Intrusive acceleration strategies for uncertainty quantification for hyperbolic systems of conservation laws

Methods for quantifying the effects of uncertainties in hyperbolic problems can be divided into intrusive and non-intrusive techniques. Non-intrusive methods allow the usage of a given deterministic solver in a black-box manner, while being embarrassingly parallel. On the other hand, intrusive modifications allow for certain acceleration techniques. Moreover, intrusive methods are expected to reach a given accuracy with a smaller number of unknowns compared to non-intrusive techniques. This effect is amplified in settings with high dimensional uncertainty. A downside of intrusive methods is the need to guarantee hyperbolicity of the resulting moment system. In contrast to stochastic-Galerkin (SG), the Intrusive Polynomial Moment (IPM) method is able to maintain hyperbolicity at the cost of solving an optimization problem in every spatial cell and every time step. In this work, we propose several acceleration techniques for intrusive methods and study their advantages and shortcomings compared to the non-intrusive Stochastic Collocation method. When solving steady problems with IPM, the numerical costs arising from repeatedly solving the IPM optimization problem can be reduced by using concepts from PDE-constrained optimization. Integrating the iteration from the numerical treatment of the optimization problem into the moment update reduces numerical costs, while preserving local convergence. Additionally, we propose an adaptive implementation and efficient parallelization strategy of the IPM method. The effectiveness of the proposed adaptations is demonstrated for multi-dimensional uncertainties in fluid dynamics applications, resulting in the observation of requiring a smaller number of unknowns to achieve a given accuracy when using intrusive methods. Furthermore, using the proposed acceleration techniques, our implementation reaches a given accuracy faster than Stochastic Collocation. (C) 2020 Elsevier Inc. All rights reserved.