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

Wu, Kailiang; Tang, Huazhong; Xiu, Dongbin

doi:10.1016/j.jcp.2017.05.027

Cited by 28 publications

(12 citation statements)

References 42 publications

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“…Efforts have been made to overcome this difficulty. In [45], a strategy to regain hyperbolicity for the quasilinear form of hyperbolic systems is proposed. Hyperbolicity can be regained by multiplying the SG formulation of the system by the left eigenvector matrix of the flux Jacobian matrix, but this approach is limited to quasilinear forms, and the numerical schemes designed for conservative forms cannot be applied directly.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Dai¹,

Epshteyn²,

Narayan³

2021

SIAM J. Sci. Comput.

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Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms of (q x ) 2 /h, (q y ) 2 /h, and (q x q y )/h in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicitypreserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.

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Section: Introductionmentioning

confidence: 99%

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Dai¹,

Epshteyn²,

Narayan³

2021

SIAM J. Sci. Comput.

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“…In conjunction with a stochastic Galerkin projection (cf. [3]) this yields a deterministic and symmetric non-linear hyperbolic system for the stochastic modes of the gPC expansion, which are discretized via standard numerical upwinding.…”

Section: Methodsmentioning

confidence: 99%

Stochastic segmentation on images with uncertain data

Theilen

Preußer

2021

Proc Appl Math & Mech

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The present work considers a stochastic segmentation method on images in the presence of noise within a PDE‐based image processing framework. Classical methods are not able to capture the error propagation of uncertain estimated input data and their impact on the final segmentation result, which can be of great importance for clinical decisions. Therefore, an intrusive generalized polynomial chaos (gPC) expansion for a stochastic level‐set based geodesic active contours method is proposed. Employing an operator splitting and a stochastic Galerkin projection a deterministic and symmetric non‐linear hyperbolic system can be obtained, which can be treated using common numerical methods.

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“…The stochastic Galerkin method has been successfully applied to solve hyperbolic equations [5][6][7][8][9][10][11][12]. It is noticeable that these methods employ finite difference or finite volume methods to discretize the balance laws of the gPC coefficients.…”

Section: Introductionmentioning

confidence: 99%

A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws

Xiao¹,

Kusch²,

Koellermeier³

2021

Preprint

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The study of uncertainty propagation poses a great challenge to design numerical solvers with high fidelity. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. Unlike the finite volume method, the treatments in physical and random space are consistent, e.g., the modal representation of solutions based on an orthogonal polynomial basis and the nodal representation based on solution collocation points. Therefore, the numerical behaviors of the scheme in the phase space can be designed and understood uniformly. A family of filters is extended to multi-dimensional cases to mitigate the well-known Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture stochastic cross-scale flow evolution where resolved and unresolved regions coexist. Numerical experiments including wave propagation, Burgers' shock, one-dimensional Riemann problem, and two-dimensional shock-vortex interaction problem are presented to validate the scheme. The order of convergence of the current scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license [1].

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

Cited by 28 publications

References 42 publications

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Stochastic segmentation on images with uncertain data

A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws

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