Uncertainty Propagation; Intrusive  Kinetic Formulations of Scalar Conservation Laws

Després, Bruno; Perthame, Benôıt

doi:10.1137/15m1018861

Cited by 28 publications

(22 citation statements)

References 23 publications

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“…A third example is from reduced modeling, typically with POD techniques [17], and from interpolation on a sparse grid [14]. Uncertainty quantification intrusive techniques are also highly demanding in terms of having polynomial approximations with a priori unconditional respect of the maximum principle [21,7].…”

Section: Introductionmentioning

confidence: 99%

Polynomials with bounds and numerical approximation

Després

2017

Numer Algor

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We discuss the generation of polynomials with two bounds -an upper bound and a lower bound-on compact setsin view on numerical approximation and scientific computing. We show that a composition formula based on a weighted 4-squares Euler identity generates all such polynomials in dimension d = 1. It yields a new algebraic or compositional approach to the classical problem related to polynomials with minimal uniform norm. Higher dimensions are discussed by means of the 8-squares Degen identity and tensorization, and the connection with quaternions algebras is made explicit. Various numerical results illustrate the potentialities of this approach and some implementation details are provided.

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Section: Introductionmentioning

confidence: 99%

Polynomials with bounds and numerical approximation

Després

2017

Numer Algor

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“…While uncertainty quantification has been a hot topic in scientific and engineering computing in the last two decades, research on uncertainty quantification for kinetic equations has been relatively recent. We refer to recent review articles [23,11] and some recent works [30,22,10,27,26,8,33,34,28,31,29,39,1] in this direction. The first sensitivity analysis similar to this paper for the linear transport equation, with uniform (in the Knudsen number) spectral convergence of the gPC-SG approximation, was given by Jin, Liu and Ma in [26].…”

Section: Introduction Consider the Initial Value Problem For Kineticmentioning

confidence: 99%

Hypocoercivity Based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs

Liu¹,

Jin²

2018

Multiscale Model. Simul.

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In this paper we provide a general framework to study general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin approximations, in both incompressible Navier-Stokes and Euler (acoustic) regimes. First, we show that the general framework put forth in [C. Mouhot and L. Neumann, Nonlinearity, 19, 969-998, 2006; M. Briant, J. Diff. Eqn., 259, 6072-6141, 2005] based on hypocoercivity for the deterministic kinetic equations can be easily adopted for sensitivity analysis for random kinetic equations, which gives rise to an exponential convergence of the random solution toward the (deterministic) global equilibrium, under suitable conditions on the collision kernel. Then we use such theory to study the stochastic Galerkin (SG) methods for the equations, establish hypocoercivity of the SG system and regularity of its solution, and spectral accuracy and exponential decay in time of the numerical error of the method in a weighted Sobolev norm.

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“…It is more crucial and interesting to find quantitative estimates on the rates of convergence, which is known as hypocoercivity theory, see for example [30,29,25,10] for kinetic equations. There have been recent efforts extended to the study of kinetic equations with random inputs, including their mathematical properties such as regularity and long-time behavior in the random space, for example refer to [20,24,23,22,27,13]. Although large-time behavior of the monospecies BGK equations were intensively studied in the literature, but they are unknown for multispecies BGK systems.…”

Section: Introductionmentioning

confidence: 99%

Hypocoercivity for a BGK model for gas mixtures

Liu

Pirner

2019

Journal of Differential Equations

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We consider a kinetic model for a two component gas mixture without chemical reactions. Our goal is to study hypocoercivity for the linearized BGK model for gas mixtures in continuous phase space. By constructing an entropy functional, we can prove exponential relaxation to equilibrium with explicit rates. Our strategy is based on the entropy and spectral methods adapting Lyapunov's direct method as presented in [1] for the one species linearized BGK model. In comparison to the one species case, we start with two partial differential equations, one for each species. These equations are coupled due to interspecies interactions, which requires additional estimates on these interspecies terms. Nonlinear BGK model for gas mixturesWe want to consider the BGK model for gas mixtures described in [21]. For the convenience of the reader, we want to briefly repeat it here. For more details, see [21]. We consider the position spacẽ T d := L 2Π T d , the d-dimensional torus of side length L. Then, we consider the following BGK

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Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws

Cited by 28 publications

References 23 publications

Polynomials with bounds and numerical approximation

Polynomials with bounds and numerical approximation

Hypocoercivity Based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs

Hypocoercivity for a BGK model for gas mixtures

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