Journal of Scientific Computing

, Volume 72, Issue 1, pp 291–313 | Cite as

Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives

  • Jian-Guo Liu
  • Zheng Ma
  • Zhennan ZhouEmail author


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.


TVD schemes Conservation law Caputo derivative Memory effect 



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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Department of PhysicsDuke UniversityDurhamUSA
  2. 2.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  3. 3.Department of MathematicsDuke UniversityDurhamUSA

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