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.

翻译：本文针对具有不确定性的非线性偏微分方程双曲系统，提出了一系列新的高阶数值方法。该新方法在半离散有限体积框架下实现，其核心在于将（多维）随机空间中的五阶加权本质无振荡插值技术与物理空间中的二阶分段线性重构相结合。与随机空间中的谱近似方法相比，所提出的方法具有本质无振荡特性，既能避免吉布斯现象，又能保持高阶精度。通过对气体动力学欧拉方程和浅水方程圣维南系统的一系列数值算例进行测试，验证了新方法的有效性。特别地，在浅水方程案例中，该方法还被证明具备良平衡性和保正性。