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Stochastic physical problems governed by nonlinear conservation laws arechallenging due to solution discontinuities in stochastic and physical space.In this paper, we present a level set method to track discontinuities instochastic space by solving a Hamilton-Jacobi equation. By introducing a speedfunction that vanishes at discontinuities, the iso-zero of the level setproblem coincide with the discontinuities of the conservation law. The levelset problem is solved on a sequence of successively finer grids in stochasticspace. The method is adaptive in the sense that costly evaluations of theconservation law of interest are only performed in the vicinity of thediscontinuities during the refinement stage. In regions of stochastic spacewhere the solution is smooth, a surrogate method replaces expensive evaluationsof the conservation law. The proposed method is tested in conjunction withdifferent sets of localized orthogonal basis functions on simplex elements, aswell as frames based on piecewise polynomials conforming to the level setfunction. The performance of the proposed method is compared to existingadaptive multi-element generalized polynomial chaos methods.

Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems