  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.
