We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom (DoFs) where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for non-linear hyperbolic PDEs based on Burgers' and Euler's equations.

翻译：我们提出了一种用于不确定性下双曲偏微分方程（PDEs）研究的预测-校正自适应方法。该方法基于随机有限体积（SFV）方法框架构建，避免了采样方案或模拟集成，同时保留了基本性质，特别是所得系统的双曲性和离散解的守恒性。此外，我们通过数值实验证明，将现有SFV理论扩展至统计量（特别是前推密度）的先验收敛结果。通过将细化指标与物理空间和随机空间的区域相关联，我们驱动离散化的各向异性细化，在认为有益的位置引入新的自由度（DoFs）。为说明所提方法，我们基于Burgers方程和Euler方程，考虑了一系列非线性双曲PDE的数值算例。