CWENO Interpolation for Non-Oscillatory Stochastic Collocation in Uncertainty Quantification Problems

TL;DR

This paper evaluates various interpolation methods within stochastic collocation for uncertainty quantification, highlighting CWENO's robustness in handling discontinuities effectively compared to traditional methods.

Contribution

It introduces CWENO interpolation as a robust alternative for stochastic collocation, especially for discontinuous functions in UQ, outperforming traditional polynomial and spline methods.

Findings

CWENO effectively captures sharp gradients without oscillations.

gPC and B-splines perform well with smooth data but oscillate near discontinuities.

Approximation B-splines and SP splines converge more slowly but avoid oscillations.

Abstract

Uncertainty quantification (UQ) in mathematical models is essential for accurately predicting system behavior under variability. This study provides guidance on method selection for reliable UQ across varied functional behaviors in engineering applications. Specifically, we compare several interpolation and approximation methods within a stochastic collocation (SC) framework, namely: generalized polynomial chaos (gPC), B-splines, shape-preserving (SP) splines, and central weighted essentially nonoscillatory (CWENO) interpolation, to reconstruct probability density functions (PDFs) and estimate statistical moments. These methods are assessed for both smooth and discontinuous functions, as well as for the solution of the 1-D Euler and shallow water equations. While gPC and interpolation B-splines perform well with smooth data, they produce oscillations near discontinuities. Approximation B-splines and SP splines, while avoiding oscillations, converge more slowly. In contrast, CWENO interpolation demonstrates high robustness, effectively capturing sharp gradients without oscillations, making it suitable for complex, discontinuous data. Overall, CWENO interpolation emerges as a versatile and effective approach for SC, particularly in handling discontinuities in UQ.