High-Order Entropy-Conservative Schemes and Kinetic Relations for van der Waals Fluids

Chalons, Christophe; LeFloch, Philippe G.

doi:10.1006/jcph.2000.6690

Cited by 30 publications

(34 citation statements)

References 33 publications

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“…This is the case, for instance, of models describing the dynamics of liquid-vapor phase transitions in compressible fluids, or of solid-solid phase transformations in materials such as memory alloys. For numerical work in this direction we refer to [18,19,10,31,32].…”

Section: State Of the Artmentioning

confidence: 99%

“…Due to the great sensitivity of nonclassical solutions with respect to small scales and numerical diffusion, it turns out that numerical results are satisfactory for shocks with moderate amplitude, but discrepancies between the exact and the numerical kinetic function arise for shocks with large amplitudes and in long-time computations. For this circle of ideas we refer the reader to [15,16], and the follow-up papers [26,9,10].…”

Section: Objectives Of This Papermentioning

confidence: 99%

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Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

Boutin¹,

Chalons²,

Lagoutìère³

et al. 2008

Interfaces Free Bound.

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We propose a new numerical approach to computing nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keeps nonclassical shock waves as sharp interfaces, unlike standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux functions.

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Section: State Of the Artmentioning

confidence: 99%

Section: Objectives Of This Papermentioning

confidence: 99%

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

Boutin¹,

Chalons²,

Lagoutìère³

et al. 2008

Interfaces Free Bound.

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“…Numerical analyses for mixed type differential systems have been carried out by precursors in the finite difference community [42,60,62]. However, to the best of our knowledge, there are very few semi-discrete schemes that have achieved entropy stability [15,17]. Among the schemes that are indeed entropy stable, the common techniques employed are intricate discrete flux terms to directly enforce the nonlinear stability condition.…”

Section: Entropy Variables and Provably Stable-in-entropymentioning

confidence: 99%

“…As is depicted in Figure 2 and 3, the van der Waals model gives a qualitatively accurate approximation of the various materials behavior in both liquid and vapor phases. It should be noted that the form of the equation of state is partially determined by the choice of critical point given by (15) and (16). Hence, there is still room to improve the van der Waals model for a specific material by tuning the parameters a and b using, for example, least squares.…”

Section: Definitionmentioning

confidence: 99%

Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations

Liu¹,

Gómez²,

Evans³

et al. 2012

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. We propose a new methodology for the numerical solution of the isothermal Navier-Stokes-Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws. AbstractWe propose a new methodology for the numerical solution of the isothermal Navier-StokesKorteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws.

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“…In practice, this method provides satisfactory results for shocks with small strength, but shocks with large strength are not properly captured due to the great sensitivity of the nonclassical shocks with respect to the numerical diffusion and small scales in general. For more details, we refer for instance to [10], [11], [17], [6], [7] or [3] and the references therein. The second strategy, known under the name of "sharp interface approach", deals directly with the kinetic function ϕ to tackle the dynamics of the nonclassical solutions.…”

Section: Introductionmentioning

confidence: 99%

Transport-equilibrium schemes for computing nonclassical shocks

Chalons

2006

Comptes Rendus. Mathématique

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This paper presents a very efficient numerical strategy for computing the weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of either concave-convex or convex-concave flux functions. In such a situation, nonclassical shocks violating the classical Oleinik entropy criterion must be taken into account since they naturally arise as limits of certain diffusive-dispersive regularizations to hyperbolic conservation laws. Such discontinuities play an important part in the resolution of the Riemann problem and their dynamics turns out to be driven by a prescribed kinetic function which acts as a selection principle. It aims at imposing the entropy dissipation rate across nonclassical discontinuities, or equivalently their speed of propagation. From a numerical point of view, the serious difficulty consists in enforcing the kinetic criterion, that is in controling the numerical entropy dissipation of the nonclassical shocks for any given discretization. This is known to be a very challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic criterion at the discrete level. The resulting scheme provides in addition sharp profiles. Numerical evidences illustrate the validity of our approach.

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High-Order Entropy-Conservative Schemes and Kinetic Relations for van der Waals Fluids

Cited by 30 publications

References 33 publications

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations

Transport-equilibrium schemes for computing nonclassical shocks

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