Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

Tryoen, Julie; Maître, Olivier Le; Ndjinga, Michaël; Ern, Alexandre

doi:10.1016/j.jcp.2010.05.007

Cited by 121 publications

(107 citation statements)

References 34 publications

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“…Further details about the formulation of the Roe average matrix are given in [22]. The scheme is a direct generalization of the deterministic MUSCL scheme.…”

Section: Expansion Of Conservative Variablesmentioning

confidence: 99%

“…Lemma 1 shows that since the eigenvalue matrixΛ P is also the eigenvalue matrix of the symmetric matrix J S defined in (22), the eigenvalues are all real. Lemma 1 also shows that the eigenvectors are distinct.…”

Section: Proposition 3 Property (Iii) Is Satisfied By (19)mentioning

confidence: 99%

“…[23], where a reduced-cost Roe solver with entropy corrector was presented, and in [22] with different localized representations of uncertainty in initial functions and problem coefficients. In previous works we investigated well-posedness and stability for an intrusive formulation of Burgers' equation with uncertainty in [14], and imposition of uncertain boundary conditions in [15].…”

Section: Introductionmentioning

confidence: 99%

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A stochastic Galerkin method for the Euler equations with Roe variable transformation

Pettersson

Iaccarino

Nordström

2014

Journal of Computational Physics

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The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square-roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables and no auxiliary quantities are needed, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

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“…Further details about the formulation of the Roe average matrix are given in [22]. The scheme is a direct generalization of the deterministic MUSCL scheme.…”

Section: Expansion Of Conservative Variablesmentioning

confidence: 99%

Section: Proposition 3 Property (Iii) Is Satisfied By (19)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A stochastic Galerkin method for the Euler equations with Roe variable transformation

Pettersson

Iaccarino

Nordström

2014

Journal of Computational Physics

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“…[30,31] for an overview), and seems to be a more promising technique in the present situation. While this approach is supported by a meanwhile well-understood theory for models posed in terms of linear PDEs, for nonlinear problems first steps have been done just recently [1,34,37,38]. The present work is focused on the important subclass of nonlinear problems formed by hyperbolic conservation laws.…”

Section: Scopementioning

confidence: 99%

“…By the method introduced in [38,39] (cf. Appendix C for details) we finally obtain from (25) or (26) a (P + 3)-dimensional system for the determination of the unknown u = (u 0 , .…”

Section: Application Of the Hybrid Stochastic Galerkin Approach To Thmentioning

confidence: 99%

A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier‐thickener unit

2013

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The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction. The flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. Typically, the feed flow cannot be given deterministically and efficient numerical simulation requires a concept for quantifying uncertainty. In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space. The new approach leads to a deterministic hyperbolic system for a finite number of stochastic moments which is however partially decoupled and thus allows efficient parallelisation. The complexity of the problem is further reduced by stochastic adaptivity. For the approximate solution of the resulting high-dimensional system a finite volume scheme is introduced. Numerical experiments cover one-and two-dimensional situations.

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Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

Liao

Zhang

2013

Water Resour. Res.

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[1] In this work, we propose a new collocation method for uncertainty quantification in strongly nonlinear problems. Based on polynomial construction, the traditional probabilistic collocation method (PCM) approximates the model output response, which is a function of the random input parameter, from the Eulerian point of view in specific locations. In some cases, especially when the advection dominates, the model response has a strongly nonlinear profile with a discontinuous shock or large gradient. This nonlinearity in the space domain is then translated to nonlinearity in the random parametric domain, which causes nonphysical oscillation and inaccurate estimation using the traditional PCM. To address this issue, a new location-based transformed probabilistic collocation method (xTPCM) is developed in this study, inspired by the Lagrangian point of view, in which model response is represented by an alternative variable, i.e., the location of a particular response value, which is relatively linear to the random parameter with a smooth profile. The location is then approximated by polynomial construction, from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples and to estimate the statistical properties. The advantage of the xTPCM is demonstrated through applications to multiphase flow and solute transport in porous media, which shows that the xTPCM achieves higher statistical accuracy than does the PCM, and produces more reasonable realizations without oscillation, while computational effort is greatly reduced compared to the direct sampling Monte Carlo method.

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Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

Cited by 121 publications

References 34 publications

A stochastic Galerkin method for the Euler equations with Roe variable transformation

A stochastic Galerkin method for the Euler equations with Roe variable transformation

A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier‐thickener unit

Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

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