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Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

Jake J. Harmon, Svetlana Tokareva, Anatoly Zlotnik, Pieter J. Swart

TL;DR

This work tackles uncertainty quantification for hyperbolic conservation laws by developing a predictor-corrector adaptive stochastic finite-volume (SFV) framework that preserves hyperbolicity and conservation without sampling ensembles. The method introduces a flux 1-irregularity–based, anisotropic refinement framework coupled with an enriched-reduced reconstruction pair to drive resource allocation in both physical and stochastic spaces, and it provides a priori convergence bounds for push-forward densities and CDFs. Key contributions include a computable SFV error indicator, a rigorous error analysis, and a practical adaptive algorithm that coarsens and equilibrates discretizations while handling shocks. Numerical experiments on stochastic Burgers' and Euler equations show substantial DoF savings and linear convergence in global quantities, highlighting the method's potential for scalable and accurate uncertainty propagation in complex hyperbolic systems.

Abstract

We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom (DoFs) where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for non-linear hyperbolic PDEs based on Burgers' and Euler's equations.

Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

TL;DR

This work tackles uncertainty quantification for hyperbolic conservation laws by developing a predictor-corrector adaptive stochastic finite-volume (SFV) framework that preserves hyperbolicity and conservation without sampling ensembles. The method introduces a flux 1-irregularity–based, anisotropic refinement framework coupled with an enriched-reduced reconstruction pair to drive resource allocation in both physical and stochastic spaces, and it provides a priori convergence bounds for push-forward densities and CDFs. Key contributions include a computable SFV error indicator, a rigorous error analysis, and a practical adaptive algorithm that coarsens and equilibrates discretizations while handling shocks. Numerical experiments on stochastic Burgers' and Euler equations show substantial DoF savings and linear convergence in global quantities, highlighting the method's potential for scalable and accurate uncertainty propagation in complex hyperbolic systems.

Abstract

We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom (DoFs) where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for non-linear hyperbolic PDEs based on Burgers' and Euler's equations.
Paper Structure (16 sections, 7 theorems, 78 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 7 theorems, 78 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. Preliminaries
  5. Refinement Framework
  6. Error Analysis and Adaptivity
  7. Convergence Results for the SFV Method
  8. Example: Transport Equation
  9. Refinement
  10. Coarsening
  11. Equilibration
  12. The Unsteady Enriched-Reduced $\mathbb{B}$ Predictor-Corrector Scheme
  13. Numerical Results
  14. Burgers' equation with initial value uncertainty
  15. Euler equations with initial value uncertainty
  16. ...and 1 more sections

Key Result

Theorem 2.2

\newlabelthm:solution_L10 Let $u$ denote the exact solution to the stochastic IBVP eq:stochastic_cons_law-eq:stochastic_boundary_con, and let $u^y$ be the numerical solution that is exact in $\mathbf{x}$ and discretized in $\mathbf{y}$ of order $\ell$. Similarly, let $u^{xy}$ denote the numerical Then, we have that

Figures (11)

  • Figure 1: An admissible discretization that satisfies the flux 1-irregularity and hierarchy constraints. (a) A multi-level perspective that illustrates the hierarchical refinement structure. (b) A 2-D perspective of the same discretization in (a). Notice that in the stochastic space, the proposed approach permits multiple levels of hanging nodes.
  • Figure 1: Numerical confirmation of the a priori convergence rates of Thms \ref{['thm:pdf_error_convergence']} and \ref{['thm:CDF_convergence_SFV']} for the transport problem in Section \ref{['sec:ex:transport']}. (a) CDF. (b) PDF. Neither quantities are normalized.
  • Figure 1: Initial and terminal states for the Burgers' equation problem of Section \ref{['sec:burger_1D']} with uncertain initial data. (a) The initial state for the problem, i.e., the initial condition \ref{['eq:herty_ic']}. (b) The terminal state at $t=0.35$.
  • Figure 2: Solution characteristics of $u$ at the terminal time $t=0.35$ for the problem of Section \ref{['sec:burger_1D']} as described by the first stochastic moments. The red shaded region denotes the first-to-third quartile confidence region.
  • Figure 3: Refined discretizations at the terminal time $t=0.35$ for the problem of Section \ref{['sec:burger_1D']} with $\mathbf{y}_1\sim\mathcal{B}(2,5)$. (a) The discretizaton for the second coarsest tolerance pair in Table \ref{['tab:tolerances']}. (b) The discretization for the finest tolerance in Table \ref{['tab:tolerances']}.
  • ...and 6 more figures

Theorems & Definitions (25)

  • Remark 2.1
  • Theorem 2.2: (7.11), tokareva2014
  • Theorem 2.3: (7.17), tokareva2014
  • Remark 3.1
  • Remark 3.2
  • Definition 3.3: Forest
  • Definition 3.4: Active cell
  • Remark 3.5
  • Definition 3.6: Flux 1-Irregularity
  • Definition 3.7: Reconstruction
  • ...and 15 more