# Haut PDF Monotonization of flux, entropy and numerical schemes for conservation laws ### Monotonization of flux, entropy and numerical schemes for conservation laws

18 Adimurthi & G. D. Veerappa Gowda & J. Jaffr´e family includes the Force and proposes an alternative for the Musta schemes. For all these schemes we proved convergence of the approximate solution to the entropy solution of the continuous problem. We also gave hints on how to extend them to systems and high resolution schemes. In forthcoming papers we will extend these schemes to higher resolution schemes and to the discontinuous flux case. We will also give an example of application to a 2 × 2 system of conservation laws representing a problem of polymer flooding.
En savoir plus ### Optimal regularity for all time for entropy solutions of conservation laws in \$BV^s\$

unknown for some triangular systems of conservation laws e.g., a pressure swing adsorption system. We answer this question for a class of 1-D triangular systems and the multi-D Keyfitz-Kranzer system in section 5 and 6 respectively. To provide an answer to the question ( Q ) we recall some of the previously constructed examples [ 2 , 3 , 4 ]. In Section 3 , Theorem 3.1 provides the direct answer to ( Q ) for a power-law type flux function f (u) = (1 + p) −1 |u| 1+p . We have discussed before that convex flux function with p- degeneracy (i.e., satisfying ( 5 )) gives a regularizing effect in BV s with s = 1/p [ 13 ]. We construct an entropy solution u to ( 1 ) such that T V s+ε (u(·, t)) = ∞ for all t > 0 and ε > 0 with s = 1/p. Following the constructions in [ 3 , 4 ] we build the entropy solution u, consisting infinitely many shock profiles in a compact interval. These shock profiles are named Asymptotically Single Shock Packet (ASSP) in [ 2 ]. Loosely speaking an ASSP is a solution with a special structure between two parallel lines in the half plane R x × R + t such that in large time only one shock curve appears
En savoir plus ### Stability estimates for scalar conservation laws with moving flux constraints

y 0 = 0. (14) Denote by R the standard (i.e., without the constraint ( 1c )) Riemann solver for ( 1a )-( 1b )-( 14 ), i.e., the (right continuous) map (t, x) 7→ R(ρ L , ρ R )(x/t) given by the standard weak entropy solution, see for instance [ 7 , Chapter 5]. Moreover, assume that ˙ y is constant and let ˇ ρ = ˇ ρ( ˙ y) and ˆ ρ = ˆ ρ( ˙ y), with ˇ ρ ≤ ˆ ρ, be the intersection points of the flux function f (ρ) with the line f α (ρ α ) + ˙ y(ρ − ρ α ) (see ### Applications of the DFLU flux to systems of conservation laws

Riemann problems associated to this system when gravity is neglected and therefore the fractional flow function is an increasing function of the unknown. In this case, the eigenvalues of the corresponding Jaco- bian matrix are positive and hence it is less difficult to construct Godunov type schemes which turn out to be upwind schemes. When the above model with gravity effects is considered, then the flux function is not necessarily monotone and hence the eigenvalues can change sign. This makes the construction of Godunov type schemes more difficult as it involves exact solutions of Riemann problems with a nonmonotonous fractional flow function. Therefore in section 3 we solve the Riemann problems in the general case when gravity terms are taken into account so the flux function is not anymore monotone. This will allow to compare our method with that using an exact Riemann solver. In section 4 we consider Godunov type finite volume schemes. We present the DFLU scheme for the system of polymer flooding and compare it to the Godunov scheme whose flux is given by the exact solution of the Riemann problem. We also present several other possible numerical fluxes, centered like Lax-Friedrichs or Force, or upstream like the upstream mobility flux commonly used in reservoir engineering [4, 5, 16]. Finally in section 5 we compare numerically the DFLU method with these fluxes.
En savoir plus ### A BGK approximation to scalar conservation laws with discontinuous flux

