A method for the uncertainty quantification of nonlinear hyperbolic equations with many uncertain parameters is presented. The method combines the stochastic finite volume method and tensor trains in a novel way: the physical space and time dimensions are kept as full tensors, while all stochastic dimensions are compressed together into a tensor train. The resulting hybrid format has one tensor train for each spatial cell and each time step. The MUSCL scheme is adapted to this hybrid format and the feasibility of the approach using several classical test cases is shown. For the Burgers' equation a convergence study and a comparison with the full tensor train format are done with three stochastic parameters. The equation is then solved for an increasing number of stochastic dimensions. The Euler equations are then considered. A parameter study and a comparison with the full tensor train format are performed with the Sod problem. For a complex application we consider the Shu-Osher problem. The presented method opens new avenues for combining uncertainty quantification with well-known numerical schemes for conservation laws.

翻译：本文提出了一种用于处理具有多个不确定参数的非线性双曲型方程不确定性量化的方法。该方法创新性地结合了随机有限体积法与张量列车格式：物理空间与时间维度保持为完整张量，而所有随机维度则被共同压缩为张量列车格式。由此生成的混合格式在每个空间单元和每个时间步长上对应一个张量列车。MUSCL格式被适配于此混合格式，并通过多个经典测试案例验证了该方法的可行性。针对Burgers方程，在三个随机参数下进行了收敛性研究，并与完整张量列车格式进行了对比。随后在递增随机维度下求解该方程。进一步考虑Euler方程组，通过Sod问题开展参数研究并与完整张量列车格式比较。在复杂应用方面，我们考察了Shu-Osher问题。所提出的方法为将不确定性量化与守恒律的经典数值格式相结合开辟了新途径。