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Fast Spectral Methods for Temporally-Distributed Fractional PDEs

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 119))

Abstract

Temporally-distributed fractional partial differential equations appear as rigorous mathematical models that solve the probability density function of non-Markovian processes coding multi-physics diffusion-to-wave and multi-rate ultra slow-to-super diffusion dynamics (Chechkin et al, Phys Rev E 66(4):046129, 2002). We develop a Petrov-Galerkin spectral method for high dimensional temporally-distributed fractional partial differential equations with two-sided derivatives in a space-time hypercube. We employ Jacobi poly-fractonomials given in (Zayernouri and Karniadakis, J Comput Phys 252:495–517, 2013) and Legendre polynomials as the temporal and spatial basis/test functions, respectively. Moreover, we formulate a fast linear solver for the corresponding Lyapunov system. Furthermore, we perform the corresponding discrete stability and error analysis of the numerical scheme. Finally, we carry out several numerical test cases to examine the efficiency and accuracy of the method.

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Acknowledgements

This work was supported by the AFOSR Young Investigator Program (YIP) award on: “Data-Infused Fractional PDE Modeling and Simulation of Anomalous Transport” (FA9550-17-1-0150).

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Authors and Affiliations

  1. Department of Computational Mathematics, Science, and Engineering, Michigan State University, 428 S Shaw Ln, East Lansing, MI, 48824, USA

    Mehdi Samiee, Ehsan Kharazmi & Mohsen Zayernouri

  2. Department of Mechanical Engineering, Michigan State University, 428 S Shaw Ln, East Lansing, MI, 48824, USA

    Mehdi Samiee, Ehsan Kharazmi & Mohsen Zayernouri

Authors
  1. Mehdi Samiee
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  2. Ehsan Kharazmi
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  3. Mohsen Zayernouri
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Corresponding author

Correspondence to Mohsen Zayernouri .

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Editors and Affiliations

  1. School of Mechanical Engineering, University of Campinas, Campinas, Brazil

    Marco L. Bittencourt

  2. Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

    Ney A. Dumont

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

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Samiee, M., Kharazmi, E., Zayernouri, M. (2017). Fast Spectral Methods for Temporally-Distributed Fractional PDEs. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_47

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