 1I−ε≤x≤ε + kR1Iε<x, ε > 0. The kinetic formulation of scalar conservation laws is well adapted to the analy- sis of the (Perthame-Tadmor) BGK approximation of scalar conservation laws. Developed in [PT91], this equation is a continuous version of the Transport- Collapse method of Brenier [Bre81, Bre83]. BGK models have also been used for gas dynamics and the construction of numerical schemes. See for example the book of Perthame [Per02] for a survey of this field.
En savoir plus ### Non-local conservation laws for traffic flow modeling

while in [ 55 , 57 ], the uniqueness of weak solutions is obtained directly from the fixed point theorem, without prescribing any kind of entropy condition. The numerical approximation of the solutions of non-local models is challenging due to the high non-linearity of the system and the dependence of the flux function on convolution terms, which highly impacts the computational cost. There are different ways to numerically integrate non-local conservation laws. In [ 2 , 5 , 10 , 11 , 49 ] a first order Lax-Friedrichs-type numerical scheme is used to approximate the problem and to prove the existence of solutions. Indeed, in this setting, the numerical scheme is not just a tool for numerically investigating the behaviour of solutions, but it is important from the analytical point of view, because it allows to construct a sequence of approximate solutions in order to apply an adapted version of Helly’s theorem, see [ 44 ]. Another first order numerical scheme is proposed in [ 46 ], where a Godunov-type numerical scheme for a specific class of non-local flux problems is presented. This scheme allows for physically reasonable solutions, meaning that negative velocities as well as negative fluxes are avoided. Moreover, in comparison with Lax-Friedrichs scheme, Godunov scheme is less diffusive. Concerning high-order numerical schemes, it is worth citing the papers [ 19 , 45 ]. In [ 19 ], the authors propose discontinuous Galerkin and finite volume WENO schemes to obtain high-order approximations of non-local scalar conservation laws in one space dimension, where the velocity function depends on a weighted mean of the conserved quantity. The discontinuos Galerkin schemes give the best results, but under a very restrictive Courant-Friedrichs-Lewy (CFL) condition. On the contrary, finite volume WENO schemes can be implemented on larger time steps. In [ 45 ], central WENO schemes are proposed, which, in contrast to the other high-order schemes for non-local conservation laws, neither require a restrictive CFL condition nor an additional reconstruction step.
En savoir plus ### A Remark on the Controllability of a System of Conservation Laws in the Context of Entropy Solutions

u t + (f (u)) x = 0 for (t, x) ∈ R + × R, (1.3) where the flux function f is regular from Ω to R n . Typically, t is the time and x is the position. The general problem of controllability is the following. Consider the problem posed in the interval [0, 1] rather than in R. In such a case one needs of course to prescribe boundary conditions on [0, T ] × {0, 1}: here boundary conditions will be considered as a control, that is, a way to influence the system to make it behave in a prescribed way. Let us call u(t, ·) the state of the system at time t. The question is: given two possible states of the system, say u 0 and u 1 , can we choose the
En savoir plus ### Numerical Methods for the solution of Hyperbolic Conservation Laws

and its numerical flux is given by g(u, v) = f (ˆ v R (0; u, v)). (15) 3.2 Van Leer’s method Since the first and thirst steps of Godunov’s methods are of a numerical nature, they can be modified without influencing the physical input, for instance by replacing the piecewise constant approximation by a piecewise linear variation inside each cell, leading to the definition of second order space-accurate schemes, as van Leer’s method. However, the straightforward replacement of the first-order scheme by appropriate second-order accurate formulae leads to the generation of oscillations around discontinuities. To overcome this limitation and achieve the goal of oscillation-free, second-order schemes able to represent accurately shock and discontinuities, there are introduced non linear components. Non linear discretizations imply that the schemes will be non linear even when applied to linear equations. This concept was introduced initially by van Leer under the form of limiters, which control the gradient of the computed solution such as to prevent the appearance of overshoots or undershoots.
En savoir plus ### Stability estimates for non-local scalar conservation laws

