafter, J.: Fractional diffusion equation for a power-law-truncated Lévy process. Phys. A 336(3), 245–251 (2004)
Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative. Appl. Numer. Math. 90, 22–37 (2015)
Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comp. 84, 1703–1727 (2015)
Varga, R.S.: Matrix Iterative Analysis. Springer, Berlin (2000)
Wang, H., Wang, K., Sircar, T.: A direct N l o g 2 N finite difference method for fractional diffusion equations. J. Comput. Phys. 229(21), 8095–8104 (2010)
Zhao, L.J., Deng, W.H.: A series of high order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives. Numer. Methods Partial Differential Equations 31, 1345–1381 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Jan Hesthaven
Rights and permissions
About this article
Cite this article
Li, C., Deng, W. High order schemes for the tempered fractional diffusion equations. Adv Comput Math 42, 543–572 (2016). https://doi.org/10.1007/s10444-015-9434-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-015-9434-z