Advertisement

Journal of Scientific Computing

, Volume 72, Issue 3, pp 1169–1195 | Cite as

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

  • Daniel BaffetEmail author
  • Jan S. Hesthaven
Article

Abstract

High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel compression scheme for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We study the local truncation error of integral deferred correction schemes for Volterra equations and present numerical results obtained with both implicit and the explicit methods applied to different problems.

Keywords

Fractional differential equations Volterra equations Kernel compression High-order numerical methods Integral deferred correction Local schemes 

Notes

Acknowledgements

The authors are grateful to Stefano Guarino for his contribution. Funding was provided by EPFL.

References

  1. 1.
    Zhang, Y., Sun, Z., Liao, H.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 256, 195–210 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. doi: 10.1093/imanum/dru063
  3. 3.
    Lubich, C.: Convolution quadrature and discretized operational calculus I. Numer. Math. 52, 129–145 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Schädle, A., López-Fernández, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28(2), 421–438 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brunner, H., Schötazau, D.: \(hp\) discontinuous Galerkin time-stepping for volterra integrodifferential equations. SIAM J. Numer. Anal. 44(1), 224–245 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mustapha, K., Brunner, H., Mustapha, H., Schötazau, D.: An \(hp\)-version discontinuous Galerkin method for integro-differential equations of parabolic type. SIAM J. Numer. Anal. 49(4), 1369–1396 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baffet, D., Hesthaven, J.S.: A kernel compression scheme for fractional differential equations. SIAM J. Numer. Anal. (accepted) (2016)Google Scholar
  8. 8.
    Li, J.R.: A fast time stepping method for evaluating fractional integrals. SIAM J. Sci. Comput. 31(6), 4696–4714 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lubich, C., Schädle, A.: Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput. 24(1), 161–182 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    López-Fernández, M., Lubich, C., Schädle, A.: Adaptive fast and oblivious convolution in evolution equations with memory. SIAM J. Sci. Comput. 30(2), 1015–1037 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Beylkin, G., Monzón, L.: Approximation by exponential sums revisited. Appl. Comput. Harmon. Anal. 28, 131–149 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Conte, D., Del Prete, I.: Fast collocation methods for volterra integral equations of convolution type. J. Comput. Appl. Math. 196, 652–663 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lubich, C.: Runge–Kutta theory for volterra and abel integral equations of the second kind. Math. Comp. 41(163), 87–102 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40(2), 241–266 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hagstrom, T., Zhou, R.: On the spectral deferred correction of splitting methods for initial value problems. Comm. Appl. Math. Comp. Sci. 1(1), 169–205 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Christlieb, A., Ong, B., Qiu, J.M.: Integral deferred correction methods constructed with high order Runge–Kutta integrators. Math. Comp. 79(270), 761–783 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Christlieb, A., Ong, B., Qiu, J.M.: Comments on high-order integrators embedded within integral deferred correction methods. Comm. Appl. Math. Comp. Sci. 4(1), 27–56 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guarino, S.: Spectral Deferred Correction Methods for Differential Integral Equations, Master Dissertation, EPFL (2016)Google Scholar
  19. 19.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Alpert, B., Greengard, L., Hagstrom, T.: Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal. 37(4), 1138–1164 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial laplace transforms. SIAM J. Numer. Anal. 44(3), 1332–1350 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Askey, R., Fitch, J.: Integral representations for jacobi polynomials and some applications. J. Math. Anal. Appl. 26, 411–437 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Baffet, D.: Kernel compression schemes for fractional differential equations. MATLAB Central File Exchange file ID: 61024 (2017)Google Scholar
  25. 25.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  26. 26.
    Garrappa, R.: The Mittag–Leffler Function, MATLAB Central File Exchange, file ID: 48154 (2014)Google Scholar
  27. 27.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comp. 28(125), 145–162 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.SB-MATHICSE-MCSSÉcole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland

Personalised recommendations