article

A Time-Spectral Algorithm for Fractional Wave Problems

Authors:

Xiaoping XieAuthors Info & Claims

Journal of Scientific Computing, Volume 77, Issue 2

Pages 1164 - 1184

https://doi.org/10.1007/s10915-018-0743-5

Published: 01 November 2018 Publication History

Abstract

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.

References

[1]

Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

[2]

Chen, C., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227(2), 886---897 (2007)

[3]

Chen, S., Shen, J., Wang, L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603---1638 (2016)

[4]

Ciarlet, P.: The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)

Digital Library

[5]

Deng, W.: Finite element method for the space and time fractional Fokker---Planck equation. SIAM J. Numer. Anal. 47(1), 204---226 (2009)

Digital Library

[6]

Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558---576 (2006)

[7]

Gao, G., Sun, Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications. J. Comput. Phys. 259, 33---50 (2014)

[8]

Huang, J., Tang, Y., Vzquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algorithms 64(4), 707---720 (2013)

Digital Library

[9]

Li, B., Luo, H., Xie, X.: Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. Submitted. arXiv:1804.10552 (2018)

[10]

Li, X., Xu, C.: A space---time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108---2131 (2009)

Digital Library

[11]

Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533---1552 (2007)

Digital Library

[12]

Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65---77 (2004)

Digital Library

[13]

Podlubny, I.: Fractional Eifferential Equations. Academic Press, San Diego (1998)

[14]

Ren, J., Long, X., Mao, S., Zhang, J.: Superconvergence of finite element approximations for the fractional diffusion-wave equation. J. Sci. Comput. 72(3), 917---935 (2017)

Digital Library

[15]

Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Philadelphia (1993)

[16]

Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms. Analysis and Applications. Springer, Berlin (2011)

Digital Library

[17]

Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193---209 (2006)

[18]

Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007)

[19]

Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703---1727 (2012)

[20]

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)

Digital Library

[21]

Yang, J., Huang, J., Liang, D., Tang, Y.: Numerical solution of fractional diffusion-wave equation based on fractional multistep method. Appl. Math. Model. 38(14), 3652---3661 (2014)

[22]

Yang, Y., Chen, Y., Huang, Y., Wei, H.: Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis. Comput. Math. Appl. 73(6), 1218---1232 (2017)

Digital Library

[23]

Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216(1), 264---274 (2006)

Digital Library

[24]

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(5), 1862---1874 (2005)

Digital Library

[25]

Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40---A62 (2014)

[26]

Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), 2976---3000 (2013)

Digital Library

[27]

Zhang, Y., Sun, Z., Liao, H.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195---210 (2014)

Digital Library

[28]

Zhang, Y., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. Siam J. Numer. Anal. 45(50), 1535---1555 (2012)

[29]

Zheng, M., Liu, F., Turner, I., Anh, V.: A novel high order space---time spectral method for the time fractional Fokker---Planck equation. Siam J. Sci. Comput. 37(2), A701---A724 (2015)

Cited By

Luo HXie X(2022)Optimal Error Estimates of a Time-Spectral Method for Fractional Diffusion Problems with Low Regularity DataJournal of Scientific Computing10.1007/s10915-022-01791-191:1Online publication date: 24-Feb-2022
https://dl.acm.org/doi/10.1007/s10915-022-01791-1
Li BLuo HXie X(2020)A space-time finite element method for fractional wave problemsNumerical Algorithms10.1007/s11075-019-00857-w85:3(1095-1121)Online publication date: 4-Jan-2020
https://dl.acm.org/doi/10.1007/s11075-019-00857-w
Li BWang TXie X(2020)Numerical Analysis of Two Galerkin Discretizations with Graded Temporal Grids for Fractional Evolution EquationsJournal of Scientific Computing10.1007/s10915-020-01365-z85:3Online publication date: 22-Nov-2020
https://dl.acm.org/doi/10.1007/s10915-020-01365-z
Show More Cited By

A Time-Spectral Algorithm for Fractional Wave Problems
1. Mathematics of computing
  1. Mathematical analysis
2. Theory of computation

Recommendations

A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations (TFSEs) subject to initial-boundary and non-local conditions. The first step ...
Least-Squares Spectral Method for the solution of a fractional advection-dispersion equation

Fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields. This article describes a Least Squares Spectral Method for solving advection-dispersion equations using Caputo or Riemann-Liouville ...
A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order

We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Scientific Computing

Journal of Scientific Computing Volume 77, Issue 2

November 2018

614 pages

ISSN:0885-7474

Issue’s Table of Contents

Copyright © Copyright © 2018 Springer Science+Business Media, LLC, part of Springer Nature.

Publisher

Plenum Press

United States

Publication History

Published: 01 November 2018

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 15 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Luo HXie X(2022)Optimal Error Estimates of a Time-Spectral Method for Fractional Diffusion Problems with Low Regularity DataJournal of Scientific Computing10.1007/s10915-022-01791-191:1Online publication date: 24-Feb-2022
https://dl.acm.org/doi/10.1007/s10915-022-01791-1
Li BLuo HXie X(2020)A space-time finite element method for fractional wave problemsNumerical Algorithms10.1007/s11075-019-00857-w85:3(1095-1121)Online publication date: 4-Jan-2020
https://dl.acm.org/doi/10.1007/s11075-019-00857-w
Li BWang TXie X(2020)Numerical Analysis of Two Galerkin Discretizations with Graded Temporal Grids for Fractional Evolution EquationsJournal of Scientific Computing10.1007/s10915-020-01365-z85:3Online publication date: 22-Nov-2020
https://dl.acm.org/doi/10.1007/s10915-020-01365-z
Luo HLi BXie X(2019)Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth DataJournal of Scientific Computing10.1007/s10915-019-00962-x80:2(957-992)Online publication date: 1-Aug-2019
https://dl.acm.org/doi/10.1007/s10915-019-00962-x

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents