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Numerical Approximations for the Tempered Fractional Laplacian: Error Analysis and Applications

  • Siwei Duo
  • Yanzhi ZhangEmail author
Article
  • 36 Downloads

Abstract

In this paper, we propose an accurate finite difference method to discretize the d-dimensional (for \(d \ge 1\)) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Our numerical method has an accuracy of \({\mathcal {O}}(h^\varepsilon )\), for \(u \in C^{0, \,\alpha + \varepsilon } (\bar{\Omega })\) if \(\alpha < 1\) (or \(u \in C^{1, \,\alpha - 1 + \varepsilon } (\bar{\Omega })\) if \(\alpha \ge 1\)) with \(\varepsilon > 0\), suggesting the minimum consistency conditions. The accuracy can be improved to \({\mathcal {O}}(h^2)\), for \(u \in C^{2, \,\alpha + \varepsilon } (\bar{\Omega })\) if \(\alpha < 1\) (or \(u \in C^{3, \,\alpha - 1 + \varepsilon } (\bar{\Omega })\) if \(\alpha \ge 1\)). Numerical experiments confirm our analytical results and provide insights in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our method only requires the solution \(u \in C^{1,1}(\bar{\Omega })\) for any \(\alpha \in (0, 2)\). Moreover, if the solution of tempered fractional Poisson problems satisfies \(u \in C^{p, s}(\bar{\Omega })\) for \(p = 0, 1\) and \(s\in (0, 1]\), our method has the accuracy of \({\mathcal {O}}(h^{p+s})\). Since our method yields a (multilevel) Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including the Allen–Cahn equation and Gray–Scott equations.

Keywords

Tempered integral fractional Laplacian Finite difference methods Error estimates Fractional Allen–Cahn equation Fractional Gray–Scott equations 

Notes

Acknowledgements

This work was supported by the US National Science Foundation under Grant No. DMS-1620465.

References

  1. 1.
    Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55, 472–495 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baeumer, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 303–325 (2002)CrossRefGoogle Scholar
  4. 4.
    Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Finance 13, 345–382 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cartea, \({\dot{\text{A}}}\)., del Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A 374, 749–763 (2007)Google Scholar
  6. 6.
    Chechkin, A.V., Gonchar, VYu., Klafter, J., Metzler, R.: Natural cutoff in Lévy flights caused by dissipative nonlinearity. Phys. Rev. E 72, 010101 (2005)CrossRefGoogle Scholar
  7. 7.
    Dubrulle, B., Laval, J.-P.: Truncated Lévy laws and 2D turbulence. Eur. Phys. J. B 4, 143–146 (1998)CrossRefGoogle Scholar
  8. 8.
    Duo, S., van Wyk, H.W., Zhang, Y.: A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem. J. Comput. Phys. 355, 233–252 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duo, S., Zhang, Y.: Computing the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Commun. Comput. Phys. 18, 321–350 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duo, S., Zhang, Y.: Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications. Comput. Method Appl. Mech. Eng. 355, 639–662 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Duo, S., Zhang, Y.: Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation. Comput. Math. Appl. 77, 2257–2271 (2016)CrossRefGoogle Scholar
  12. 12.
    Javanainen, M., Hammaren, H., Monticelli, L., Jeon, J.-H., Miettinen, M.S., Martinez-Seara, H., Metzler, R., Vattulainen, I.: Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes. Faraday Discuss. 161, 397–417 (2013)CrossRefGoogle Scholar
  13. 13.
    Khan, A.R., Pečarić, J., Praljak, M.: Weighted Montgomery’s identities for higher order differentiable functions of two variables. Rev. Anal. Numér. Théor. Approx. 42, 49–71 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kirkpatrick, K., Zhang, Y.: Fractional Schrödinger dynamics and decoherence. Phys. D 332, 41–54 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koponen, I.: Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E 52, 1197–1199 (1995)CrossRefGoogle Scholar
  16. 16.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mantegna, R.N., Stanley, H.E.: Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys. Rev. Lett. 73, 2946–2949 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L17403 (2008)CrossRefGoogle Scholar
  19. 19.
    Minden, V., Ying, L.: A simple solver for the fractional Laplacian in multiple dimensions. arXiv:1802.03770
  20. 20.
    Pearson, J.E.: Complex patterns in a simple system. Science 261, 189–192 (1993)CrossRefGoogle Scholar
  21. 21.
    Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117, 677–707 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sokolov, I.M., Chechkin, A.V., Klafter, J.: Fractional diffusion equation for a power-law-truncated Lévy process. Phys. A 336, 245251 (2004)CrossRefGoogle Scholar
  23. 23.
    Sun, J., Nie, D., Deng, W.: Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian. preprint (2018)Google Scholar
  24. 24.
    Tang, T., Wang, L., Yuan, H., Zhou, T.: Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains. arXiv:1905.02476
  25. 25.
    Zhang, Y., Meerschaert, M.M., Packman, A.I.: Linking fluvial bed sediment transport across scales. Geophys. Res. Lett. 39, L20404 (2012)Google Scholar
  26. 26.
    Zhang, Z., Deng, W., Fan, H.: Finite difference schemes for the tempered fractional Laplacian. Numer. Math. Theor. Meth. Appl. 12, 492–516 (2019)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zhang, Z., Deng, W., Karniadakis, G.E.: A Riesz basis Galerkin method for the tempered fractional Laplacian. SIAM J. Numer. Anal. 56, 3010–3039 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA

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