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Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems

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Abstract

Finite difference methods for approximating fractional derivatives are often analyzed by determining their order of consistency when applied to smooth functions, but the relationship between this measure and their actual numerical performance is unclear. Thus in this paper several wellknown difference schemes are tested numerically on simple Riemann-Liouville and Caputo boundary value problems posed on the interval [0, 1] to determine their orders of convergence (in the discrete maximum norm) in two unexceptional cases: (i) when the solution of the boundary-value problem is a polynomial (ii) when the data of the boundary value problem is smooth. In many cases these tests reveal gaps between a method’s theoretical order of consistency and its actual order of convergence. In particular, numerical results show that the popular shifted Grünwald-Letnikov scheme fails to converge for a Riemann-Liouville example with a polynomial solution p(x), and a rigorous proof is given that this scheme (and some other schemes) cannot yield a convergent solution when p(0) = 0.

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Authors and Affiliations

  1. Institute of Mathematics and Applications Department of Applied Mathematics, University of Zaragoza, Spain

    José Luis Gracia

  2. Applied Mathematics Division, Beijing Computational Science Research Center, Haidian District, Beijing, China

    Martin Stynes

  3. Department of Mathematics, National University of Ireland, Western Road, Cork, Ireland

    Martin Stynes

Authors
  1. José Luis Gracia
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  2. Martin Stynes
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Correspondence to José Luis Gracia.

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Gracia, J.L., Stynes, M. Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems. FCAA 18, 419–436 (2015). https://doi.org/10.1515/fca-2015-0027

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  • DOI: https://doi.org/10.1515/fca-2015-0027

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