On the quantification of discretization uncertainty: comparison of two paradigms

TL;DR

This paper compares two paradigms for quantifying discretization uncertainty in numerical PDE models: the traditional grid convergence index and a Bayesian Gaussian process approach, evaluating their effectiveness on standard test cases.

Contribution

It provides a comparative analysis of GCI and Bayesian methods for discretization uncertainty, highlighting the Bayesian approach's potential for better probabilistic foundations.

Findings

Bayesian approach offers improved probabilistic interpretation.

GCI remains a standard but less probabilistically rigorous method.

Both methods are validated on classical test cases.

Abstract

Numerical models based on partial differential equations (PDE), or integro-differential equations, are ubiquitous in engineering and science, making it possible to understand or design systems for which physical experiments would be expensive-sometimes impossible-to carry out. Such models usually construct an approximate solution of the underlying continuous equations, using discretization methods such as finite differences or the finite elements method. The resulting discretization error introduces a form of uncertainty on the exact but unknown value of any quantity of interest (QoI), which affects the predictions of the numerical model alongside other sources of uncertainty such as parametric uncertainty or model inadequacy. The present article deals with the quantification of this discretization uncertainty.A first approach to this problem, now standard in the V\&V (Verification and Validation) literature, uses the grid convergence index (GCI) originally proposed by P. Roache in the field of computational fluid dynamics (CFD), which is based on the Richardson extrapolation technique. Another approach, based on Bayesian inference with Gaussian process models, was more recently introduced in the statistical literature. In this work we present and compare these two paradigms for the quantification of discretization uncertainty, which have been developped in different scientific communities, and assess the potential of the-younger-Bayesian approach to provide a replacement for the well-established GCI-based approach, with better probabilistic foundations. The methods are illustrated and evaluated on two standard test cases from the literature (lid-driven cavity and Timoshenko beam).