Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function

Abstract

This work is devoted to the Galerkin projection of highly nonlinear random quantities. The dependency on a random input is described by Haar-type wavelet systems. The classical Haar sequence has been used by Pettersson, Iaccarino, Nordström (2014) for a hyperbolic stochastic Galerkin formulation of the one-dimensional Euler equations. This work generalizes their approach to several multi-dimensional systems with Lipschitz continuous and non-polynomial flux functions. Theoretical results are illustrated numerically by a genuinely multidimensional CWENO reconstruction.

Energy stable and structure-preserving schemes for the stochastic Galerkin shallow water equations

“…Despite this challenge, recent work has investigated numerical methods for stochastic Galerkin methods for various types of hyperbolic conservation laws. Some advances include SG-type analysis and algorithms for scalar conservation laws [46] including well-balanced methods [22], Haar wavelet-based SG approaches [20], hyperbolicity preservation through a non-equivalent Roe variables formulation [19], filtering strategies for SG systems [26], limiter-type methods to maintain hyperbolicity [33], hyperbolicity formulations for linear problems [31] or using linearization techniques [38], splitting type approaches for SWE models [7], non-conservative formulations of SG SWE systems [5] and approximate representations through entropic variables [30].…”