Random compressible Euler equations: Numerics and its Analysis

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Mathematical models arising in science and engineering inherit several sources of uncertainties, such as model parameters, initial and boundary conditions. In order to predict reliable results, deterministic models are insufficient and more sophisticated methods are needed to analyse the influence of uncertainties on numerical solutions. Progress in analytical results and corresponding design of numerical schemes for elliptic and parabolic equations, have, however, not yet been fully expanded towards the hyperbolic problems. A main obstacle is posed by the nonlinear transport that causes the loss of regularity of a solution that propagates also in the random space and the possible loss of hyperbolicity in intrusive Galerkin methods. Uncertainty quantification of the Euler equations is intrinsically connected to statistical hydrodynamics. The idea of considering random or statistical solutions of compressible fluid flow models is natural in order to describe turbulent fluid behaviour. Our aim in this proposal is to deepen the understanding of random/statistical solutions of compressible Euler equations both from the theoretical as well as numerical point of view. The proposed project aims to achieve three goals: Firstly, we introduce and analyse a new concept of ensemble-averaged solutions, the random dissipative solutions. Their existence will be proved via convergence of suitable uncertainty quantification methods, based on polynomial chaos expansion. To this end, we work with inherently stochastic compactness arguments. In order to quantify errors of numerical approximations the random relative energy inequality will be derived. Applying a set-valued compactness framework, K-convergence, we approximate turbulent Reynolds stress and energy dissipation. Secondly, using the formulation of random dissipative solutions we propose and analyse novel numerical schemes. Here, moment approximations will be used to derive effective equations for the evolution of statistical moments. Thirdly, we study the low Mach number limit of the random weakly-compressible Euler equations by means of asymptotic preserving numerical schemes. Consequently, we address multiscale phenomena in hyperbolic problems, investigate a delicate interplay between randomness and hyperbolic transport and contribute to overarching goals of SPP 2410 to design entropy stable and structure-preserving numerical schemes.