Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916103497711616 |
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| author | Harmon, Jake J. Tokareva, Svetlana Zlotnik, Anatoly Swart, Pieter J. |
| author_facet | Harmon, Jake J. Tokareva, Svetlana Zlotnik, Anatoly Swart, Pieter J. |
| contents | We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom (DoFs) where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for non-linear hyperbolic PDEs based on Burgers' and Euler's equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws Harmon, Jake J. Tokareva, Svetlana Zlotnik, Anatoly Swart, Pieter J. Numerical Analysis 35L60, 35L67, 65C30, 65M50, 65M60 We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom (DoFs) where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for non-linear hyperbolic PDEs based on Burgers' and Euler's equations. |
| title | Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws |
| topic | Numerical Analysis 35L60, 35L67, 65C30, 65M50, 65M60 |
| url | https://arxiv.org/abs/2401.12880 |