Algebraic construction of a third order difference approximations for fractional derivatives and applications

Abstract

Finite difference approximations for fractional derivatives based on Grunwald formula are well known to be of first order accuracy, but display unstable solutions with known numerical methods. The shifted form of the Grunwald approximation removes this instability and keeps the same first order accuracy. Higher order approximations have been obtained by convex combinations of various shifted Grunwald approximations. Recently, a second order shifted Grunwald type approximation was constructed algebraically through a generating function. In this paper, we derive a new third order approximation from this second order approximation by preconditioning the fractional differential operator. This approximation is used with Crank-Nicolson numerical scheme to approximate the solutions of space-fractional diffusion equations by the same preconditioning. Stability and convergence of the numerical scheme are analysed, supported by numerical results showing third order convergence. References

• Boris Baeumer, Mihaly Kovacs, and Harish Sankaranarayanan. Higher order grunwald approximations of fractional derivatives and fractional powers of operators. Transactions of the American Mathematical Society, 367(2):813–834, 2015. doi:10.1090/S0002-9947-2014-05887-X
• E. Barkai, R. Metzler, and J. Klafter. From continuous time random walks to the fractional fokker-planck equation. Physical Review E, 61(1):132, 2000. doi:10.1103/PhysRevE.61.132
• Z. Hao, Z. Sun, and W. Cao. A fourth-order approximation of fractional derivatives with its applications. Journal of Computational Physics, 281:787–805, 2015. doi:10.1016/j.jcp.2014.10.053
• Ch Lubich. Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3):704–719, 1986. doi:10.1137/0517050
• M. M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1):65–77, 2004. doi:10.1016/j.cam.2004.01.033
• H. M. Nasir, B. L. K. Gunawardana, and H. M. N. P. Aberathna. A second order finite difference approximation for the fractional diffusion equation. International Journal of Applied Physics and Mathematics, 3(4):237, 2013. doi:10.7763/IJAPM.2013.V3.212
• H. M. Nasir and K. Nafa. A new second order approximation for fractional derivatives with applications. SQU Journal of Science, 23(1):43–55, 2018. doi:10.24200/squjs.vol23iss1pp43-55
• W. Tian, H. Zhou, and W. Deng. A class of second order difference approximations for solving space fractional diffusion equations. Mathematics of Computation, 84(294):1703–1727, 2015. doi:10.1090/S0025-5718-2015-02917-2
• Y. Yu, W. Deng, and Y. Wu. Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations. arXiv preprint arXiv:1408.6364, 2014. doi:10.4310/CMS.2017.v15.n5.a1
• Y. Yu, W. Deng, Y. Wu, and J. Wu. Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations. Applied Numerical Mathematics, 112:126–145, 2017. doi:10.1016/j.apnum.2016.10.01
• L. Zhao and W. Deng. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives. Numerical Methods for Partial Differential Equations, 31(5):1345–1381, 2015. doi:10.1002/num.21947