Essentially Non-oscillatory Stencil Selection and Subcell Resolution in Uncertainty Quantification

Abstract

The essentially non-oscillatory stencil selection and subcell resolution robustness concepts from finite volume methods for computational fluid dynamics are extended to uncertainty quantification for the reliable approximation of discontinuities in stochastic computational problems. These two robustness principles are introduced into the simplex stochastic collocation uncertainty quantification method, which discretizes the probability space using a simplex tessellation of sampling points and piecewise higher-degree polynomial interpolation. The essentially non-oscillatory stencil selection obtains a sharper refinement of discontinuities by choosing the interpolation stencil with the highest polynomial degree from a set of candidate stencils for constructing the local response surface approximation. The subcell resolution approach achieves a genuinely discontinuous representation of random spatial discontinuities in the interior of the simplexes by resolving the discontinuity location in the probability space explicitly and by extending the stochastic response surface approximations up to the predicted discontinuity location. The advantages of the presented approaches are illustrated by the results for a step function, the linear advection equation, a shock tube Riemann problem, and the transonic flow over the RAE 2822 airfoil.