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 Strong bounded variation estimates for the multi--dimensional finite volume approximation of scalar conservation laws and  application to a tumour growth model

2021 ◽  
Author(s):  
Gopikrishnan Chirappurathu Remesan
Keyword(s):  
Finite Volume ◽  
Bounded Variation ◽  
Tumour Cell ◽  
Conservation Law ◽  
Cartesian Grids ◽  
Finite Volume Schemes ◽  
Existence Of A Solution ◽  
Hyperbolic Conservation Law ◽  
Scalar Conservation Law ◽  
Scalar Conservation

A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law $\partial_t \alpha + \mathrm{div}(\boldsymbol{u}f(\alpha)) = 0$ in two and three spatial dimensions with an initial data of bounded variation is established.  We assume that the divergence of the velocity $\mathrm{div}(\boldsymbol{u})$ is of bounded variation instead of the classical assumption that $\mathrm{div}(\boldsymbol{u})$ is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law $\partial_t \alpha + \mathrm{div}(\boldsymbol{F}(t,\boldsymbol{x},\alpha)) = 0$, where $\mathrm{div}_{\boldsymbol{x}}\boldsymbol{F} \not=0$ on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in $L^p$ spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in $\alpha$, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by S. J. Franks et al.~\cite{Franks2003424}. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity--pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively.

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 L1-Theory of Scalar Conservation Law with Continuous Flux Function

2000 ◽  
Vol 171 (1) ◽  
pp. 15-33 ◽  
Author(s):  
Boris P. Andreianov ◽  
Philippe Bénilan ◽  
Stanislav N. Kruzhkov
Keyword(s):  
Conservation Law ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Scalar Conservation

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.

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