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Entropy–Based Methods for Uncertainty Quantification of Hyperbolic Conservation Laws

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Recent Advances in Numerical Methods for Hyperbolic PDE Systems

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 28))

Abstract

Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as non-hyperbolic moment systems. Entropy-based Stochastic Galerkin methods, on the other hand, guarantee hyperbolicity and entropy decay. A key challenge facing these methods is computational cost, since they require repeatedly solving a non-linear optimization problem. Furthermore, the spatial and temporal discretization needs to preserve realizability, meaning that the existence of a unique solution to the optimization problem must be ensured. We review strategies to guarantee realizability, which use a special choice of the numerical flux while considering errors from the optimization solve. Most importantly, we indicate how intrusive entropy-based closures can be made competitive. We show several numerical test cases and discuss the advantages and disadvantages of several uncertainty propagation methods.

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Notes

  1. 1.

    Note that we have prescribed \(\boldsymbol{u}\) to be in \(\mathbb {R}^m\), i.e. strictly speaking we have \(\boldsymbol{u}(\boldsymbol{v}) = \left( \nabla _{\boldsymbol{u}} s \right) ^{-T}(\boldsymbol{v})\).

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Acknowledgment

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – FR 2841/6-1.

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Correspondence to Martin Frank .

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Frank, M., Kusch, J., Wolters, J. (2021). Entropy–Based Methods for Uncertainty Quantification of Hyperbolic Conservation Laws. In: Muñoz-Ruiz, M.L., Parés, C., Russo, G. (eds) Recent Advances in Numerical Methods for Hyperbolic PDE Systems. SEMA SIMAI Springer Series, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-030-72850-2_2

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