Numerical methods for conservation laws with rough flux

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{ö}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.

58 pages, 12 figures