## Discrete Kinetic Schemes For Multidimensional Systems Of Conservation Laws (2000)

Venue: | SIAM J. Numer. Anal |

Citations: | 33 - 11 self |

### BibTeX

@ARTICLE{Aregba-driollet00discretekinetic,

author = {Denise Aregba-driollet and Roberto Natalini},

title = {Discrete Kinetic Schemes For Multidimensional Systems Of Conservation Laws},

journal = {SIAM J. Numer. Anal},

year = {2000},

volume = {37},

pages = {1973--2004}

}

### OpenURL

### Abstract

We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whitham's stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

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Citation Context ...ur purpose here is to construct numerical schemes for system (1.3) in order to obtain a numerical approximation of (1.1) in the relaxed limit # = 0. As well known for general relaxation problems, see =-=[71, 44, 55]-=-, approximation (1.3) needs a suitable stability condition to produce the correct limits. In the framework of general 2 2 quasi-linear hyperbolic relaxation problems, this condition is known as the su... |

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Citation Context ..., if f # converges in some strong topology to a limit f and if Pf # 0 converges to u 0 , then Pf is a solution of problem (1.1), (1.2). In fact system (1.3) is just a BGK approximation for (1.1); see =-=[5, 12]-=- and references therein. The interaction term on the right-hand side is given by the di#erence between a nonlinear function, which describes the equilibria of the system, in our case M(Pf ), and the u... |

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Citation Context ...have been established for special systems or partially kinetic approximations [42, 29, 9, 40]. Related numerical schemes can be found in [19, 59]; for a general overview and many other references see =-=[24]-=-. Discrete velocities models and their fluid dynamical limits have also been considered by many people; see the review paper [61]. In particular we mention the studies on the Broadwell model [11, 72].... |

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Citation Context ...ce results. Our approximation framework generalizes to systems the construction presented in [54] for the scalar case, and shares most of the advantages of the relaxation approximation as proposed in =-=[30]-=- (see also [53, 2, 70]): simple formulation even for general multidimensional systems of conservation laws and easy numerical implementation, hyperbolicity, regular approximating solutions. Actually t... |

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Citation Context ..., if f # converges in some strong topology to a limit f and if Pf # 0 converges to u 0 , then Pf is a solution of problem (1.1), (1.2). In fact system (1.3) is just a BGK approximation for (1.1); see =-=[5, 12]-=- and references therein. The interaction term on the right-hand side is given by the di#erence between a nonlinear function, which describes the equilibria of the system, in our case M(Pf ), and the u... |

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Citation Context ...ur purpose here is to construct numerical schemes for system (1.3) in order to obtain a numerical approximation of (1.1) in the relaxed limit # = 0. As well known for general relaxation problems, see =-=[71, 44, 55]-=-, approximation (1.3) needs a suitable stability condition to produce the correct limits. In the framework of general 2 2 quasi-linear hyperbolic relaxation problems, this condition is known as the su... |

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Citation Context ... purposes; - we can easily change the number and the geometry of the velocities involved in our construction to improve the accuracy of the method. In this sense our work shares most of the spirit of =-=[56, 34, 45]-=-, where very flexible and simple schemes, which do not need Riemann solvers in their construction, were proposed to approximate general multidimensional systems of conservation laws. Let us also obser... |

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Citation Context ... could be useful, for example, in the numerical investigation of large systems like those arising in the extended thermodynamics and other generalized moment closures hierarchies for kinetic theories =-=[52, 38, 1]-=-. Further investigations will be addressed to the construction of high order schemes. The plan of the article is as follows: In section 2 we establish the stability condition and define the monotone M... |

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Citation Context ...eak) solution, was proven in [7] (see also [22]). Another convergence result was given later in [60], using a continuous velocities BGK model. An important related kinetic formulation can be found in =-=[41]-=-. Other results have been established for special systems or partially kinetic approximations [42, 29, 9, 40]. Related numerical schemes can be found in [19, 59]; for a general overview and many other... |

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Citation Context ... approximation framework generalizes to systems the construction presented in [54] for the scalar case, and shares most of the advantages of the relaxation approximation as proposed in [30] (see also =-=[53, 2, 70]-=-): simple formulation even for general multidimensional systems of conservation laws and easy numerical implementation, hyperbolicity, regular approximating solutions. Actually the main advantage, esp... |

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Citation Context ...0], using a continuous velocities BGK model. An important related kinetic formulation can be found in [41]. Other results have been established for special systems or partially kinetic approximations =-=[42, 29, 9, 40]-=-. Related numerical schemes can be found in [19, 59]; for a general overview and many other references see [24]. Discrete velocities models and their fluid dynamical limits have also been considered b... |

