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Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives

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Abstract

In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally \(\ell ^1\) contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, \( (\Delta t)^{\alpha } = O(\Delta x)\), where \(\alpha \in (0,1]\) is the fractional exponent in the derivative. When \(\alpha \) is small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally \(\ell ^1\) contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.

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References

  1. Allen, M., Caffarelli, L., Vasseur, A.: Porous medium flow with both a fractional potential pressure and fractional time derivative. arXiv preprint (2015)

  2. Cao, J., Xu, C.: A high order schema for the numerical solution of the fractional ordinary differential equations. J. Comput. Phys. 238, 154–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caputo, M.: Diffusion of fluids in porous media with memory. Geothermics 28(1), 113–130 (1999)

    Article  Google Scholar 

  4. del Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Fractional diffusion in plasma turbulence. Phys. Plasmas 11(8), 3854–3864 (2004)

    Article  Google Scholar 

  5. del Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Phys. Rev. Lett. 94(6), 065003 (2005)

    Article  Google Scholar 

  6. Gao, G., Sun, H.: Three-point combined compact alternating direction implicit difference schemes for two-dimensional time-fractional advection-diffusion equations. Commun. Comput. Phys. 17(02), 487–509 (2015)

    Article  MathSciNet  Google Scholar 

  7. Kumar, P., Agrawal, O.P.: An approximate method for numerical solution of fractional differential equations. Signal Process. 86(10), 2602–2610 (2006). Special Section: Fractional Calculus Applications in Signals and Systems

    Article  MATH  Google Scholar 

  8. 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(2), 719–736 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lin, R., Liu, F.: Fractional high order methods for the nonlinear fractional ordinary differential equation. Nonlinear Anal. 66(4), 856–869 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

  11. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191(1), 12–20 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Lv, C., Xu, C.: Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations. Int. J. Numer. Anal. Model. 12(2), 384–400 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 229(1), 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shu, C.: A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting. J. Comput. Phys. 100(2), 424–429 (1992)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Tsai, R., Cheng, L., Osher, S., Zhao, H.: Fast sweeping algorithms for a class of hamilton-jacobi equations. SIAM J. Numer. Anal. 41(2), 673–694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, C., Liu, J.: Positivity property of second-order flux-splitting schemes for the compressible euler equations. Discret. Contin. Dyn. Syst. Ser. B 3(2), 201–228 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xu, D.: Alternating direction implicit galerkin finite element method for the two-dimensional time fractional evolution equation. Numer. Math. Theory Methods Appl. 7(01), 41–57 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371(6), 461–580 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zayernouri, M., Karniadakis, G.E.: Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252(C), 495–517 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66(5), 693–701 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhao, X., Sun, Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293(C), 184–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Jianfeng Lu for helpful discussions. J. Liu is partially supported by KI-Net NSF RNMS Grant No. 1107444 and NSF Grant DMS 1514826. Z. Zhou is partially supported by RNMS11-07444 (KI-Net). Z. Ma is partially supported by the NSF Grant DMS–1522184, DMS–1107291: RNMS (KI-Net) and Natural Science Foundation of China Grant 91330203.

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Authors and Affiliations

  1. Department of Mathematics and Department of Physics, Duke University, Box 90320, Durham, NC, 27708, USA

    Jian-Guo Liu

  2. Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240, China

    Zheng Ma

  3. Department of Mathematics, Duke University, Box 90320, Durham, NC, 27708, USA

    Zhennan Zhou

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  1. Jian-Guo Liu
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  2. Zheng Ma
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Correspondence to Zhennan Zhou.

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Liu, JG., Ma, Z. & Zhou, Z. Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives. J Sci Comput 72, 291–313 (2017). https://doi.org/10.1007/s10915-017-0356-4

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  • DOI: https://doi.org/10.1007/s10915-017-0356-4

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