scholarly journals An explicit finite volume algorithm for vanishing viscosity solutions on a network

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
John D. Towers
Keyword(s):  
Finite Volume ◽  
Viscosity Solutions ◽  
Entropy Solution ◽  
Finite Volume Scheme ◽  
Vanishing Viscosity ◽  
Single Junction ◽  
Vehicle Density ◽  
Scalar Conservation Law ◽  
Volume Algorithm ◽  
Scalar Conservation

<p style='text-indent:20px;'>In [Andreianov, Coclite, Donadello, Discrete Contin. Dyn. Syst. A, 2017], a finite volume scheme was introduced for computing vanishing viscosity solutions on a single-junction network, and convergence to the vanishing viscosity solution was proven. This problem models <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula> incoming and <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> outgoing roads that meet at a single junction. On each road the vehicle density evolves according to a scalar conservation law, and the requirements for joining the solutions at the junction are defined via the so-called vanishing viscosity germ. The algorithm mentioned above processes the junction in an implicit manner. We propose an explicit version of the algorithm. It differs only in the way that the junction is processed. We prove that the approximations converge to the unique entropy solution of the associated Cauchy problem.</p>

Existence and uniqueness result for an hyperbolic scalar conservation law with a stochastic force using a finite volume approximation

2020 ◽  
Vol 17 (02) ◽  
pp. 213-294
Author(s):  
Caroline Bauzet ◽  
Vincent Castel ◽  
Julia Charrier
Keyword(s):  
Finite Volume ◽  
Existence And Uniqueness ◽  
Entropy Solution ◽  
Stability Result ◽  
Uniqueness Result ◽  
Finite Volume Scheme ◽  
Uniqueness Of Solutions ◽  
Stochastic Force ◽  
Scalar Conservation Law ◽  
Scalar Conservation

We are interested in multi-dimensional nonlinear scalar conservation laws forced by a multiplicative stochastic noise with a general time and space dependent flux-function. We address simultaneously theoretical and numerical issues in a general framework and consider a larger class of flux functions in comparison to the one in the literature. We establish existence and uniqueness of a stochastic entropy solution together with the convergence of a finite volume scheme. The novelty of this paper is the use of a numerical approximation (instead of a viscous one) in order to get, both, the existence and the uniqueness of solutions. The quantitative bounds in our uniqueness result constitute a preliminary step toward the establishment of strong error estimates. We also provide an [Formula: see text] stability result for the stochastic entropy solution.

scholarly journals ON A STOCHASTIC FIRST-ORDER HYPERBOLIC EQUATION IN A BOUNDED DOMAIN

2009 ◽  
Vol 12 (04) ◽  
pp. 613-651 ◽  
Author(s):  
GUY VALLET ◽  
PETRA WITTBOLD
Keyword(s):  
Hyperbolic Equation ◽  
Bounded Domain ◽  
Entropy Solution ◽  
Stochastic Perturbation ◽  
Dirichlet Condition ◽  
First Order ◽  
Scalar Conservation Law ◽  
Order Hyperbolic Equation ◽  
Scalar Conservation ◽  
Entropy Formulation

In this paper, we are interested in the stochastic perturbation of a first-order hyperbolic equation of nonlinear type. In order to illustrate our purposes, we have chosen a scalar conservation law in a bounded domain with homogeneous Dirichlet condition on the boundary. Using the concept of measure-valued solutions and Kruzhkov's entropy formulation, a result of existence and uniqueness of the entropy solution is given.

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

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.

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 Optimal jump set in hyperbolic conservation laws

2020 ◽  
Vol 17 (04) ◽  
pp. 765-784
Author(s):  
Shyam Sundar Ghoshal ◽  
Animesh Jana
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Entropy Solution ◽  
Hyperbolic Systems ◽  
Scalar Case ◽  
Qualitative Properties ◽  
Hyperbolic Conservation ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
Jump Set

We investigate qualitative properties of entropy solutions to hyperbolic conservation laws, and construct an entropy solution to a scalar conservation law for which the jump set is not closed, in particular, it is dense in a space-time domain. In a second part, we establish a similar result for hyperbolic systems. We give two different approaches for scalar conservation laws and hyperbolic systems in order to obtain these results. For the scalar case, the solutions are explicitly calculated.

scholarly journals Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network

2017 ◽  
Vol 37 (11) ◽  
pp. 5913-5942 ◽  
Author(s):  
Boris P. Andreianov ◽  
◽  
Giuseppe Maria Coclite ◽  
Carlotta Donadello ◽  
◽  
...  
Keyword(s):  
Conservation Laws ◽  
Viscosity Solutions ◽  
Well Posedness ◽  
Scalar Conservation Laws ◽  
Vanishing Viscosity ◽  
Scalar Conservation

scholarly journals On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

2021 ◽  
Vol 88 (1) ◽  
Author(s):  
Saray Busto ◽  
Michael Dumbser ◽  
Sergey Gavrilyuk ◽  
Kseniya Ivanova
Keyword(s):  
Numerical Methods ◽  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Finite Volume ◽  
Finite Volume Scheme ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Viscosity Limit

AbstractIn this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

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