Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives


Journal Article

© 2017, Springer Science+Business Media New York. 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 ℓ1 contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, (Δ t) α= O(Δ x) , where α∈ (0 , 1 ] is the fractional exponent in the derivative. When α 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 ℓ1 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.

Full Text

Duke Authors

Cited Authors

  • Liu, JG; Ma, Z; Zhou, Z

Published Date

  • July 1, 2017

Published In

Volume / Issue

  • 72 / 1

Start / End Page

  • 291 - 313

International Standard Serial Number (ISSN)

  • 0885-7474

Digital Object Identifier (DOI)

  • 10.1007/s10915-017-0356-4

Citation Source

  • Scopus