We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of √ ∆x in L 1 , whenever the flux is strictly monotone in u and the spatial dependency of the flux is piecewise constant with finitely many discontinuities. We also present numerical experiments to illustrate the main result. To the best of our knowledge, this is the first proof of any type of convergence rate for numerical methods for conservation laws with discontinuous flux.Our proof relies on convergence rates for conservation laws with initial and boundary value data. Since those are not readily available in the literature we establish convergence rates in that case en passant in the Appendix.

We prove convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky-Hunter equation on a bounded domain with non-homogeneous Dirichlet boundary conditions. Our scheme is an extension of monotone schemes for conservation laws to the equation at hand. The convergence result at the center of this article also proves existence of entropy solutions for the initial-boundary value problem for the general Ostrovsky-Hunter equation. Additionally, we show uniqueness using Kružkov's doubling of variables technique. We also include numerical examples to confirm the convergence results and determine rates of convergence experimentally.

We consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows the flux to change across finitely many points in space whose positions are uncertain. We propose a single- and multilevel Monte Carlo method based on a finite volume approximation for each sample. Our analysis includes convergence rate estimates of the resulting Monte Carlo and multilevel Monte Carlo finite volume methods as well as error versus work rates showing that the multilevel variant outperforms the single-level method in terms of efficiency. We present numerical experiments motivated by two-phase reservoir simulations for reservoirs with varying geological properties.

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, Lip + -bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of Lip +unbounded initial data is worse than first-order. IntroductionIn their 1994 paper, Nessyahu, Tadmor and Tassa [8] showed that a large class of monotone finite volume methods converge to the entropy solution of the hyperbolic conservation lawat a rate of O(∆x) in the 1-Wasserstein distance W 1 (using the different name Lip ′ ) under the assumption that f is strictly convex (f ′′ α > 0) and the initial datum u 0 is compactly supported and Lip + -bounded, i.e.Recently, Fjordholm and Solem [2] showed a convergence rate of O(∆x 2 ) in W 1 for initial data consisting of finitely many shocks. This raises the question whether the first-order rate in W 1 of [8] can be improved. In this paper we show that this is not possible. We construct a compactly supported and Lip + -bounded initial datum for which the convergence rate in W 1 is no better than first-order. In other words, the rate O(∆x) in [8] is optimal.

We consider a random scalar hyperbolic conservation law in one spatial dimension with bounded random flux functions which are discontinuous in the spatial variable. We show that there exists a unique random entropy solution to the conservation law corresponding to the specific entropy condition used to solve the deterministic case. Assuming the empirical convergence rates of the underlying deterministic problem over a broad range of parameters, we present a convergence analysis of a multilevel Monte Carlo Finite Volume Method (MLMC-FVM). It is based on a pathwise application of the finite volume method for the deterministic conservation laws. We show that the work required to compute the MLMC-FVM solutions is an order lower than the work required to compute the Monte Carlo Finite Volume Method solutions with equal accuracy.