Hyperbolicity-Preserving Stochastic Galerkin Methods for Conservation Laws Based on Associative Truncated Products on Polynomial Spaces
Stochastic Galerkin discretizations of nonlinear hyperbolic conservation laws may lose hyperbolicity because the standard pseudospectral product is generally nonassociative, leading to non-commuting blocks in the flux Jacobian matrix. We develop a novel framework for constructing hyperbolicity-preserving stochastic Galerkin systems based on associative truncated products on polynomial spaces. In one stochastic dimension, we characterize associative truncated products through a single polynomial datum and identify examples with useful symmetry, positivity, and spectral properties, including collocation products and an associative symmetric product based on Gaussian quadrature nodes. We prove a consistency result showing that, under suitable projection-error assumptions, these products converge to the classical product as the polynomial degree grows. For systems with rational fluxes, we derive sufficient conditions under which the resulting stochastic Galerkin flux remains hyperbolic on the corresponding admissible set. Applications to the one-dimensional isothermal and compressible Euler equations show accurate statistical approximation and robust hyperbolicity preservation of the computed stochastic Galerkin states.