Numerical methods for conservation laws with rough flux

TL;DR

This paper develops finite volume numerical methods for conservation laws with rough flux functions, showing improved convergence rates by exploiting flux oscillation cancellations, supported by numerical evidence.

Contribution

It introduces novel finite volume schemes for rough flux conservation laws and demonstrates enhanced convergence rates by leveraging flux oscillation cancellations.

Findings

Convergence rate improves from OST^{- ext{some } ext{rate}} to OST^{- ext{min}(1/4, ext{H"older exponent}/2)}.

Numerical examples confirm the theoretical convergence improvements.

Rough path oscillations can lead to cancellations, improving numerical accuracy.

Abstract

Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with a rough path dependent flux function. For a convex flux, it is demonstrated that rough path oscillations may lead to "cancellations" in the solution. Making use of this property, we show that for $\alpha$-H{\"o}lder continuous rough paths the convergence rate of the numerical methods can improve from $\mathcal{O}(\text{COST}^{-\gamma})$, for some $\gamma \in \left[\alpha/(12-8\alpha), \alpha/(10-6\alpha)\right]$, with $\alpha\in (0, 1)$, to $\mathcal{O}(\text{COST}^{-\min(1/4,\alpha/2)})$. Numerical examples support the theoretical results.