Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Numerical Algorithms
Erschienen in: Numerical Algorithms 3/2020

18.05.2019 | Original Paper

Sharp H1-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations

verfasst von: Xin Li, Luming Zhang, Hong-lin Liao

Erschienen in: Numerical Algorithms | Ausgabe 3/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A finite difference cosine pseudo-spectral scheme is presented for solving a linear reaction-subdiffusion problem with Neumann boundary conditions. The nonuniform version of L1 formula is employed for approximating the Caputo fractional derivative, and a cosine pseudo-spectral approximation is utilized in spatial discretization. With the help of discrete fractional Grönwall inequality and global consistency analysis, sharp H1-norm error estimate reflecting the regularity of solution is verified for the proposed method. A fast algorithm is implemented in computation and numerical results confirm the sharpness of our analysis.
Vorheriger Artikel How to compute the minimum norm least squares solution of singular linear system by using the preconditioned HSS method?
Nächster Artikel Backward errors of the linear complementarity problem

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren
Anhänge
Nur mit Berechtigung zugänglich
Literatur
1.
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)MATH
2.
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific (2000)
3.
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRef
4.
Meerschaert, M.M., Benson, D., Baeumer, B.: Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63, 1112–1117 (2001)CrossRef
5.
Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Multiscaling fractional advection-dispersion equations and their solutions. Water Resour. Res. 39, 1022–1032 (2003)
6.
Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)MathSciNetCrossRef
7.
Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetCrossRef
8.
Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)MathSciNetCrossRef
9.
Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)MathSciNetCrossRef
10.
Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D frational subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010)MathSciNetCrossRef
11.
12.
Jin, B.T., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36(1), 197–221 (2016)MathSciNetMATH
13.
Sakamoto, K., Yamamoto, M.: Initial value/boundary value prolems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)MathSciNetCrossRef
14.
Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)MathSciNetCrossRef
15.
Zhang, Y.N., Sun, Z.Z., Liao, H. -L.: Finite difference methods for the time fractional diffusion equation on nonuniform meshes. J. Comput. Phys. 265, 195–210 (2014)MathSciNetCrossRef
16.
Liao, H.-L., Li, D.F., Zhang, J.W.: Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56, 1112–1133 (2018)MathSciNetCrossRef
17.
Liao, H.-L., McLean, W., Zhang, J.W.: A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218–237 (2019)MathSciNetCrossRef
18.
19.
Liao, H.-L., McLean, W., Zhang, J.W.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. (2018) arXiv:1803.​09873v2
20.
Povstenko, Y.: Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn. 59, 593–605 (2010)CrossRef
21.
Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)MathSciNetCrossRef
22.
Zhao, X., Sun, Z.Z.: A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6601–6074 (2011)MathSciNet
23.
Ren, J.C., Sun, Z.Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456–467 (2013)MathSciNetCrossRef
24.
Cao, J.X., Li, C.P., Chen, Y.Q.: Compact difference method for solving the fractional reaction subdiffusion equation with Neumann boundary value condition. Int. J. Comput. Math. 92, 167–180 (2015)MathSciNetCrossRef
25.
Chen, C.M., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007)MathSciNetCrossRef
26.
Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)MathSciNetCrossRef
27.
Lv, C.W., Xu, C.J.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699–A2724 (2016)MathSciNetCrossRef
28.
Bhrawy, A.H.: A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer. Algo. 73, 91–113 (2016)MathSciNetCrossRef
29.
Gong, Y.Z., Cai, J.X., Wang, Y.S.: Multi-symplectic Fourier pseudospectral method for the Kawahara equation. Commun. Comput. Phys. 16, 35–55 (2014)MathSciNetCrossRef
30.
Gong, Y.Z., Wang, Q., Wang, Y.S., Cai, J.X.: A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation. J. Comput. Phys. 328, 354–370 (2017)MathSciNetCrossRef
31.
Jiang, S.D., Zhang, J.W., Zhang, Q., Zhang, Z.M.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650–678 (2017)MathSciNetCrossRef
32.
Ren, J.C., Liao, H.-L., Zhang, J.W., Zhang, Z.M.: Sharp H1-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems. (2018) arXiv:1811.​08059v1
33.
Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67–86 (1982)MathSciNetCrossRef
Metadaten
Titel
Sharp H1-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations
verfasst von
Xin Li
Luming Zhang
Hong-lin Liao
Publikationsdatum
18.05.2019
Verlag
Springer US
Erschienen in
Numerical Algorithms / Ausgabe 3/2020
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00722-w

Weitere Artikel der Ausgabe 3/2020

Numerical Algorithms 3/2020 Zur Ausgabe

Original Paper

How to compute the minimum norm least squares solution of singular linear system by using the preconditioned HSS method?

Original Paper

Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems

Original Paper

A computational approach for the non-smooth solution of non-linear weakly singular Volterra integral equation with proportional delay

Original Paper

A new coarse space for overlapping Schwarz algorithms for H(curl) problems in three dimensions with irregular subdomains

Original Paper

A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems

Original Paper

Optimal inverse projection of floating-point addition