From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: A quantitative error estimate

Briant, Marc

doi:10.1016/j.jde.2015.07.022

Cited by 74 publications

(154 citation statements)

References 37 publications

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“…H k x,v , by using Sobolev embeddings and the Cauchy-Schwarz inequality, exactly the same as discussed in [36,1]. The following assumptions on the collision kernels σi (i = 1, 2) and σI are needed:…”

Section: Regularity and Local Sensitivity Resultsmentioning

confidence: 99%

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

Liu

2019

Journal of Computational Physics

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In this paper, we study the bipolar Boltzmann-Poisson model, both for the deterministic system and the system with uncertainties, with asymptotic behavior leading to the drift diffusion-Poisson system as the Knudsen number goes to zero. The random inputs can arise from collision kernels, doping profile and initial data. We adopt a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method. Sensitivity analysis is conducted using hypocoercivity theory for both the analytical solution and the gPC solution for a simpler model that ignores the electric field, and it gives their convergence toward the global Maxwellian exponentially in time. A formal proof of the stochastic asymptotic-preserving (s-AP) property and a uniform spectral convergence with error exponentially decaying in time in the random space of the scheme is given. Numerical experiments are conducted to validate the accuracy, efficiency and asymptotic properties of the proposed method.

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Section: Regularity and Local Sensitivity Resultsmentioning

confidence: 99%

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

Liu

2019

Journal of Computational Physics

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“…Theorem 4.1. If all the assumptions in Lemma 6.1 are satisfied for each n = 0, 1, · · · , N − 1, then 9) where d N (u H (I z )) is the N -width of the functional manifold u H (I z ), and the constant…”

Section: The Projection Errormentioning

confidence: 99%

A bi-fidelity method for the multiscale Boltzmann equation with random parameters

Liu

Zhu

2020

Journal of Computational Physics

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In this paper, we study the multiscale Boltzmann equation with multi-dimensional random parameters by a bi-fidelity stochastic collocation (SC) method developed in [51,65,66]. By choosing the compressible Euler system as the low-fidelity model, we adapt the bi-fidelity SC method to combine computational efficiency of the lowfidelity model with high accuracy of the high-fidelity (Boltzmann) model. With only a small number of high-fidelity asymptotic-preserving solver runs for the Boltzmann equation, the bi-fidelity approximation can capture well the macroscopic quantities of the solution to the Boltzmann equation in the random space. A priori estimate on the accuracy between the high-and bi-fidelity solutions together with a convergence analysis is established. Finally, we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method.

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“…One of the important analysis in UQ is the so-called local sensitivity analysis, in which one aims to understand how sensitive the solution depends on the input parameters [26]. For kinetic equations, a major tool to conduct sensitivity analysis for random kinetic equations has been the coercivity, or more generally, hypocoercivity, which originated in the study of long-time behavior of kinetic equations (see [9,11,23,28]). In such analysis, by using the hypocoercivity of the kinetic operator, in a perturbative setting, namely, considering solutions near the global equilibrium (see [13]), one can establish the long time convergence toward the local equilibrium with an exponential time decay rate.…”

Section: Introductionmentioning

confidence: 99%

Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

Arnold¹,

Jin

Wöhrer³

2020

Journal of Differential Equations

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We review the Lyapunov functional method for linear ODEs and give an explicit construction of such functionals that yields sharp decay estimates, including an extension to defective ODE systems. As an application, we consider three evolution equations, namely the linear convection-diffusion equation, the two velocity BGK model and the Fokker-Planck equation. Adding an uncertainty parameter to the equations and analyzing its linear sensitivity leads to defective ODE systems. By applying the Lyapunov functional construction, we prove sharp long time behavior of order (1 + t M )e −µt , where M is the defect and µ is the spectral gap of the system. The appearance of the uncertainty parameter in the three applications makes it important to have decay estimates that are uniform in the non-defective limit.

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From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: A quantitative error estimate

Cited by 74 publications

References 37 publications

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

A bi-fidelity method for the multiscale Boltzmann equation with random parameters

Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

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