Hyperbolic conservation laws with stiff relaxation terms and entropy

Chen, Gui–Qiang; Levermore, C. David; Liu, Tai-Ping

doi:10.1002/cpa.3160470602

Cited by 607 publications

(685 citation statements)

References 24 publications

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“…Such systems arise in reacting gases, shallow water equations, discrete-velocity kinetic models, etc. [57], and have been mathematically studied extensively in recent years (see for example [4,12,43,50]). A prototype example is the following 2 Â 2 nonlinear hyperbolic system with relaxation:…”

Section: ð1:4þmentioning

confidence: 99%

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

Filbet

Jin

2010

Journal of Computational Physics

317

432

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a b s t r a c tIn this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.

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Section: ð1:4þmentioning

confidence: 99%

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

Filbet

Jin

2010

Journal of Computational Physics

317

432

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“…The limiting process → 0 in systems in the form (55) was extensively analysed by Liu [24] and Chen et al [4], with a particular focus on the relationship between stability and wave propagation. It is of high interest to obtain good numerical methods for systems in the form (55) when the relaxation source term is stiff; i.e.…”

Section: Hyperbolic Relaxation Systemsmentioning

confidence: 99%

“…as analysed in detail by Chen et al [4]. The parameter is typically small, imposing a high degree of stiffness in the system (2).…”

Section: Introductionmentioning

confidence: 99%

An exponential time-differencing method for monotonic relaxation systems

Aursand

Evje

Flåtten

et al. 2014

Applied Numerical Mathematics

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Abstract. We present first and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation.We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge-Kutta and exponential methods. Through operator splitting, an application to granulargas flow is provided.

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“…It is well known that even for general relaxation models [9,24,26,40] an approximation of the type (2.12) has to obey some stability criterion so as to possess the correct hydrodynamic limit. In the case of 2 × 2 relaxation systems it is the well known sub-characteristic condition [9,24], see [5] for a survey of different stability conditions for relaxation problems. We use the entropy extension condition of Bouchut [4] so that the BGK model (2.12) is compatible with the entropies of (2.1).…”

Section: Relaxation System For Euler Equationsmentioning

confidence: 99%

“…However, it is a characteristic of the Maxwellian equilibrium distributions of the type M k (w) to give an inviscid system of conservation laws in the hydrodynamic limit, see [4,9]. Therefore, in order to get a dissipative flux like term ∂D/∂x we need to change the Maxwellian distribution to a ChapmanEnskog distribution.…”

Section: Second Order Accuracy In Timementioning

confidence: 99%

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

Arun

Lukáčová-Medviďová

Prasad

et al. 2013

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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Abstract. In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accurate method is conservative, preserves the positivity of mass density and pressure and entropy stable. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. The results of numerical experiments on some benchmark problems confirm the efficiency and robustness of the proposed scheme.

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Hyperbolic conservation laws with stiff relaxation terms and entropy

Cited by 607 publications

References 24 publications

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

An exponential time-differencing method for monotonic relaxation systems

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

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