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Citation Context ...n example of numerical (relaxed, i.e., # = 0) first-order discretization of our construction in the scalar case. Other numerical investigations for hyperbolic problems with relaxation can be found in =-=[37, 57, 58, 73, 4, 3, 18, 70, 31]-=-. Our numerical schemes are constructed by splitting (1.3) into a homogeneous linear part and an ordinary di#erential system, which is exactly solved thanks to the particular structure of the source t... |

55 | A kinetic equation with kinetic entropy functions for scalar conservation laws
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Citation Context ...fractional step BGK approximation with continuous velocities, with an entropy condition for the limit (weak) solution, was proven in [7] (see also [22]). Another convergence result was given later in =-=[60]-=-, using a continuous velocities BGK model. An important related kinetic formulation can be found in [41]. Other results have been established for special systems or partially kinetic approximations [4... |

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Citation Context ... is compatible with any convex entropy # of (1.1): there exists a kinetic entropy for (3.1) associated with # and Lax entropy inequalities are satisfied in the hydrodynamic limit # # 0. As well known =-=[36, 10]-=- these inequalities characterize the admissible weak solutions of (1.1). Moreover, in this case, (2.6) is parabolic. More general results for Maxwellian functions not in the form (3.4) can be found in... |

38 |
Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics
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Citation Context ...n example of numerical (relaxed, i.e., # = 0) first-order discretization of our construction in the scalar case. Other numerical investigations for hyperbolic problems with relaxation can be found in =-=[37, 57, 58, 73, 4, 3, 18, 70, 31]-=-. Our numerical schemes are constructed by splitting (1.3) into a homogeneous linear part and an ordinary di#erential system, which is exactly solved thanks to the particular structure of the source t... |

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Citation Context ...lso been considered by many people; see the review paper [61]. In particular we mention the studies on the Broadwell model [11, 72]. Convergence for various relaxation models was also investigated in =-=[15, 14, 17, 47, 68, 74, 32, 33, 67]-=-. The analysis of the stability DISCRETE KINETIC SCHEMES 1975 of various nonlinear waves for relaxation models, and in particular for the Jin--Xin relaxation approximation, can be found in [44, 16, 50... |

37 |
Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions
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Citation Context ...elated kinetic formulation can be found in [41]. Other results have been established for special systems or partially kinetic approximations [42, 29, 9, 40]. Related numerical schemes can be found in =-=[19, 59]-=-; for a general overview and many other references see [24]. Discrete velocities models and their fluid dynamical limits have also been considered by many people; see the review paper [61]. In particu... |

36 |
Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates
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Citation Context ...0], using a continuous velocities BGK model. An important related kinetic formulation can be found in [41]. Other results have been established for special systems or partially kinetic approximations =-=[42, 29, 9, 40]-=-. Related numerical schemes can be found in [19, 59]; for a general overview and many other references see [24]. Discrete velocities models and their fluid dynamical limits have also been considered b... |

35 |
The Unique Limit of the Glimm Scheme
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Citation Context ... is compatible with any convex entropy # of (1.1): there exists a kinetic entropy for (3.1) associated with # and Lax entropy inequalities are satisfied in the hydrodynamic limit # # 0. As well known =-=[36, 10]-=- these inequalities characterize the admissible weak solutions of (1.1). Moreover, in this case, (2.6) is parabolic. More general results for Maxwellian functions not in the form (3.4) can be found in... |

35 |
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Citation Context ...ound in [19, 59]; for a general overview and many other references see [24]. Discrete velocities models and their fluid dynamical limits have also been considered by many people; see the review paper =-=[61]-=-. In particular we mention the studies on the Broadwell model [11, 72]. Convergence for various relaxation models was also investigated in [15, 14, 17, 47, 68, 74, 32, 33, 67]. The analysis of the sta... |

35 |
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Citation Context ...em. We consider the system # # t u + # x v = 0 , # t v + # x #(u) = 0 , (6.17) where #(u) = u # . In all our computations we take # = 1.4. Exact solutions are known for this system; see, for example, =-=[63]-=-. We compute the 1-shock, 2-rarefaction solution of the Riemann problem with data: # # # # # # # u 0 (x) = # 0.1 for xs0, 0.4 for x > 0, v 0 (x) = # 0.5 for xs0, 0.60466 for x > 0. Here we discretize ... |

32 | Convergence of relaxation schemes for conservation laws
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Citation Context ... approximation framework generalizes to systems the construction presented in [54] for the scalar case, and shares most of the advantages of the relaxation approximation as proposed in [30] (see also =-=[53, 2, 70]-=-): simple formulation even for general multidimensional systems of conservation laws and easy numerical implementation, hyperbolicity, regular approximating solutions. Actually the main advantage, esp... |

