scholarly journals Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition

2017 ◽  
Vol 14 (04) ◽  
pp. 671-701 ◽  
Author(s):  
K. H. Karlsen ◽  
J. D. Towers
Keyword(s):  
Conservation Laws ◽  
Viscosity Solution ◽  
Discontinuous Coefficients ◽  
Entropy Inequality ◽  
Discrete Version ◽  
Vanishing Viscosity ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Discontinuous Flux ◽  
Scalar Conservation

We study a scalar conservation law whose flux has a single spatial discontinuity. There are many notions of (entropy) solution, the relevant concept being determined by the application. We focus on the so-called vanishing viscosity solution. We utilize a Kružkov-type entropy inequality which generalizes the one in [K. H. Karlsen, N. H. Risebro and J. D. Towers, [Formula: see text]-stability for entropy solutions of nonlinear degenerate parabolic convection–diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk. 3 (2003) 1–49], singles out the vanishing viscosity solution whether or not the crossing condition is satisfied, and has a discrete version satisfied by the Godunov variant of the finite difference scheme of [S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal. 26(6) (1995) 1425–1451]. We show that the solutions produced by that scheme converge to the unique vanishing viscosity solution. The scheme does not require a Riemann solver for the discontinuous flux problem. This makes its implementation simple even when the flux is multimodal, and there are multiple flux crossings.

scholarly journals Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function

2009 ◽  
Vol 2009 ◽  
pp. 1-33 ◽  
Author(s):  
H. Holden ◽  
K. H. Karlsen ◽  
D. Mitrovic
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Flux Function ◽  
Discontinuous Flux Function ◽  
Scalar Conservation Law ◽  
Discontinuous Flux ◽  
Dispersion Terms ◽  
Scalar Conservation

We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach ofH-measures to investigate the zero diffusion-dispersion-smoothing limit.

scholarly journals EXISTENCE RESULT FOR THE COUPLING PROBLEM OF TWO SCALAR CONSERVATION LAWS WITH RIEMANN INITIAL DATA

2010 ◽  
Vol 20 (10) ◽  
pp. 1859-1898 ◽  
Author(s):  
BENJAMIN BOUTIN ◽  
CHRISTOPHE CHALONS ◽  
PIERRE-ARNAUD RAVIART
Keyword(s):  
Conservation Laws ◽  
Weak Sense ◽  
Scalar Conservation Laws ◽  
Existence Of A Solution ◽  
Coupling Condition ◽  
Coupling Problem ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Fixed Interface ◽  
Scalar Conservation

This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at x = 0. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely x < 0 or x > 0. At interface x = 0 we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed variable, which may differ from the conservative unknown (and the flux functions as well). We prove the existence of a solution to the coupled Riemann problem using a constructive approach. The latter allows in particular to highlight interesting features like non-uniqueness of both continuous and discontinuous (at interface x = 0) solutions. The behavior of some numerical scheme is also investigated.

scholarly journals ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS

2003 ◽  
Vol 13 (02) ◽  
pp. 221-257 ◽  
Author(s):  
NICOLAS SEGUIN ◽  
JULIEN VOVELLE
Keyword(s):  
Finite Volume ◽  
Conservation Law ◽  
Discontinuous Coefficients ◽  
Finite Volume Methods ◽  
Finite Volume Schemes ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Line Of Discontinuity ◽  
Scalar Conservation ◽  
Industrial Context

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

scholarly journals Numerical Study for Two-Phase Flow with Gravity in Homogeneous and Piecewise-Homogeneous Porous Media

2020 ◽  
Vol 21 (1) ◽  
pp. 21
Author(s):  
Isamara L. N. Araujo ◽  
Panters Rodríguez-Bermúdez ◽  
Yoisell Rodríguez-Núñez
Keyword(s):  
Two Phase Flow ◽  
Numerical Study ◽  
Point Of View ◽  
Phase Flow ◽  
Two Phase ◽  
Flux Function ◽  
Discontinuous Flux Function ◽  
Scalar Conservation Law ◽  
Discontinuous Flux ◽  
Scalar Conservation

In this work we study two-phase flow with gravity either in 1-rock homogeneous media or 2-rocks composed media, this phenomenon can be modeled by a non-linear scalar conservation law with continuous flux function or discontinuous flux function, respectively. Our study is essentially from a numerical point of view, we apply the new Lagrangian-Eulerian finite difference method developed by Abreu and Pérez  and the Lax-Friedrichs classic method to obtain numerical entropic solutions. Comparisons between numerical and analytical solutions show the efficiency of the methods even for discontinuous flux function.

scholarly journals High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1079
Author(s):  
Tingting Xiang ◽  
Guodong Wang ◽  
Suping Zhang
Keyword(s):  
Conservation Laws ◽  
High Order ◽  
Splitting Scheme ◽  
Scalar Conservation Laws ◽  
Flux Function ◽  
Flux Splitting ◽  
Discontinuous Flux Function ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Accurate Scheme

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme.

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