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.
Cite
@article{arxiv.2312.08280,
title = {New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties},
author = {Alina Chertock and Michael Herty and Arsen S. Iskhakov and Safa Janajra and Alexander Kurganov and Maria Lukacova-Medvidova},
journal= {arXiv preprint arXiv:2312.08280},
year = {2024}
}