Abstract
A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new high-order formulas can be regarded as the modification of the classical (or shifted) Lubich’s difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order \({\mathcal {O}}(\tau ^2+h^2)\), where \(\tau \) and h are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.
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References
Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
Bailey, W.N.: Generalized Hypergeometric Series. Stechert-Hafner Inc, New York (1964)
Çelik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)
Chan, R.H.: Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions. IMA J. Numer. Anal. 11, 333–345 (1991)
Chan, R.H., Jin, X.Q.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007)
Chen, M.H., Deng, W.H.: WSLD operators: a class of fourth order difference approximations for space Riemann–Liouville derivative. SIAM J. Numer. Anal. 52, 1418–1438 (2014)
Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 79, 807–827 (2016)
Chen, S., Jiang, X., Liu, F., Turner, I.: High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation. J. Comput. Appl. Math. 278, 119–129 (2015)
Ding, H.F., Li, C.P., Chen, Y.Q.: High-order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218–237 (2015)
Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)
Feller, W.: On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them. Commun. Sém. Math. Univ. Lund. 1, 73–81 (1952)
Garrappaa, R., Moretb, I., Popolizioc, M.: Solving the time-fractional Schrödinger equation by Krylov projection methods. J. Comput. Phys. 293, 115–134 (2015)
Huang, J., Tang, Y., Vázquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algorithms 64, 707–720 (2013)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)
Li, C.P., Ding, H.F.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38, 3802–3821 (2014)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)
Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Springer, Berlin (1924)
Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, Article ID 48391, 1–2 (2006)
Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
Podlubny, I.: Fractional Differential Equations. Academin Press, San Diego (1999)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, New York (2007)
Rall, L.B.: Perspectives on Automatic Differentiation: Past, Present, and Future? Automatic Differentiation: Applications, Theory, and Implementations. Springer, Berlin (2006)
Srivastava, H.M.: An explicit formula for the generalized Bernoulli polynomials. J. Math. Anal. Appl. 130, 509–513 (1988)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, London (1993)
Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)
Tricomi, F.G., Erdélyi, A.: The asymptotic expansion of a ratio of Gamma functions. Pac. J. Math. 1, 133–142 (1951)
Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)
Wang, H., Du, N.: Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258, 305–318 (2014)
Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)
Xu, Q., Hesthaven, J.S., Chen, F.: A parareal method for time-fractional differential equations. J. Comput. Phys. 293, 173–183 (2015)
Yuste, S.B., Acedo, L.: An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)
Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, A40–A62 (2014)
Zeng, F.H., Li, C.P., Liu, F.W., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, A2976–A3000 (2013)
Zhao, X., Sun, Z.Z.: Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, 747–771 (2015)
Zhao, X., Sun, Z.Z., Hao, Z.P.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation. SIAM J. Sci. Comput. 36, A2865–A2886 (2014)
Acknowledgements
The work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11372170, 11561060 and 11671251), the Scientific Research Program for Young Teachers of Tianshui Normal University (Grant Nos. TSA1405) and Tianshui Normal University Key Construction Subject Project (Big data processing in dynamic image).
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Ding, H., Li, C. High-Order Numerical Algorithms for Riesz Derivatives via Constructing New Generating Functions. J Sci Comput 71, 759–784 (2017). https://doi.org/10.1007/s10915-016-0317-3
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DOI: https://doi.org/10.1007/s10915-016-0317-3