New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

doi:10.1007/s42967-024-00392-z

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (3): 2011-2044.doi: 10.1007/s42967-024-00392-z

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New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

Alina Chertock¹, Michael Herty², Arsen S. Iskhakov¹, Safa Janajra¹, Alexander Kurganov³, Mária Lukáčová-Medvid'ová⁴

1 Department of Mathematics, North Carolina State University, Raleigh, NC, USA;
2 Department of Mathematics, RWTH Aachen University, Aachen, Germany;
3 Department of Mathematics, Shenzhen International Center for Mathematics, and Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China;
4 Institute of Mathematics, Johannes Gutenberg University Mainz, Mainz, Germany

Received:2023-12-12 Revised:2024-02-19 Accepted:2024-02-20 Published:2024-12-20
Contact: Alexander Kurganov,alexander@sustech.edu.cn;Alina Chertock,chertock@math.ncsu.edu;Michael Herty,herty@igpm.rwth-aachen.de;Arsen S. Iskhakov,aiskhak@ncsu.edu;Safa Janajra,sjanajr@ncsu.edu;Mária Lukáčová-Medvid’ová,lukacova@uni-mainz.de E-mail:alexander@sustech.edu.cn;chertock@math.ncsu.edu;herty@igpm.rwth-aachen.de;aiskhak@ncsu.edu;sjanajr@ncsu.edu;lukacova@uni-mainz.de
Supported by:
The work of A. Chertock was supported in part by the NSF grant DMS-2208438. The work of M. Herty was supported in part by the DFG (German Research Foundation) through 20021702/GRK2326, 333849990/IRTG-2379, HE5386/ 18-1, 19-2, 22-1, 23-1, and under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612. The work of A. Kurganov was supported in part by the NSFC grant 12171226 and the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design, China (No. 2019B030301001). The work of M. Lukáčová-Medvid’ová has been funded by the DFG under the SFB/TRR 146 Multiscale Simulation Methods for Soft Matter Systems. M. Lukáčová-Medvid’ová gratefully acknowledges the support of the Gutenberg Research College of University Mainz and the Mainz Institute of Multiscale Modeling. A. Chertock and A. S. Iskhakov also acknowledge the support of the LeRoy B. Martin, Jr. Distinguished Professorship Foundation. M. Herty and M. Lukáčová- Medvid’ová also acknowledge the support of the DFG through the project 525853336 funded within the Focused Programme SPP 2410 “Hypebrolic Balance Laws: Complexity, Scales and Randomness”.

Abstract

Abstract: In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving.

Key words: Hyperbolic conservation and balance laws with uncertainties, Finite-volume methods, Central-upwind schemes, Weighted essentially non-oscillatory (WENO) interpolations

CLC Number:

Alina Chertock, Michael Herty, Arsen S. Iskhakov, Safa Janajra, Alexander Kurganov, Mária Lukáčová-Medvid'ová. New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 2011-2044.

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New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

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