CONVERGENCE OF THIN FILM APPROXIMATION FOR A SCALAR CONSERVATION LAW

In this paper we consider the thin film approximation of a 1D scalar conservation law with strictly convex flux. We prove that the sequence of approximate solutions converges to the unique Kružkov solution.

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Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2016.m-s1 ◽

2017 ◽

Vol 9 (3) ◽

pp. 515-542

Author(s):

K. H. Karlsen ◽

J. D. Towers

Keyword(s):

Boundary Conditions ◽

Conservation Law ◽

Entropy Solution ◽

Approximate Solutions ◽

Convergence Result ◽

Simple Modification ◽

Monotone Schemes ◽

Scalar Conservation Law ◽

Flux Boundary Conditions ◽

Scalar Conservation

AbstractWe consider a scalar conservation law with zero-flux boundary conditions imposed on the boundary of a rectangular multidimensional domain. We study monotone schemes applied to this problem. For the Godunov version of the scheme, we simply set the boundary flux equal to zero. For other monotone schemes, we additionally apply a simple modification to the numerical flux. We show that the approximate solutions produced by these schemes converge to the unique entropy solution, in the sense of [7], of the conservation law. Our convergence result relies on a BV bound on the approximate numerical solution. In addition, we show that a certain functional that is closely related to the total variation is nonincreasing from one time level to the next. We extend our scheme to handle degenerate convection-diffusion equations and for the one-dimensional case we prove convergence to the unique entropy solution.

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Convergence of the Godunov scheme for a scalar conservation law with time and space discontinuities

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891618500078 ◽

2018 ◽

Vol 15 (02) ◽

pp. 175-190 ◽

Cited By ~ 3

Author(s):

John D. Towers

Keyword(s):

Conservation Law ◽

Approximate Solutions ◽

Godunov Scheme ◽

Time And Space ◽

Multiplicative Coefficient ◽

Scalar Conservation Law ◽

Fixed Interface ◽

Scalar Conservation ◽

Space Dependence ◽

Spatial Discontinuity

We consider the Godunov scheme as applied to a scalar conservation law whose flux has discontinuities in both space and time. The time-and space-dependence of the flux occurs through a positive multiplicative coefficient. That coefficient has a spatial discontinuity along a fixed interface at [Formula: see text]. Time discontinuities occur in the coefficient independently on either side of the interface. This setup applies to the Lighthill–Witham–Richards (LWR) traffic model in the case where different time-varying speed limits are imposed on different segments of a road. We prove that the approximate solutions produced by the Godunov scheme converge to the unique entropy solution, as defined by Coclite and Risebro in 2005. Convergence of the Godunov scheme in the presence of spatial flux discontinuities alone is a well-established fact. The novel aspect of this paper is convergence in the presence of additional temporal flux discontinuities.

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Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

Archive for Rational Mechanics and Analysis ◽

10.1007/s00205-021-01682-z ◽

2021 ◽

Author(s):

Konstantinos Dareiotis ◽

Benjamin Gess ◽

Manuel V. Gnann ◽

Günther Grün

Keyword(s):

Thin Film ◽

Energy Estimates ◽

Approximation Procedure ◽

Thin Film Flow ◽

Thin Film Equations ◽

Scalar Conservation Law ◽

Thin Film Equation ◽

Degenerate Parabolic ◽

Scalar Conservation ◽

Martingale Solutions

AbstractWe prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

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