## Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives

Publication ,  Journal Article
Liu, JG; Ma, Z; Zhou, Z
Published in: Journal of Scientific Computing
July 1, 2017

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.

## Published In

Journal of Scientific Computing

0885-7474

July 1, 2017

72

1

## Start / End Page

291 / 313

• Applied Mathematics
• 4903 Numerical and computational mathematics
• 4901 Applied mathematics
• 0802 Computation Theory and Mathematics
• 0103 Numerical and Computational Mathematics
• 0102 Applied Mathematics

### Citation

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Liu, J. G., Ma, Z., & Zhou, Z. (2017). Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives. Journal of Scientific Computing, 72(1), 291–313. https://doi.org/10.1007/s10915-017-0356-4
Liu, J. G., Z. Ma, and Z. Zhou. “Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives.” Journal of Scientific Computing 72, no. 1 (July 1, 2017): 291–313. https://doi.org/10.1007/s10915-017-0356-4.
Liu JG, Ma Z, Zhou Z. Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives. Journal of Scientific Computing. 2017 Jul 1;72(1):291–313.
Liu, J. G., et al. “Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives.” Journal of Scientific Computing, vol. 72, no. 1, July 2017, pp. 291–313. Scopus, doi:10.1007/s10915-017-0356-4.
Liu JG, Ma Z, Zhou Z. Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives. Journal of Scientific Computing. 2017 Jul 1;72(1):291–313.

## Published In

Journal of Scientific Computing

0885-7474

July 1, 2017

72

1

291 / 313