Explicit and implicit TVD schemes for conservation laws with Caputo derivatives

TL;DR

This paper develops explicit and implicit TVD schemes for scalar conservation laws with Caputo derivatives, addressing stability issues caused by the fractional order and demonstrating their effectiveness through numerical tests.

Contribution

It introduces the first explicit and implicit upwind schemes for conservation laws with Caputo derivatives, including stability analysis and practical implementation insights.

Findings

Explicit schemes are conditionally stable with modified CFL conditions.

Implicit schemes are unconditionally stable and TVD.

Numerical tests validate the schemes and illustrate the memory effect.

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.