Monotonicity - preserving uncertainty quantification

The need for accurate resolution of shock waves in Computational Fluid Dynamics (CFD) is a major driver in the development of robust numerical methods for approximating discontinuities. The Local Extremum Diminishing (LED) robustness concept has, for example, been introduced into the Finite Volume Method (FVM) for preventing overshoots at discontinuities (Jameson 1993). However, LED schemes have the disadvantage that they reduce to first-order accuracy at physical local extrema in the solution (Jame-son 1995). Monotonicity-Preserving (MP) limiters have therefore been developed which do not affect the accuracy at smooth extrema (Balsara & Shu 2000). A straightforward approach to avoid the clipping of extrema is to turn off the convexity-preserving constraints in cells with a non-monotonic solution (Leonard et al. 1995). These cells can be identified by a change in the sign of the discrete derivatives over the volume, in combination with a tolerance for distinguishing physical extrema from spurious numerical oscillations. Because of the dependence on this threshold value, it can be difficult to differentiate between smooth and non-smooth extrema. Therefore, a second MP approach has been proposed which automatically enlarges the limiter intervals near physical ex-trema in such a way that the solution is continuously dependent on the data (Suresh & Huynh 1997). The robust approximation of discontinuities is also important in Uncertainty Quantifi-cation (UQ), since nonlinearities can result in strong amplification of input uncertainties during their propagation through a computational model. In that respect, the introduction of FVM robustness concepts into multi-element UQ methods seems promising, because their local tessellation of the probability space into multiple subdomains is comparable to spatial FVM discretizations in physical space. These multi-element methods (Babuška et al. Global UQ methods for approximating stochastic discontinuities are, for example, based on entropic variables (Poëtte et al. 2009) and iterative formulations (Poëtte & Lucor 2012). The LED concept has also been extended successfully to UQ in terms of the Local Ex-tremum Conserving (LEC) limiter for interpolation in the Simplex Stochastic Collocation (SSC) method (Witteveen & Iaccarino 2012a,b) to avoid overshoots in the approximation of discontinuous response surfaces. However, it has been observed that the LEC limiter leads to a linear first-order approximation at local extrema in the probability space as well. The MP concept is here introduced into UQ to achieve higher-order approximations of local minima and maxima in the SSC method, while at the same time improving robustness at discontinuities.