arXiv subjects

Erlend Briseid Storrøsten

Publications and source records attributed to Erlend Briseid Storrøsten.

4 recordsLinked to original sources

arXiv2018

Numerical methods for conservation laws with rough flux

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.

math.NA↗

arXiv2017

Path-dependent convex conservation laws

For scalar conservation laws driven by a rough path $z(t)$, in the sense of Lions, Perthame and Souganidis in arXiv:1309.1931, we show that it is possible to replace $z(t)$ by a piecewise linear path, and still obtain the same solution at a given time, under the assumption of a convex flux function in one spatial dimension. This result is connected to the spatial regularity of solutions. We show that solutions are spatially Lipschitz continuous for a given set of times, depending on the path and the initial data. Fine properties of the map $z \mapsto u(τ)$, for a fixed time $τ$, are studied. We provide a detailed description of the properties of the rough path $z(t)$ that influences the solution. This description is extracted by a "factorization" of the solution operator (at time $τ$). In a companion paper, we make use of the observations herein to construct computationally efficient numerical methods.

math.AP↗

arXiv2016

On stochastic conservation laws and Malliavin calculus

For stochastic conservation laws driven by a semilinear noise term, we propose a generalization of the Kružkov entropy condition by allowing the Kružkov constants to be Malliavin differentiable random variables. Existence and uniqueness results are provided. Our approach sheds some new light on the stochastic entropy conditions put forth by Feng and Nualart [J. Funct. Anal., 2008] and Bauzet, Vallet, and Wittbold [J. Hyperbolic Differ. Equ., 2012].

math.AP↗

arXiv2015

On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions

We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local $L^1$-error between the exact and numerical solutions is $\mathcal{O}(Δx^{2/(19+d)})$, where $d$ is the spatial dimension and $Δx$ is the grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof makes effective use of specific kinetic formulations of the difference method and the convection-diffusion equation.

math.AP↗