Numerical methods for conservation laws with rough flux
Provides numerical methods with provably better convergence for a class of PDEs with rough coefficients, relevant to stochastic and multiscale modeling.
Proposed finite volume methods for conservation laws with rough flux achieve improved convergence rates by exploiting cancellations from rough path oscillations, with rates up to O(COST^{-min(1/4, α/2)}).
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 $α$-H{ö}lder continuous rough paths the convergence rate of the numerical methods can improve from $\mathcal{O}(\text{COST}^{-γ})$, for some $γ\in \left[α/(12-8α), α/(10-6α)\right]$, with $α\in (0, 1)$, to $\mathcal{O}(\text{COST}^{-\min(1/4,α/2)})$. Numerical examples support the theoretical results.