Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics

Bennoune, Mounir; Lemou, Mohammed; Mieussens, Luc

doi:10.1016/j.jcp.2007.11.032

Cited by 173 publications

(253 citation statements)

References 27 publications

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“…However, a similar assumption is done in [2]. We only prove theoretical results when the initial data is close enough to the local Maxwellian.…”

Section: ð4:9þmentioning

confidence: 87%

“…This result can be extended to essentially all AP schemes, although the specific proof is problem dependent. We refer to AP schemes for kinetic equations in the fluid dynamic or diffusive regimes [2,7,14,32,[40][41][42]44,45,[47][48][49]. The AP framework has also been extended in [15,16] for the study of the quasi-neutral limit of Euler-Poisson and Vlasov-Poisson systems, and in [19,21,34] for all-speed (Mach number) fluid equations bridging the passage from compressible flows to the incompressible flows.…”

Section: ð1:4þmentioning

confidence: 99%

“…In Fig. 4, we only show the results obtained in the kinetic regime (10 À2 ) using a spectral scheme for the discretization of the collision operator [27] (with n v = 32 2 and a truncation of the velocity domain v max = 7) and second order explicit Runge-Kutta and second order method (4.14) for the time discretization with a time step Dt = 0.005 satisfying the CFL condition for the transport part (with n x = 100). For such a value of e, the problem is not stiff and this test is only performed to compare the accuracy of our second order scheme (4.14) with the classical (second order) Runge-Kutta method.…”

mentioning

confidence: 99%

“…6, we compare the numerical solution of the Boltzmann Eq. (1.1) with the numerical solution to the compressible Navier-Stokes system derived from the BGK model since the viscosity and heat conductivity are in that case explicitly known [2]. To approximate the compressible Navier-Stokes system, we apply a second order Lax-Friedrich scheme using a large number of points (n x = 1000) whereas we only used n x = 100, and 200 points in space and n 2 v ¼ 32 2 points in velocity for the approximation of the kinetic Eq.…”

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confidence: 99%

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

Filbet

Jin

2010

Journal of Computational Physics

320

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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“…However, a similar assumption is done in [2]. We only prove theoretical results when the initial data is close enough to the local Maxwellian.…”

Section: ð4:9þmentioning

confidence: 87%

Section: ð1:4þmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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

Filbet

Jin

2010

Journal of Computational Physics

320

432

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“…In these works, the kinetic model considered is BGK, because the decoupling of the macroscopic solution from the kinetic correction is simplified exploiting the linearity of the collision integral. See also [5].…”

Section: Introductionmentioning

confidence: 99%

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

Alaia

Puppo

2012

Journal of Computational Physics

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In this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic-kinetic problems, in which the Knudsen number may span from equilibrium to highly rarefied regimes. Our approach is characterized by using the full Boltzmann equation for the kinetic regime, the Compressible Euler equations for equilibrium, with a buffer zone in which the BGK-ES equation is used to represent the transition between fully kinetic to equilibrium flows.In this fashion, the Boltzmann solver is used only when the collision integral is non-stiff, and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities. Thus, in principle, the same mesh size and time steps can be used in the whole computation. Moreover, the time step is limited only by convective terms.Since the Boltzmann solver is applied only in wholly kinetic regimes, we use the reduced noise DSMC scheme we have proposed in Part I of the present work. This ensures a smooth exchange of information across the different domains, with a natural way to construct interface numerical fluxes. Several tests comparing our hybrid scheme with full Boltzmann DSMC computations show the good agreement between the two solutions, on a wide range of Knudsen numbers.

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Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

Yang

Chen

Yang

et al. 2019

Numerical Methods in Fluids

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Summary In the conventional discrete velocity method (DVM), the local solution of collisionless Boltzmann equation with a piecewise constant distribution for the distribution function is utilized to reconstruct distribution function at the cell interface and then calculate numerical flux of Boltzmann equation for updating the distribution function at cell center. In this process, a numerical dissipation will be introduced into the solution due to neglecting of the collision effect at the cell interface. This numerical dissipation may deteriorate the solution accuracy of conventional DVM in the continuum flow regime, in which the particle collision happens frequently. To overcome this defect, two improved schemes are first presented in this work, in which the local discrete solution of Boltzmann equation with Shakhov model is adopted to evaluate the distribution function at the cell interface, while the equilibrium state of the local solution is computed by different ways. One of the improved schemes evaluates the equilibrium state exactly by the moments of distribution functions according to the compatibility condition, while the other computes the equilibrium state approximately by a simple average at the cell interface. Since the collision effect is incorporated in evaluation of numerical flux, the improved schemes can provide reasonable solutions in all flow regimes. On the other hand, they introduce some extra computational efforts for determining the collision term at the cell interface as compared with the conventional DVM. To assess the performance of different methods for simulation of flows in all flow regimes, a comprehensive study is then carried out in this work.

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Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics

Cited by 173 publications

References 27 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

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

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