This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.

中文翻译：

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随机守恒定律系统的后验误差分析和自适应非侵入式数值方案

本文考虑双曲守恒定律的一维随机系统。建立了涉及随机初始数据和随机通量函数的守恒定律初值问题随机熵容许解的存在性和唯一性。基于这些结果，提出了随机熵解的数值近似的后验误差分析。对于随机离散化，使用了一种非侵入式方法，即随机搭配方法。时空离散化依赖于 Runge-Kutta Discontinuous Galerkin 方法。使用离散解的平滑重建推导出后验估计量。结合相对熵稳定性框架，这产生了整个空间随机离散化误差的可计算误差界限。估计器允许拆分为随机和确定性（时空）部分，从而实现一种新颖的基于残差的空间随机自适应网格细化算法。研究了残差的缩放特性，并在各种数值示例中说明了所提出的自适应算法的效率。