Non-intrusive Uncertainty Propagation with Error Bounds for Conservation Laws Containing Discontinuities

Abstract

The propagation of statistical model parameter uncertainty in the numerical approximation of nonlinear conservation laws is considered. Of particular interest are nonlinear conservation laws containing uncertain parameters resulting in stochastic solutions with discontinuities in both physical and random variable dimensions. Using a finite number of deterministic numerical realizations, our objective is the accurate estimation of output uncertainty statistics (e.g. expectation and variance) for quantities of interest such as functionals, graphs, and fields. Given the error in numerical realizations, error bounds for output statistics are derived that may be numerically estimated and included in the calculation of output statistics. Unfortunately, the calculation of output statistics using classical techniques such as polynomial chaos, stochastic collocation, and sparse grid quadrature can be severely compromised by the presence of discontinuities in random variable dimensions. An alternative technique utilizing localized piecewise approximation combined with localized subscale recovery is shown to significantly improve the quality of calculated statistics when discontinuities are present. The success of this localized technique motivates the development of the HYbrid Global and Adaptive Polynomial (HYGAP) method described in Sect. 4.4. HYGAP employs a high accuracy global approximation when the solution data varies smoothly in a random variable dimension and local adaptive polynomial approximation with local postprocessing when the solution is non-smooth. To illustrate strengths and weaknesses of classical and newly proposed uncertainty propagation methods, a number of computational fluid dynamics (CFD) model problems containing various sources of parameter uncertainty are calculated including 1-D Burgers’ equation, subsonic and transonic flow over 2-D single-element and multi-element airfoils, transonic Navier-Stokes flow over a 3-D ONERA M6 wing, and supersonic Navier-Stokes flow over a greatly simplified Saturn-V rocket.