Goal-based adaptive control of stochastic and deterministic errors in compressible CFD

The purpose of this paper is to control both deterministic discretisation error and stochastic error for compressible CFD simulations when a quantity of interest is targeted. Recent works [2,4] have extended goal-based methodologies to problems with uncertain data, mostly by extending the wellknown dual-weighted a posteriori error estimation to the stochastic error. Nevertheless several nontrivial questions remain to be adressed. Our goal is to build a goal-oriented method for anisotropic adaptive control of both deterministic and stochastic errors. An a priori error estimator is used as refinement criterion for anisotropic meshes. Anisotropy in the deterministic space is controled for a functional of interest via a riemannian metric (hessian) based method (see [1]), while h and h-p refinement will be applied to the stochastic space where an anisotropic refinement criterion based on the local variance is considered (see [3]). Performance of the proposed approach is illustrated on 2D and 3D numerical experiments for shock-dominated flows. We represent the stochastic component of the solution by a piecewise polynomial approximation in a finite probability space. The results obtained in an initial investigation for a transonic flow around a NACA0012 airfoil are shown in the images below: goal-based adapted mesh (left image), stochastic response on uniform meshes (middle image) and stochastic response for adaptive meshes (right image) with uncertain Mach number and angle of attack. The improvement of the stochastic response is visible not only as a decrease of the overall error on the quantity of interest but also by an improvement of the statistics