Stability estimates for non-local scalar conservation laws Felisia Angela Chiarello 1 Paola Goatin 1 Elena Rossi 1 Abstract We prove the stability of entropy weak solutions of a class of scalar conservation laws with non-local flux arising in traffic modelling. We obtain an estimate of the dependence of the solution with respect to the kernel function, the speed and the initial datum. Stability is obtained from the entropy condition through doubling of variable technique. We finally provide some numerical simulations illustrating the dependencies above for some cost functionals derived from traffic flow applications.
En savoir plus ### A Local Entropy Minimum Principle for Deriving Entropy Preserving Schemes

1. Introduction. The numerical approximation of the weak solutions of hyper- bolic systems of conservation laws was widely studied during the last three decades, with a special attention to the so-called Euler equations. Several strategies, coming from finite volume methods, have been introduced. Our purpose is not to detail these techniques, but let us refer to the most famous of them: the Godunov scheme [18, 25], the HLL scheme , the HLLC scheme , the Roe scheme , the Osher scheme , the relaxation schemes [7, 22, 12, 3, 1], the VFRoe scheme [9, 5, 17, 26], BGK scheme [18, 23, 8]... Of course, this list stays exhaustive and the reader is referred, for instance, to Godlewski-Raviart  or Toro  or LeVeque  and references therein. These kind of schemes have also been applied to general fluid equations, for instance to 10-moment equations system [3, 28] or radiative transfer equations .
En savoir plus ### GLOBAL ENTROPY STABILITY FOR A CLASS OF UNLIMITED HIGH-ORDER SCHEMES FOR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS

A weak solution of (1.1) is called an entropy satisfying solution if and only if the entropy inequality (1.4) holds for any pair entropy-entropy flux (η, G). Integrating in space the entropy inequality (1.4) results in a global entropy stability inequality, d dt ### High order numerical schemes for one-dimension non-local conservation laws

1 x ≥ 1, with zero-flux boundary conditions in the interval [0, 1], and compute the solution at time T = 1, with parameters α = 1, n = 3 and a = 0.025. We set ∆x = 1/400 and compute the solutions with different RKDG and FV-WENO schemes, including the first-order Lax-Friedrichs scheme used in . The results displayed in Fig. 3 are compared to a reference solution computed with FV-WENO5 and ∆x = 1/3200. Compared to the reference solution, we observe that RKDG1 is more accurate than FV-WENO3, and FV-WENO5 more accurate than RKDG3 and RKDG2 (Fig. 3). In particular, we observe that the numerical solutions obtained with the high-order schemes provide better approximations of the oscillatory shape of the solution than the first order scheme. These oscillations can possibly be explained as layering phenomenon in sedimentation , which denotes a traveling staircases pattern, looking as several distinct
En savoir plus ### Finite volume schemes for constrained conservation laws

In the present paper, we characterize and approximate entropy solutions of ( 1 - 3 ) in the L ∞ setting. The core of the paper is the convergence analysis for finite volume schemes adapted to the constrained problem ( 1 - 3 ). The schemes are constructed as follows. We consider a classical monotone three-point finite volume scheme (see [ EGH00 ]) and denote by g(u, v) the associated numerical flux; at the interface of the mesh which corresponds to the obstacle position {x = 0}, the numerical flux is replaced by min(g(u, v), F ) in order to comply with the constraint ( 3 ) (see Section 4 for more details). Our approach is simpler than the wave-front tracking algorithm devised in [ CG07 ], because we do not need to define explicitly the Riemann solvers at the interface {x = 0} which would fit the constraint at time t. Notice that with our approach, existing finite volume codes for the non-constrained conservation law ( 1 ) are trivially combined with the constraint ( 3 ).
En savoir plus ### Optimal Regularizing Effect for Scalar Conservation Laws

All these works use at some point tools from harmonic analysis: Fourier trans- form, Hardy-Littlewood decomposition, Radon transform. Moreover, in all these results, the regularity of moments in v of the function f depends on γ. In this section, we give an example of velocity averaging result where the reg- ularity of the moments in v of f is independent of γ, at the expense of an extra assumption on the v dependence in f . Also, the proof of this result is based on the interaction identities (7)-(8) and uses only elementary techniques in physical space. Theorem 3.1. Let a ∈ C 1 (R); assume that there exists β ≥ 1 such that, for each M > 0, there exists αM > 0 for which
En savoir plus ### Switching rules for stabilization of linear systems of conservation laws

