Robust Uncertainty Propagation in Systems of Conservation Laws with the Entropy Closure Method

Després, Bruno; Poëtte, Gaël; Lucor, Didier

doi:10.1007/978-3-319-00885-1_3

Cited by 58 publications

(109 citation statements)

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“…In particular, in [8], the authors show how to model uncertainty in the coefficients F of the equations (1)-(2), using an additional variable. Even if the theory of existence, uniqueness and the theory of numerical approximation is well established for scalar conservation laws like (1), there is no such theory for systems of partial differential equations like (4) or for the convergence of the solutions of (4) towards solutions of (1) parametrized by the parameter ω, see [8].…”

Section: Introductionmentioning

confidence: 99%

“…Even if the theory of existence, uniqueness and the theory of numerical approximation is well established for scalar conservation laws like (1), there is no such theory for systems of partial differential equations like (4) or for the convergence of the solutions of (4) towards solutions of (1) parametrized by the parameter ω, see [8]. Some results may be found nevertheless in [10] for the advection equation, in [8] where spectral convergence is proved with a weakstrong method but before any shock, and in [20] for Monte-Carlo methods applied to conservation laws. Note also the convergence proof in the recent work [4], for the method of moments in the framework of kinetic equations.…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws

Després¹,

Perthame²

2016

SIAM/ASA J. Uncertainty Quantification

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We study two intrusive methods for uncertainty propagation in scalar conservation laws based on their kinetic formulations. The first one is based on expansions on an orthogonal family of polynomials.The first method uses convolutions based on Jackson kernels and we prove that it satisfies BV bounds and converges to the entropy solution but with a spurious damping phenomenon. Therefore we introduce a second method, which is based on projection on layered Maxellians, and which arises as a minimization of entropy. Our construction of layered Maxwellians relies on the Bojavic-Devore theorem about best L 1 polynomial approximation. This new method, denoted below as a kinetic polynomial method, satisfies the maximum principle by construction as well as partial entropy inequalities and thus provides an alternative to the standard method of moments which, in general, does not satisfy the maximum principle.Simple numerical simulations for the Burgers equation illustrate these theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws

Després¹,

Perthame²

2016

SIAM/ASA J. Uncertainty Quantification

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“…4.3, where the same peak and valley values are observed in the deterministic problems with different w. and Γ = 1.4. The same setup is considered in [20,8,21,6]. The gPC-SG method may easily fail in this test due to the appearance of negative density caused by the oscillations [20].…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…[5,11,22,14]. However, for a general system of quasilinear hyperbolic conservation laws, the resulting gPC-SG system may be not globally hyperbolic [8]. The lack of hyperbolicity means that the Jacobian matrix may contain complex eigenvalues, which lead to ill-posedness of the initial or boundary problem and instability of the numerical computations.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, some efforts were made to obtain well-behaved gPC-SG system for a system of hyperbolic conservation laws. Després et al used the gPC approximation to the entropy variables instead of the conservative variables in the Euler equations [8], and proved that the resulting gPC-SG system is hyperbolic, based on the fact that the Euler equations can be reformulated in a symmetrically hyperbolic form in term of the entropy variables. This method, however, can not be extended to a general quasilinear hyperbolic system without a convex entropy pair or a non-symmetrically hyperbolic system.…”

Section: Introductionmentioning

confidence: 99%

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A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

Tang

Xiu

2017

Journal of Computational Physics

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This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Stochastic Galerkin method is then applied to derive the equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is proved to be symmetrically hyperbolic. This important property then allows one to use a variety of numerical schemes for spatial and temporal discretization. Here a higher-order and path-conservative finite volume WENO scheme is adopted in space, along with a third-order total variation diminishing Runge-Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional system is carried over via the operator splitting technique. Several 1D and 2D numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

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Sensitivity equation method for the Navier‐Stokes equations applied to uncertainty propagation

Fiorini¹,

Després²,

Puscas

2020

Numerical Methods in Fluids

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This works deals with sensitivity analysis for the Navier-Stokes equations. The aim is to provide an estimate of the variance of the velocity field when some of the parameters are uncertain and then to use the variance to compute confidence intervals for the output of the model. First, we introduce the physical model and analyse its stability. The sensitivity equations are derived, and their stability analysed as well. We propose a finite elementvolume numerical scheme for the state and the sensitivity, which is integrated into the open-source industrial code TrioCFD. Finally, we present some numerical results: a steady and an unsteady test case for the channel flow problem are investigated. For the steady case, we compare the results to the Monte Carlo method and show how the sensitivity analysis technique succeeds in providing very accurate estimates of the variance. For the unsteady case, a new filtering procedure is proposed to deal with a sensitivity that grows in time. The filtered sensitivity is then used to compute the variance of the output and to provide confidence intervals. problems, particularly in the case of PDEs models: Monte Carlo method [RC13], polynomial chaos [Wal03, XK03, KM06, DPL13], random space partition [AC13], to name but a few. A review of these methods applied to fluid dynamics problems can be found in [WH02]. Methods of uncertainty propagation based on SA are particularly efficient in terms of computational time if compared, for instance, to methods like Monte Carlo. However, since SA is based on Taylor expansions of the state variable with respect to the parameter of interest, these methods are intrinsically local [Del14] : they can be used only for random variables with a small variance. The Monte Carlo approach does not require this assumption; however, it is not applicable for realistic unsteady test cases in 2D and 3D, due to its high computational cost. In this work we propose an approach based on the sensitivity equation method: under the hypothesis of small variance of the input parameters, we can provide a first-order estimate of the variance of the solution at a reasonable computational cost. Two declinations of the channel flow test case (i.e. study of the flow past an obstacle) are investigated: a steady (Re " 25) and an unsteady (Re " 100) test case. For the steady test case, a detailed comparison with the Monte Carlo method is performed: the variance of the output is first estimated using a classical Monte Carlo technique and then with the sensitivity analysis method. The results are extremely accurate. The unsteady test case presents a sensitivity which grows in time: this is caused by the fact that the parameter considered has an infuence on the frequency of vortex shedding. This problem is also described in [HEPB04] for a similar test case to the one considered here: to deal with it, they chose a pulsating inflow velocity, which imposes the frequency of vortex shedding. However, this choice is not suitable for all applications. Therefore we propose a new filtering te...

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Robust Uncertainty Propagation in Systems of Conservation Laws with the Entropy Closure Method

Cited by 58 publications

References 31 publications

Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws

Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws

A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

Sensitivity equation method for the Navier‐Stokes equations applied to uncertainty propagation

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