Explicit and Implicit TVD Schemes for Conservation

Jian-Guo Liu Duke University Zheng Ma Shanghai Jiao Tong University Zhennan Zhou Duke University

Numerical Analysis and Scientific Computing mathscidoc:1702.25020

Journal of Scientific Computing, 1-23, 2017.1
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 $\alpha$ 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
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  title={Explicit and Implicit TVD Schemes for Conservation},
  author={Jian-Guo Liu, Zheng Ma, and Zhennan Zhou},
  booktitle={Journal of Scientific Computing},
Jian-Guo Liu, Zheng Ma, and Zhennan Zhou. Explicit and Implicit TVD Schemes for Conservation. 2017. In Journal of Scientific Computing. pp.1-23. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170206151108654539191.
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