In this paper, the exponential convergence in L 2 -norm is analyzed for a class of switched linear systems of conservation laws. The boundary conditions are subject to switches. We investigate the problem of synthesizing stabilizing switching controllers. By means of Lyapunov techniques, three control strategies are developed based on steepest descent selection, possibly combined with a hysteresis and a low-pass filter. For the first strategy we show the global exponential stabilizability, but no result for the existence and uniqueness of trajectories can be stated. For the other ones, the problem is shown to be well posed and global exponential convergence can be obtained. Moreover, we consider the robustness issues for these switching rules in presence of measurement noise. Some numerical examples illustrate our approach and show the merits of the proposed strategies. Particularly, we have developped a model for a network of open channels, with switching controllers in the gate operations.
En savoir plus ### Some aspects of high resolution numerical methods for hyperbolic systems of conservation laws,with applications to gas dynamics

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r] ### Microscopic local conservation laws for classical fluids

Due to the non-locality of the potential, J P (r, t) involves contributions from the two- body function f 2 taken at points located on each side of r, at distances of the order of the range of W . The above results for the local balance of momentum can alternatively be derived through Noether’s approach. The conservation of the total momentum results from the invariance of the Lagrangian under the translation r j 7→ r j + δr j by a small ### A Bhatnagar-Gross-Krook approximation to scalar conservation laws with discontinuous flux

tinuous with respect to x, several criteria are possible . For B(x, u) = k(x)A(u) as above, the choice of entropy solution is unambiguous [1, remark 4.4] and we consider here the selection criterion ﬁrst given in . A kinetic formulation (in the spirit of ) equivalent to the entropy formulation in  has been given in . In particular, solutions given by this criterion are limits (almost everywhere (a.e.) and in L 1 ) of the solutions obtained by monotone regularization of the coeﬃcient k in ### L1 and L8 intermediate asymptotics for scalar conservation laws

During the completion of this paper, we became aware of a study by Y.-J. Kim, who kindly communicated us a preliminary version of his work . His approach is based on a detailed study of special self-similar solutions. Y.-J. Kim obtains a sharp L 1 -norm convergence rate and we do not. Nevertheless, we believe that our method, based on qualitative results and global integral estimates, and our results on the convergence in weighted norms are of interest. ### Conservation laws and generalized isometric embeddings

Chapter 1 Cartan’s structure equations The goal of this chapter is to introduce the fundamental notions of differential geometry expressed in moving frames, and to establish Cartan’s structure equations. Almost everything revolves around differential forms, and thus is expressed in the Cartan formalism. The first section is dedicated to introducing and defining the notion of a connection and its curvature on an arbitrary vector bundle, where the dimension of the fiber is not necessarily equal to the dimension of the base manifold, and the relationship between the connection and its curvature is given by Cartan’s second-structure equation. Also shown is an interesting property of the connection when the vector bundle is endowed with a metric, and, as always in differential geometry, all of the transformation rules for these objects will be demonstrated. In the second section, a special and fundamental class of vector bundles is investigated: the tangent bundle of a differentiable manifold. The notion of torsion of a connection appears and leads to Cartan’s first-structure equation. An important technical result, the Cartan lemma, is stated because it is useful in several applications. The proofs from the first and second sections are given in appendix 1 not only because the calculations are not that difficult and almost all of the results derive from the definitions, but also to lighten the reading. Finally, since Cartan’s structure equations are ”intensely” used from chapters 3 to 6, the modest purpose of section 3 is to pro- vide some useful applications to Cartan’s structure equations in the study of surfaces, such as computing Christoffel symbols, computing the Gauss curvature and presenting a problem studied by Poincaré pertaining to the conformal metrics of constant curvature.
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