Preserving Hyperbolicity in Stochastic Galerkin Method for Uncertainty Quantification

TL;DR

This paper develops a framework to preserve hyperbolicity in stochastic Galerkin methods for nonlinear hyperbolic systems with random inputs, enabling effective model reduction without restrictions on basis functions or random dimensions.

Contribution

It introduces a novel framework for model reduction that maintains hyperbolicity in stochastic Galerkin systems for nonlinear hyperbolic equations, regardless of basis functions or random variable dimensions.

Findings

The framework successfully reduces nonlinear hyperbolic systems to symmetric hyperbolic systems.

It allows flexible basis functions beyond traditional polynomial chaos.

The method is applicable to systems with multiple random variables.

Abstract

We first investigate the structure of the systems derived from the gPC based stochastic Galerkin method for the nonlinear hyperbolic systems with random inputs. This method adopts a generalized Polynomial Chaos (gPC) approximations in the stochastic Galerkin framework, but such approximations to the nonlinear hyperbolic systems do not necessarily yield hyperbolic systems \cite{Lucor2013}. Thus based on the work in \cite{framework}, we propose a framework to carry out the model reduction for the general nonlinear hyperbolic system to derive a final global system. Within this framework, the nonlinear hyperbolic system in one space dimension and the symmetric hyperbolic system in multiple space dimensions are reduced into a symmetric hyperbolic system based on the stochastic Galerkin method. We note that the basis functions in the expansion are not restricted to the random-dependent polynomials as that in gPC method and there is no restriction on the dimensions of the random variables neither.