A variety of real-world applications are modeled via hyperbolic conservation laws. To account for uncertainties or insufficient measurements, random coefficients may be incorporated. These random fields may depend discontinuously on the state space, e.g., to represent permeability in a heterogeneous or fractured medium. We introduce a suitable admissibility criterion for the resulting stochastic discontinuous-flux conservation law and prove its well-posedness. Therefore, we ensure the pathwise existence and uniqueness of the corresponding deterministic setting and present a novel proof for the measurability of the solution, since classical approaches fail in the discontinuous-flux case. As an example of the developed theory, we present a specific advection coefficient, which is modeled as a sum of a continuous random field and a pure jump field. This random field is employed in the stochastic conservation law, in particular a stochastic Burgers' equation, for numerical experiments. We approximate the solution to this problem via the Finite Volume method and introduce a new meshing strategy that accounts for the resulting standing wave profiles caused by the flux-discontinuities. The ability of this new meshing method to reduce the sample-wise variance is demonstrated in numerous numerical investigations.
翻译:各种现实世界应用都通过超曲调保护法进行模型化。为了说明不确定因素或测量不足的原因,可以纳入随机系数。这些随机字段可能不连贯地依赖于国家空间,例如,代表异质或断裂介质中的渗透性。我们为由此产生的随机不连续流动保护法引入了适当的可接受性标准,并证明了其妥善的适应性。因此,我们确保相应的确定性设置的路径性存在和独特性,并为解决办法的可衡量性提出新的证据,因为传统的方法在不连续流动情况下失败。作为发展理论的一个实例,我们提出了一个特定的适应系数,作为连续随机字段和纯跳跃场的总和。这个随机字段用于随机保护法,特别是随机汉堡的方程式等,用于数字实验。我们通过精度量法来比较解决这一问题的解决方案,并引入一种新的模拟战略,用以计算由此造成的恒定波剖面变化造成的常态波谱。在抽样调查中展示的这种新模型能力是众多的。