arXiv subjects

Adrian Montgomery Ruf

Publications and source records attributed to Adrian Montgomery Ruf.

8 recordsLinked to original sources

arXiv2021

Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws

We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model" which was recently introduced by Du, Huang, and LeFloch. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper from Du, Huang, and LeFloch concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that, if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.

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

Flux-stability for conservation laws with discontinuous flux and convergence rates of the front tracking method

We prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in u and the spatial dependency is piecewise constant with finitely many discontinuities. We use this stability result to prove a convergence rate for the front tracking method -- a numerical method which is widely used in the field of conservation laws with discontinuous flux. To the best of our knowledge, both of these results are the first of their kind in the literature on conservation laws with discontinuous flux. We also present numerical experiments verifying the convergence rate results and comparing numerical solutions computed with the front tracking method to finite volume approximations.

math.NA↗

arXiv2020

Convergence rates of monotone schemes for conservation laws with discontinuous flux

We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of $\sqrt{Δx}$ in $\mathrm{L}^1$, 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, nonlinear 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.

math.NA↗

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

The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance

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.

math.NA↗

arXiv2019

A convergent finite difference scheme for the Ostrovsky--Hunter equation with Dirichlet boundary conditions

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 prob lem 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.

math.AP↗

arXiv2018

Convergence of a Full Discretization for a Second-Order Nonlinear Elastodynamic Equation in Isotropic and Anisotropic Orlicz Spaces

In this paper, we study a second-order, nonlinear evolution equation with damping arising in elastodynamics. The nonlinear term is monotone and possesses a convex potential but exhibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation. Moreover, we show uniqueness in a class of sufficiently smooth solutions and provide an a priori error estimate for the temporal semidiscretization.

math.AP↗