An active flux method for the Euler equations

It is our strong belief that efficient, accuracy-controlled and structure-preserving schemes provide an inevitable tool for simulations of complex transport-dominated compressible flows, in particular in their high Reynolds number limit. A desirable property of numerical methods for fluid mechanical processes, in addition to obtaining high order accurate approximations on highly resolved grids, is to obtain useful approximations even on coarse grids. Active Flux methods, which are truly multi-dimensional fully discrete methods of odd order with compact stencils in space and time, are interesting candidates to achieve this goal. One goal of this project is to combine our expertise in the development of structure-preserving high-order finite volume methods to derive new third order accurate Active Flux methods for multi-dimensional hyperbolic systems that are based on a truly multi-dimensional evolution operator.

Motivated by recent progress in theoretical analysis of the Euler equations, new probabilistic solution concepts are emerging. Our second aim in this project is therefore devoted to a rigorous convergence and error analysis of the proposed Active Flux methods via dissipative measure-valued solutions and the relative energy. By means of K- and statistical-convergences we will approximate the Reynolds stress, turbulent energy dissipation as well as the mean flow quantities and their variances and explore connections to statistical turbulence models studied within SPP 2410.

In addition to rigorous numerical analysis an important outcome of our project will be an efficient parallel computational code for the Active Flux method for multi-dimensional hyperbolic conservation.