Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

TL;DR

This paper introduces a level set method to efficiently detect and track solution discontinuities in stochastic nonlinear conservation laws, improving computational efficiency through adaptive refinement and surrogate modeling.

Contribution

It presents a novel level set approach for discontinuity detection in stochastic spaces, integrating adaptive refinement and surrogate models for enhanced efficiency.

Findings

Effective discontinuity detection in stochastic space.

Improved computational efficiency over existing methods.

Successful application with various basis functions.

Abstract

Stochastic physical 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-zero 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.