Level set methods for stochastic discontinuity detection in nonlinear problems

Per Pettersson, Alireza Doostan, Jan Nordström

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)511-531
Number of pages21
JournalJournal of Computational Physics
Publication statusPublished - 1 Sept 2019
Externally publishedYes


  • Discontinuity tracking
  • Hyperbolic PDEs
  • Level set methods
  • Polynomial chaos
  • Uncertainty quantification

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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