Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T02:23:16.960Z Has data issue: false hasContentIssue false
  • English
  • Français

Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators

Published online by Cambridge University Press:  03 June 2015

Minghua Chen  and
Weihua Deng
Minghua Chen
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China
Weihua Deng*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China
*
*Corresponding author.Email:dengwh@lzu.edu.cn
Get access

Abstract

High order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains the v-th order (v < 6) approximations of the α-th derivative (α > 0) or integral (α < 0) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly applied to the space fractional operator with α Є (1,2) for time dependent problem. By weighting and shifting Lubich’s 2nd order discretization scheme, in [Chen & Deng, SINUM, arXiv:1304.7425] we derive a series of effective high order discretizations for space fractional derivative, called WSLD operators there. As the sequel of the previous work, we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich’s 3rd and 4th order discretizations. In particular, we prove that the obtained 4th order approximations are effective for space fractional derivatives. And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.

Keywords

26A3303F3547B3765M12Fractional derivativeshigh order schemeweighted and shifted Lubich difference (WSLD) operatorsnumerical stability.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 516 - 540
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chan, R. H. and Jin, X. Q., An Introduction to Iterative Toeplitz Solvers, SIAM, 2007.Google Scholar
[2]Chen, M. H. and Deng, W. H., WSLD operators: A class of fourth order difference approximations for space Riemann-Liouville derivative, arXiv: 1304.7425 [math.NA].Google Scholar
[3]Chen, M. H., Wang, Y. T., Cheng, X. and Deng, W. H., Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation, BIT Numer. Math., (2014), doi:10.1007/s10543-014-0477-1.Google Scholar
[4]Chen, M. H., Deng, W. H., A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation, Appl. Math. Model., 38 (2014), 3244–3259.Google Scholar
[5]Chen, M. H., Deng, W. H., Wu, Y. J., Second order finite difference approximations for the two dimensional time-space Caputo-Riesz fractional diffusion equation, Appl. Numer. Math., 70 (2013), 22–41.CrossRefGoogle Scholar
[6]Deng, W. H., Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys., 227 (2007), 1510–1522.Google Scholar
[7]Deng, W. H., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), 204–226.Google Scholar
[8]Ervin, V. J. and Roop, J. P., Variational formulation for the stationary fractional advection dis-persion equation, Numer. Methods Partial Differential Equations., 22 (2006), 558–576.Google Scholar
[9]Fan, S. J., A new extracting formula and a new distinguishing means on the one variable cubic equation, Natural science journal of Hainan teacheres college, China., 2 (1989), 91–98.Google Scholar
[10]Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, New York: John Wiley, 1962.Google Scholar
[11]Kilbas, A., Srivastava, M. and Trujillo, J., Theory and Applications of Fractional Differential Equations, New York, 2006.Google Scholar
[12]Li, X. J. and Xu, C. J., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131.Google Scholar
[13]Li, X. J. and Xu, C. J., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016–1051.Google Scholar
[14]Lubich, Ch., Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), 704–719.Google Scholar
[15]Meerschaert, M. M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77.Google Scholar
[16]Ortigueira, M. D., Riesz potential operators and inverses via fractional centred derivatives, International J. Math. Math. Sci., 2006 (2006), 1–12.Google Scholar
[17]Podlubny, I., Fractional Differential Equations, New York: Academic Press, 1999.Google Scholar
[18]Polyanin, A. D. and Manzhirov, A. V., Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, New York, 2007.Google Scholar
[19]Quarteroni, A., Sacco, R. and Saleri, F., Numerical Mathematics, 2nd ed, Springer, 2007.Google Scholar
[20]Sousa, E. and Li, C., A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville drivative, arXiv:1109.2345v1 [math.NA].Google Scholar
[21]Sun, Z. Z. and Wu, X. N., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209.Google Scholar
[22]Tian, W. Y., Zhou, H. and Deng, W. H., A class of second order difference approximations for solving space fractional diffusion Equations, (2014), Math. Comp. (in press) (arXiv:1201.5949v3 [math.NA]).Google Scholar
[23]Zhang, N., Deng, W. H. and Wu, Y. J., Finite difference/element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech., 4 (2012), 496–518.Google Scholar
[24]Zhou, H., Tian, W. Y. and Deng, W. H., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput., 56 (2013), 45–66.Google Scholar
[25]Zhuang, P., Liu, F., Anh, V. and Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), 1760–1781.Google Scholar