Kolloquium: Michael Herty (RWTH Aachen): „Uncertainty Quantification for Nonlinear Hyperbolic Differential Equations“

Uncertainty Quantification for Nonlinear Hyperbolic Differential Equations – Vortragender: Michael Herty, Rheinisch-Westfälische Technische Hochschule Aachen – Einladender: M. Gugat

Abstract: We are interested in quantifying uncertainties that appear in nonlinear hyperbolic partial differential equations arising in a variety of applications from fluid flow to traffic modeling. A common approach to treat the stochastic components of the solution is by using generalized polynomial chaos expansions. This method was successfully applied in particular for general elliptic and parabolic PDEsas well as linear hyperbolic stochastic equations. More recently, gPC methods have been successfully applied to particular hyperbolic PDEs using the explicit form of nonlinearity or the particularity of the studied system structure as, e.g., in the p-system. While such models arise in many applications, e.g., in atmospheric flows, fluid flows under uncertain gas compositions and shallow water flows, a general gPC theory with corresponding numerical methods are still at large. Typical analytical and numerical challenges that appear for the gPC expanded systems are loss of hyperbolicity and positivity of solutions (like gas density or water depth). Any of those effects might trigger severe instabilities within classical finite-volume or discontinuous Galerkin methods. We will discuss properties and conditions to guarantee stability and present numerical results on selected examples.