Level set methods for stochastic discontinuity detection in nonlinear problems

•Novel level set formulation for discontinuity tracking in stochastic space for problems exhibiting solution discontinuities.•Adaptive level set method on multiple grids in stochastic space.•Surrogate model to reduce the number of expensive evaluations of the conservation law of interest.•Simplex multi-element stochastic basis functions for robust computation of solution statistics.•Application of frames as a more general alternative to classical orthogonal basis functions. Stochastic problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zeros of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods.