M 1011

The talk will present advances on efficient numerical methods for solving fluid flows with two focus areas, the solution of incompressible turbulent flow and high-Mach number flows with shocks. For turbulent incompressible flows, high-order discontinuous Galerkin methods are used. Based on mathematical ingredients that ensure robustness for marginally resolved flow structures, the talk will present techniques to enforce the incompressibility constraint as well as inter-element continuity of the velocity field, using techniques of H(div)-conforming Raviart-Thomas finite elements. For the efficient implementation, matrix-free operator evaluation with fast evaluation of the finite-element integrals is developed, using the concept of sum factorization. This method utilizes the properties of modern multi-core processors and GPUs in an optimal way by replacing memory-intensive sparse matrices of traditional solvers by redundant computations of integrals with a higher proportion of arithmetic work and less memory transfer. In the second part of my talk, I will discuss the efficient implementation of robust methods for the compressible Euler and Navier-Stokes equations. These methods preserve the invariant domain of the Euler equation and are therefore provably stable also near shocks, at the cost of expensive operations to evaluate limiters and shock indicators along stencils. For the viscous effects, matrix-free multigrid solvers are used.

Prof. Dr. Martin Kronbichler

Ruhruniversität Bochum