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 α-H{ö}lder continuous rough paths the convergence rate of the numerical methods can improve from O(COST−γ), for some γ∈[α/(12−8α),α/(10−6α)], with α∈(0,1), to O(COST−min(1/4,α/2)). Numerical examples support the theoretical results.