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arXiv · 1802.00708

Numerical methods for conservation laws with rough flux

Håkon Hoel·Kenneth Hvistendahl Karlsen·Nils Henrik Risebro·Erlend Briseid Storrøsten
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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.

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Håkon Hoel, Kenneth Hvistendahl Karlsen, Nils Henrik Risebro, Erlend Briseid Storrøsten. 2018-02-02. Numerical methods for conservation laws with rough flux. https://arxiv.org/abs/1802.00708

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