Intrusive methods in uncertainty quantification and their connection to kinetic theory

Uncertainty quantification for hyperbolic equations is a challenging task, since solutions exhibit discontinuities and sharp gradients. The commonly used stochastic-Galerkin (SG) Method uses polynomials to represent the solution, leading to oscillatory approximations due to Gibbs phenomenon. Additionally, the SG moment systems can loose hyperbolicity. The intrusive polynomial moment method (IPMM) yields a general framework for intrusive methods while ensuring hyperbolicity of the moment system and restricting oscillatory over- and undershoots to specified bounds. In this contribution, similarities as well as differences of the IPMM to minimal entropy closures used in transport theory will be discussed. We apply filters, which are well-known in kinetic theory to the SG Method to limit oscillatory overshoots. By investigating Burgers’ equation, we demonstrate that the use of filters improves the results of stochastic Galerkin. The IPMM method shows better approximation results than the filter, but comes at a higher computational cost.