通量系数由时空随机场给出的双曲守恒律的不确定性量化
Uncertainty quantification for hyperbolic conservation laws with flux coefficients given by spatiotemporal random fields
AI总结：
本文针对通量系数由时空随机场（空间相关高斯场、时间Ornstein-Uhlenbeck过程）描述的双曲守恒律，提出基于Monte Carlo有限体积法的离散框架，用于稳健计算解的各阶矩，并在磁感应方程和线性声学中验证。
摘要：
本文讨论具有随机系数的双曲型偏微分方程。我们考虑通量函数系数由时空随机场建模的挑战性问题。这些场在空间上由相关高斯随机场给出，在时间上由Ornstein-Uhlenbeck过程给出。由此得到的方程组由每个随机参数的随机微分方程与双曲守恒律耦合组成。我们在该设定下定义了适当的解概念，并分析了离散化方法的误差与收敛性。提出了一种基于Monte Carlo Finite Volume方法的新型离散化框架，用于稳健计算这些随机双曲型偏微分方程解的各阶矩。我们在两个应用中出现的例子上展示了该方法：磁感应方程和线性声学，两者均具有时空随机背景速度场。
英文摘要：
In this paper hyperbolic partial differential equations with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein-Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equation for each random parameter coupled to the hyperbolic conservation law. We define an appropriate solution concept in his setting and analyze errors and convergence of discretization methods. A novel discretization framework, based on Monte Carlo Finite Volume methods, is presented for the robust computation of moments of solutions to those random hyperbolic partial differential equations. We showcase the approach on two examples which appear in applications: The magnetic induction equation and linear acoustics, both with a spatiotemporal random background velocity field.
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