  We propose a predictor-corrector adaptive method for the simulation of
hyperbolic partial differential equations (PDEs) on networks under general
uncertainty in parameters, initial conditions, or boundary conditions. The
approach is based on the stochastic finite volume (SFV) framework that
circumvents sampling schemes or simulation ensembles while also preserving
fundamental properties, in particular hyperbolicity of the resulting systems
and conservation of the discrete solutions. The initial boundary value problem
(IBVP) on a set of network-connected one-dimensional domains that represent a
pipeline is represented using active discretization of the physical and
stochastic spaces, and we evaluate the propagation of uncertainty through
network nodes by solving a junction Riemann problem. The adaptivity of our
method in refining discretization based on error metrics enables
computationally tractable evaluation of intertemporal uncertainty in order to
support decisions about timing and quantity of pipeline operations to maximize
delivery under transient and uncertain conditions. We illustrate our
computational method using simulations for a representative network.
