Searcharxiv⌕ Search

arXiv subjects

Nils Henrik Risebro

Publications and source records attributed to Nils Henrik Risebro.

At least 19 recordsLinked to original sources
arXiv2025

Upwind filtering of scalar conservation laws

We study a class of multi-dimensional non-local conservation laws of the form $\partial_t u = \operatorname{div}^Φ \mathbf{F}(u)$, where the standard local divergence $\operatorname{div}$ of the flux vector $\mathbf{F}(u)$ is replaced by an average upwind divergence operator $\operatorname{div}^Φ$ acting on the flux along a continuum of directions given by a reference measure and a filter $Φ$. The non-local operator $\operatorname{div}^Φ$ applies to a general non-monotone flux $\mathbf{F}$, and is constructed by decomposing the flux into monotone components according to wave speeds determined by $\mathbf{F}'$. Each monotone component is then consistently subjected to a non-local derivative operator that utilizes an anisotropic kernel supported on the "correct" half of the real axis. We establish well-posedness, derive a priori and entropy estimates, and provide an explicit continuous dependence result on the kernel. This stability result is robust with respect to the "size" of the kernel, allowing us to specify $Φ$ as a Dirac delta $δ_0$ to recover entropy solutions of the local conservation law $\partial_t u = \operatorname{div} \mathbf{F}(u)$ (with an error estimate). Other choices of $Φ$ (and the reference measure) recover known numerical methods for (local) conservation laws. This work distinguishes itself from many others in the field by developing a consistent non-local approach capable of handling non-monotone fluxes.

math.AP↗
arXiv2024

The continuum limit of non-local Follow-the-Leader models

We study a generalized Follow-the-Leader model where the driver considers the position of an arbitrary but finite number of vehicles ahead, as well as the position of the vehicle directly behind the driver. It is proved that this model converges to the classical Lighthill-Whitham-Richards model for traffic flow when traffic becomes dense. This also underscores the robustness of the Lighthill-Whitham-Richards model.

math.AP↗
arXiv2024

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the $L^1$ sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the ``zero-filter'' (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the ``filter size'' and (ii) the dissipation of an arbitrary convex entropy.

math.AP↗
arXiv2021

Well-posedness theory for nonlinear scalar conservation laws on networks

We consider nonlinear scalar conservation laws posed on a network. We establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case -- for monotone fluxes with an upwind difference scheme -- we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.

math.NA↗
arXiv2020

Multilevel Monte Carlo Finite Volume Methods for Random Conservation Laws with Discontinuous Flux

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.

math.NA↗
arXiv2020

Singular Diffusion with Neumann boundary conditions

In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion $\partial_t u = \text{div}(k(x)\nabla G(u))$, $u|_{t=0}=u_0$ with Neumann boundary conditions $k(x)\nabla G(u)\cdot ν= 0$. Here $x\in B\subset \mathbb{R}^d$, a bounded open set with locally Lipchitz boundary, and with $ν$ as the unit outer normal. The function $G$ is Lipschitz continuous and nondecreasing, while $k(x)$ is diagonal matrix. We show that any two weak entropy solutions $u$ and $v$ satisfy $\Vert{u(t)-v(t)}\Vert_{L^1(B)}\le \Vert{u|_{t=0}-v|_{t=0}}\Vert_{L^1(B)}e^{Ct}$, for almost every $t\ge 0$, and a constant $C=C(k,G,B)$. If we restrict to the case when the entries $k_i$ of $k$ depend only on the corresponding component, $k_i=k_i(x_i)$, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.

math.AP↗
arXiv2019

Numerical investigations into a model of partially incompressible two-phase flow in pipes

We consider a model for flow of liquid and gas in a pipe. We assume that the gas is ideal and that the liquid is incompressible. Under this assumption the resulting equations, expressing conservation of mass and momentum, splits into two subsystems such that the gas flow is independent of the liquid flow, and the liquid flow is described by a conservation law parametrized by the mass fraction of gas. When solving these equations numerically, we propose to stagger the gas and liquid variables with respect to each other. The advantage of this is that in finite volume methods one can use numerical flux functions designed for 2x2 systems of hyperbolic conservation laws to solve both the gas flow and the liquid flow, rather than a much more complicated numerical flux for the whole 4x4 system. We test this using the Roe numerical flux for both subsystems, and compare the results with results produced by using the second-order Nessyahu--Tadmor scheme for the second subsystem.

math.NA↗
arXiv2019

Well-posedness of the initial value problem for the Ostrovsky-Hunter equation with spatially dependent flux

In this paper we study the Ostrovsky-Hunter equation for the case where the flux function $f(x, u)$ may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (specified later), then there exists a unique entropy solution. To show the existence, after proving some a priori estimates we have used the method of compensated compactness and to prove the uniqueness we have employed the method of doubling of variables.

math.AP↗
arXiv2018

The Ostrovsky Hunter equation with a space dependent flux function

We study the periodic Ostrovsky-Hunter equation in the case where the flux function may depend on the spatial variable. Our main results are that if the flux function is twice differentiable, then there exists a unique entropy solution. This entropy solution may be constructed as a limit of approximate solutions generated by a finite volume scheme, and the finite volume approximations converge to the entropy solution at a rate 1/2.

math.NA↗
arXiv2018

Models for dense multilane vehicular traffic

We study vehicular traffic on a road with multiple lanes and dense, unidirectional traffic following the traditional Lighthill-Whitham-Richards model where the velocity in each lane depends only on the density in the same lane. The model assumes that the tendency of drivers to change to a neighboring lane is proportional to the difference in velocity between the lanes. The model allows for an arbitrary number of lanes, each with its distinct velocity function. The resulting model is a well-posed weakly coupled system of hyperbolic conservation laws with a Lipschitz continuous source. We show several relevant bounds for solutions of this model that are not valid for general weakly coupled systems. Furthermore, by taking an appropriately scaled limit as the number of lanes increases, we derive a model describing a continuum of lanes, and show that the $N$-lane model converges to a weak solution of the continuum model.

math.AP↗
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↗
arXiv2017

Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow

We show how to view the standard Follow-the-Leader (FtL) model as a numerical method to compute numerically the solution of the Lighthill--Whitham--Richards (LWR) model for traffic flow. As a result we offer a simple proof that FtL models converge to the LWR model for traffic flow when traffic becomes dense. The proof is based on techniques used in the analysis of numerical schemes for conservation laws, and the equivalence of weak entropy solutions of conservation laws in the Lagrangian and Eulerian formulation.

math.AP↗
arXiv2016

Convergence of finite difference schemes for the Benjamin-Ono equation

In this paper, we analyze finite difference schemes for Benjamin-Ono equation, u_t = uu_x + Hu_{xx}, where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data are sufficiently regular, fully discrete finite difference schemes shown to converge to a classical solution. Finally, the convergence is illustrated by several examples.

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↗
arXiv2014

Convergence of a higher-order scheme for Korteweg-de Vries equation

We study the convergence of higher order schemes for the Cauchy problem associated to the KdV equation. More precisely, we design a Galerkin type implicit scheme which has higher order accuracy in space and first order accuracy in time. The convergence is established for initial data in L^2, and we show that the scheme converges strongly in L^2(0,T; L^2_loc(\R)) to a weak solution. Finally, the convergence is illustrated by several examples.

math.AP↗
arXiv2014

On an inverse problem for scalar conservation laws

We study in what sense one can determine the function $k=k(x)$ in the scalar hyperbolic conservation law $u_t+(k(x)f(u))_x=0$ by observing the solution $u(t,\dott)$ of the Cauchy problem with initial data $u|_{t=0}=u_o$.

math.AP↗