In this paper, we study the stochastic collocation (SC) methods for uncertainty quantification (UQ) in hyperbolic systems of nonlinear partial differential equations (PDEs). In these methods, the underlying PDEs are numerically solved at a set of collocation points in random space. A standard SC approach is based on a generalized polynomial chaos (gPC) expansion, which relies on choosing the collocation points based on the prescribed probability distribution and approximating the computed solution by a linear combination of orthogonal polynomials in the random variable. We demonstrate that this approach struggles to accurately capture discontinuous solutions, often leading to oscillations (Gibbs phenomenon) that deviate significantly from the physical solutions. We explore alternative SC methods, in which one can choose an arbitrary set of collocation points and employ shape-preserving splines to interpolate the solution in a random space. Our study demonstrates the effectiveness of spline-based collocation in accurately capturing and assessing uncertainties while suppressing oscillations. We illustrate the superiority of the spline-based collocation on two numerical examples, including the inviscid Burgers and shallow water equations.

翻译：本文研究了随机配置（SC）方法在非线性双曲型偏微分方程（PDE）系统不确定性量化（UQ）中的应用。在该方法中，基础PDE在随机空间中的一组配置点上进行数值求解。标准SC方法基于广义多项式混沌（gPC）展开，其关键在于根据预设概率分布选择配置点，并通过随机变量的正交多项式线性组合逼近计算解。我们证明该方法难以准确捕捉不连续解，常导致显著偏离物理解的振荡（吉布斯现象）。我们探索了替代性SC方法，允许选择任意配置点集，并采用保形样条函数在随机空间中对解进行插值。研究表明，基于样条函数的配置方法能在抑制振荡的同时，有效捕捉和评估不确定性。我们通过无黏Burgers方程和浅水方程两个数值算例，阐明了基于样条函数配置方法的优越性。