Roe solver with entropy corrector for uncertain hyperbolic systems

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

doi:10.1016/j.cam.2010.05.043

Cited by 28 publications

(13 citation statements)

References 11 publications

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“…The extension of the present Roe solvers to include entropy correctors is possible. Details are reported elsewhere [35].…”

Section: Resultsmentioning

confidence: 99%

“…It is possible to diagonalize A and to compute jAj using (35), but in practice this method is extremely costly. A more interesting method has been proposed in [29], which consists in computing a sequence of polynomial iterations based on the exact knowledge of the eigenvalues (or at last an explicit bound), and converging to the matrix sign if all the eigenvalues are real.…”

Section: An Efficient Methods For Approximating the Absolute Value Of mentioning

confidence: 99%

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Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

Tryoen

Maître

Ndjinga

et al. 2010

Journal of Computational Physics

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121

106

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a b s t r a c tThis paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.

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“…The extension of the present Roe solvers to include entropy correctors is possible. Details are reported elsewhere [35].…”

Section: Resultsmentioning

confidence: 99%

Section: An Efficient Methods For Approximating the Absolute Value Of mentioning

confidence: 99%

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

Tryoen

Maître

Ndjinga

et al. 2010

Journal of Computational Physics

Self Cite

121

106

View full text Add to dashboard Cite

show abstract

“…4.1 is obtained by assuming that the Y -derivatives of the solution up to order s, which appear in the coefficients C andĈ in the error estimates (24) and (28), are uniformly bounded with respect to h. In the absence of such assumption, we can employ the estimate (40) and find the coefficients in the error bounds. For instance, for the full tensor interpolation, the coefficient C in the interpolation error (24) is We now define for every Y ∈ the power series u h :…”

Section: Remark 10mentioning

confidence: 99%

“…In this more general case, the solution of linear hyperbolic problems may have lower regularity than those of elliptic, parabolic and hyperbolic problems with constant random coefficients. There are also recent works on stochastic nonlinear conservation laws, see for instance [25,26,34,39,40].…”

Section: Introductionmentioning

confidence: 99%

A stochastic collocation method for the second order wave equation with a discontinuous random speed

2012

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In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general This work was supported by the King Abdullah University of Science and Technology (AEA project "Bayesian earthquake source validation for ground motion simulation"), the VR project "Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar", and the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). The second author was partially supported by the Italian grant FIRB-IDEAS (Project no. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".

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“…[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%

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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Roe solver with entropy corrector for uncertain hyperbolic systems

Cited by 28 publications

References 11 publications

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

A stochastic collocation method for the second order wave equation with a discontinuous random speed

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

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