A Realizable Filtered Intrusive Polynomial Moment Method

DOI10.1016/J.CAM.2021.114055arXiv2105.07473MaRDI QIDQ6367766

Ryan G. McClarren, Graham Alldredge, Jonas Kusch, Martin Frank

Publication date: 16 May 2021

Abstract: Intrusive uncertainty quantification methods for hyperbolic problems exhibit spurious oscillations at shocks, which leads to a significant reduction of the overall approximation quality. Furthermore, a challenging task is to preserve hyperbolicity of the gPC moment system. An intrusive method which guarantees hyperbolicity is the intrusive polynomial moment (IPM) method, which performs the gPC expansion on the entropy variables. The method, while still being subject to oscillations, requires solving a convex optimization problem in every spatial cell and every time step. The aim of this work is to mitigate oscillations in the IPM solution by applying filters. Filters reduce oscillations by damping high order gPC coefficients. Naive filtering, however, may lead to unrealizable moments, which means that the IPM optimization problem does not have a solution and the method breaks down. In this paper, we propose and analyze two separate strategies to guarantee the existence of a solution to the IPM problem. First, we propose a filter which maintains realizability by being constructed from an underlying Fokker-Planck equation. Second, we regularize the IPM optimization problem to be able to cope with non-realizable gPC coefficients. Consequently, standard filters can be applied to the regularized IPM method. We demonstrate numerical results for the two strategies by investigating the Euler equations with uncertain shock structures in one- and two-dimensional spatial settings. We are able to show a significant reduction of spurious oscillations by the proposed filters.

Mathematics Subject Classification ID

Inference from stochastic processes and prediction (62M20) Numerical optimization and variational techniques (65K10) Smoothness and regularity of solutions to PDEs (35B65) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs (35B05) Hyperbolic conservation laws (35L65) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) A priori estimates in context of PDEs (35B45) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60) PDEs with randomness, stochastic partial differential equations (35R60) Numerical solutions to stochastic differential and integral equations (65C30) Kinetic theory of gases in time-dependent statistical mechanics (82C40) Euler equations (35Q31) Fokker-Planck equations (35Q84)

This page was built for publication: A Realizable Filtered Intrusive Polynomial Moment Method

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6367766)