Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

TL;DR

This paper introduces an adaptive predictor-corrector method for hyperbolic PDEs under uncertainty, improving efficiency and accuracy by avoiding sampling and refining discretizations based on physical and stochastic regions.

Contribution

It develops a novel adaptive SFV-based approach that preserves hyperbolicity, provides convergence results for statistical quantities, and employs anisotropic refinement driven by physical and stochastic indicators.

Findings

Successfully applied to Burgers' and Euler's equations.

Achieved accurate uncertainty quantification without sampling.

Demonstrated convergence of statistical estimates.

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.