32 | A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws
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Citation Context ... j,d=1 (P# d # j M # (u)# j , # d ) # D # j,d=1 # A # d (u)A # j (u)# j , # d # , (1.6) for all # = (# 1 , . . . , # D ) # (R K ) D and every u belonging to some fixed rectangle# # R K . Actually, in =-=[54]-=- and for the scalar case K = 1, convergence of Pf # towards the Kruzkov entropy solution of (1.1), (1.2) has been obtained under a slightly stronger version of condition (1.6): every component of the ... |

31 | Construction of BGK models with a family of kinetic entropies for a given system of conservation laws
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Citation Context ...not verified for nontrivial examples in the general case K > 1. Therefore, for the systems, we shall use just condition (1.6). Note that for certain families of kinetic approximations, the results of =-=[6]-=- show that (1.6) is also a necessary condition for (1.3) to be compatible with the entropies of (1.1) (see section 3). The continuous kinetic approximation of systems of conservation laws in gas dynam... |

30 |
Zero relaxation and dissipation limits for hyperbolic conservation laws
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Citation Context ...lso been considered by many people; see the review paper [61]. In particular we mention the studies on the Broadwell model [11, 72]. Convergence for various relaxation models was also investigated in =-=[15, 14, 17, 47, 68, 74, 32, 33, 67]-=-. The analysis of the stability DISCRETE KINETIC SCHEMES 1975 of various nonlinear waves for relaxation models, and in particular for the Jin--Xin relaxation approximation, can be found in [44, 16, 50... |

28 |
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Citation Context ... see [24]. Discrete velocities models and their fluid dynamical limits have also been considered by many people; see the review paper [61]. In particular we mention the studies on the Broadwell model =-=[11, 72]-=-. Convergence for various relaxation models was also investigated in [15, 14, 17, 47, 68, 74, 32, 33, 67]. The analysis of the stability DISCRETE KINETIC SCHEMES 1975 of various nonlinear waves for re... |

28 |
Numerical passage from kinetic to fluid equations
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Citation Context ...elated kinetic formulation can be found in [41]. Other results have been established for special systems or partially kinetic approximations [42, 29, 9, 40]. Related numerical schemes can be found in =-=[19, 59]-=-; for a general overview and many other references see [24]. Discrete velocities models and their fluid dynamical limits have also been considered by many people; see the review paper [61]. In particu... |

28 |
Relaxation semi linéaire et cinétique des systèmes de lois de conservation
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Citation Context ...ion 5 is devoted to the convergence results in the scalar multidimensional case. Some numerical experiments are given in section 6. After the completion of this work we received a preprint from Serre =-=[64]-=-, where he proves, by using the methods of compensated compactness, the convergence for the Jin--Xin relaxation approximation and some of the discrete kinetic approximations contained in the present p... |

26 | Recent results on hyperbolic relaxation problems, in Analysis of systems of conservation laws(Aachen,1997 - Natalini - 1999 |

25 |
A kinetic construction of global solutions of first order quasilinear equations
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Citation Context ...he scalar case. The first result of convergence of a fractional step BGK approximation with continuous velocities, with an entropy condition for the limit (weak) solution, was proven in [7] (see also =-=[22]-=-). Another convergence result was given later in [60], using a continuous velocities BGK model. An important related kinetic formulation can be found in [41]. Other results have been established for s... |

25 | On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term
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Citation Context ...lso been considered by many people; see the review paper [61]. In particular we mention the studies on the Broadwell model [11, 72]. Convergence for various relaxation models was also investigated in =-=[15, 14, 17, 47, 68, 74, 32, 33, 67]-=-. The analysis of the stability DISCRETE KINETIC SCHEMES 1975 of various nonlinear waves for relaxation models, and in particular for the Jin--Xin relaxation approximation, can be found in [44, 16, 50... |

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21 |
Non-oscillatory central di erencing for hyperbolic conservation laws
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Citation Context ... purposes; - we can easily change the number and the geometry of the velocities involved in our construction to improve the accuracy of the method. In this sense our work shares most of the spirit of =-=[56, 34, 45]-=-, where very flexible and simple schemes, which do not need Riemann solvers in their construction, were proposed to approximate general multidimensional systems of conservation laws. Let us also obser... |

19 |
Hyperbolic conservation laws with sti relaxation and entropy
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Citation Context ...work of general 2 2 quasi-linear hyperbolic relaxation problems, this condition is known as the subcharacteristic condition. In section 2, we shall argue in the spirit of the Chapman--Enskog analysis =-=[71, 44, 14]-=-, to find the following stability condition for (1.3): D # j,d=1 (P# d # j M # (u)# j , # d ) # D # j,d=1 # A # d (u)A # j (u)# j , # d # , (1.6) for all # = (# 1 , . . . , # D ) # (R K ) D and every ... |

