A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

Rohde, Christian; Giesselmann, Jan

doi:10.1093/imanum/drz004

Cited by 9 publications

(7 citation statements)

References 43 publications

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“…• We make use of adaptivity in stochastic space, which is one core advantage of intrusive methods [24,31,36,54]. To ensure that a large number of time iterations is performed on a low refinement level (i.e.…”

Section: Multi-element Intrusive Polynomial Moment Methodsmentioning

confidence: 99%

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Oscillation Mitigation of Hyperbolicity-Preserving Intrusive Uncertainty Quantification Methods for Systems of Conservation Laws

Kusch¹,

Schlachter²

2020

Preprint

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In this article we study intrusive uncertainty quantification schemes for systems of conservation laws with uncertainty. Standard intrusive methods lead to oscillatory solutions which sometimes even cause the loss of hyperbolicity. We consider the stochastic Galerkin scheme, in which we filter the coefficients of the polynomial expansion in order to reduce oscillations. We further apply the multi-element approach and ensure the preservation of hyperbolic solutions through the hyperbolicity limiter. In addition to that, we study the intrusive polynomial moment method, which guarantees hyperbolicity at the cost of solving an optimization problem in every spatial cell and every time step. To reduce numerical costs, we apply the multi-element ansatz to IPM. This ansatz decouples the optimization problems of all multi elements. Thus, we are able to significantly decrease computational costs while improving parallelizability. We finally evaluate these oscillation mitigating approaches on various numerical examples such as a NACA airfoil and a nozzle test case for the two-dimensional Euler equations. In our numerical experiments, we observe the mitigation of spurious artifacts. Furthermore, using the multi-element ansatz for IPM significantly reduces computational costs.

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Section: Multi-element Intrusive Polynomial Moment Methodsmentioning

confidence: 99%

“…the truncation order of the gPC polynomials adapts locally to the smoothness of the solution, cf. [24,31,36,54].…”

Section: Introductionmentioning

confidence: 99%

Oscillation Mitigation of Hyperbolicity-Preserving Intrusive Uncertainty Quantification Methods for Systems of Conservation Laws

Kusch¹,

Schlachter²

2020

Preprint

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“…They correspond P almost everywhere to entropy solutions in the deterministic case which is discussed in Kružkov's theorem [35]. For more information on this topic, we refer to [39,42].…”

Section: Discontinuous Galerkinmentioning

confidence: 96%

Parameter identification in uncertain scalar conservation laws discretized with the discontinuous stochastic Galerkin Scheme

Schlachter¹,

Totzeck²

2019

Preprint

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We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.

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“…The filtered SG and IPM methods are proposed in [11,35], where a filtering step is applied to the solution in between time steps. In addition, stochastic adaptivity [36][37][38][39] can be employed to increase the truncation order in oscillatory regions. For the IPM method, certain choices of the entropy mitigate oscillations [40].…”

Section: Challengementioning

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 posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

Cited by 9 publications

References 43 publications

Oscillation Mitigation of Hyperbolicity-Preserving Intrusive Uncertainty Quantification Methods for Systems of Conservation Laws

Oscillation Mitigation of Hyperbolicity-Preserving Intrusive Uncertainty Quantification Methods for Systems of Conservation Laws

Parameter identification in uncertain scalar conservation laws discretized with the discontinuous stochastic Galerkin Scheme

A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws

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