The stochastic finite volume method offers an efficient one-pass approach for assessing uncertainty in hyperbolic conservation laws. Still, it struggles with the curse of dimensionality when dealing with multiple stochastic variables. We introduce the stochastic finite volume method within the tensor-train framework to counteract this limitation. This integration, however, comes with its own set of difficulties, mainly due to the propensity for shock formation in hyperbolic systems. To overcome these issues, we have developed a tensor-train-adapted stochastic finite volume method that employs a global WENO reconstruction, making it suitable for such complex systems. This approach represents the first step in designing tensor-train techniques for hyperbolic systems and conservation laws involving shocks.

翻译：随机有限体积法为评估双曲守恒律中的不确定性提供了一种高效的单次通过方法。然而，当处理多个随机变量时，该方法面临维数灾难的挑战。为克服这一局限，我们将随机有限体积法引入张量列框架。然而，这种集成也带来了自身的一系列困难，主要归因于双曲系统中激波形成的倾向。为应对这些问题，我们提出了一种适用于张量列的随机有限体积法，该方法采用全局WENO重构，使其适用于此类复杂系统。该方法是设计面向含激波双曲系统及守恒律的张量列技术的第一步。