17 | A Roe-type Riemann solver for hyperbolic systems with relaxation based on time-dependent wave decomposition
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Citation Context ...n example of numerical (relaxed, i.e., # = 0) first-order discretization of our construction in the scalar case. Other numerical investigations for hyperbolic problems with relaxation can be found in =-=[37, 57, 58, 73, 4, 3, 18, 70, 31]-=-. Our numerical schemes are constructed by splitting (1.3) into a homogeneous linear part and an ordinary di#erential system, which is exactly solved thanks to the particular structure of the source t... |

17 | Stiff systems of hyperbolic conservation laws: convergence and error estimates
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17 | L 1 nonlinear stability of traveling waves for a hyperbolic system with relaxation
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Citation Context ...32, 33, 67]. The analysis of the stability DISCRETE KINETIC SCHEMES 1975 of various nonlinear waves for relaxation models, and in particular for the Jin--Xin relaxation approximation, can be found in =-=[44, 16, 50, 43, 46, 49]-=-. A general survey of recent results on relaxation hyperbolic problems is given in [55]. Let us also point out some numerical references related to our approach. A lot of computational work has been d... |

16 | Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws
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(Show Context)
Citation Context ... purposes; - we can easily change the number and the geometry of the velocities involved in our construction to improve the accuracy of the method. In this sense our work shares most of the spirit of =-=[56, 34, 45]-=-, where very flexible and simple schemes, which do not need Riemann solvers in their construction, were proposed to approximate general multidimensional systems of conservation laws. Let us also obser... |

16 |
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Citation Context ... could be useful, for example, in the numerical investigation of large systems like those arising in the extended thermodynamics and other generalized moment closures hierarchies for kinetic theories =-=[52, 38, 1]-=-. Further investigations will be addressed to the construction of high order schemes. The plan of the article is as follows: In section 2 we establish the stability condition and define the monotone M... |

16 | Pointwise error estimates for relaxation approximations to conservation laws
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Citation Context |

15 | Hanouzet B.; Weakly coupled systems of quasilinear hyperbolic equations
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Citation Context ...xwellian function M is monotone nondecreasing on the interval I. The main tool in that case is the fact that under this condition the right-hand side in system (1.3) is quasi-monotone in the sense of =-=[25]-=- and this implies special comparison and stability properties on the corresponding system. Unfortunately, similar properties are not verified for nontrivial examples in the general case K > 1. Therefo... |

15 |
The fluid dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks
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Citation Context ... see [24]. Discrete velocities models and their fluid dynamical limits have also been considered by many people; see the review paper [61]. In particular we mention the studies on the Broadwell model =-=[11, 72]-=-. Convergence for various relaxation models was also investigated in [15, 14, 17, 47, 68, 74, 32, 33, 67]. The analysis of the stability DISCRETE KINETIC SCHEMES 1975 of various nonlinear waves for re... |

14 |
Monotone di erence approximations for scalar conservation laws
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(Show Context)
Citation Context ... - f # (t # )# 1 # C|t - t # |# . (5.12) As a corollary we obtain global existence of the solution of (1.3), (1.4) (see [54] for a direct proof). Proof of Theorem 5.6. We follow exactly the method of =-=[20]-=- and just give a sketch of the proof: for all t # 0, {f # # (t), #t > 0} is bounded in L 1 # L # . Moreover, by Proposition 5.3, we can apply Frechet--Kolmogorov theorem and obtain a relatively compac... |

14 |
Recent advances in lattice Boltzmann computing, inAnnual
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Citation Context ...also point out some numerical references related to our approach. A lot of computational work has been done in the last ten years in the very closed framework of lattice Boltzmann and BGK models; see =-=[21, 62]-=- and references therein. Let us also mention the monotone schemes of [8], which are an example of numerical (relaxed, i.e., # = 0) first-order discretization of our construction in the scalar case. Ot... |

12 |
Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography
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11 | Kinetic formulation for chromatography and some other hyperbolic systems
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Citation Context ...0], using a continuous velocities BGK model. An important related kinetic formulation can be found in [41]. Other results have been established for special systems or partially kinetic approximations =-=[42, 29, 9, 40]-=-. Related numerical schemes can be found in [19, 59]; for a general overview and many other references see [24]. Discrete velocities models and their fluid dynamical limits have also been considered b... |

10 |
On modelling phase transitions by means of rate-type constitutive equations, shock wave structure
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(Show Context)
Citation Context ... # = # # M 1 M 2 M 3 # # . (3.25) 1984 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI For A 2 (u) = p(u 1 ) we have a p-system, and in this case the present approximation was first introduced by Suliciu =-=[65, 66]-=- to study instability problems in phase transitions described by elastic or viscoelastic constitutive equations. In this case the Chapman-- Enskog analysis gives the stability condition # 2 > p # (u 1... |

9 |
Convergence of relaxing schemes for conservations laws
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Citation Context |