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

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.

Funding

Division of Mathematical Sciences(DMS-2208438); Deutsche Forschungsgemeinschaft(HE5386/ 18-1, 19-2, 22-1, 23-1); Germany’s Excellence Strategy EXC-2023 Internet of Production(390621612); National Natural Science Foundation of China(12171226); Guangdong Provincial Key Laboratory Of Computational Science And Material Design(2019B030301001); Deutsche Forschungsgemeinschaft(525853336 funded within the Focused Programme SPP 2410 “Hypebrolic Balance Laws: Complexity, Scales and Randomness”); LeRoy B. Martin, Jr. Distinguished Professorship Foundation; LeRoy B. Martin, Jr. Distinguished Professorship Foundation