Discontinuous Galerkin methods for incompressible flows


Niklas Fehn, Benjamin Krank and Martin Kronbichler

Discontinuous Galerkin methods combine advantages of both the Finite Volume Method and the Finite Element Method and are well-suited for the numerical solution of problems in the field of Computational Fluid Dynamics (CFD). Discontinuous Galerkin methods are an active field of research due to several promising properties of DG methods as compared to well-established discretization approaches. The most important properties are:

  • High-order accuracy and h/p-adaptivity
  • Geometric flexibility (applicable to arbitrary geometries with curved boundaries)
  • Stability for convection-dominated problems by using consistent and conservative numerical fluxes

We pay particular attention to the performance and computational efficiency of our high-order DG solver. High-order methods have the potential to significantly improve the efficiency of the flow solver as compared to low-order methods. To achieve this goal, an efficient implementation for the evaluation of discretized finite element operators is inevitable. For high polynomial degrees of the shape functions, classical matrix-based approaches often used for low-order methods are no longer feasible. Instead, we developed a highly-efficient matrix-free implementation for tensor product elements that significantly reduces the computational complexity  by the use of so-called sum-factorization techniques. Our implementation exhibits outstanding performance numbers with a throughput of approximately 10^9 DoFs/sec for operator evaluation on a single node of typical HPC clusters. Furthermore, the throughput is almost independent of the polynomial degree of the shape functions rendering high-order methods highly efficient. The main building blocks of our high-performance DG solver for incompressible flows are

  • efficient time integration schemes, e.g., projection-type solution techniques for incompressible flows
  • robust, accurate, and efficient discontinuous Galerkin discretizations
  • efficient solvers and preconditioners including robust multigrid methods with matrix-free smoothing
  • high-performance matrix-free implementation for finite element operator evaluation

Our code specifically targets modern, cache-based computer architecture and shows excellent scaling behavior on current HPC systems such as the SuperMUC in Garching, Germany. We have demonstrated the remarkable parallel efficiency of our code through scaling to the complete phase 1 of SuperMUC on 147,456 compute cores where we achieved a parallel speedup of a factor of 9 to 12 when increasing the computational resources by a factor of 18 which has been awarded with "Extreme Scaling Award 2016 (Finalist)".

Our generic DG discretization approach is not only applicable to laminar flow problems, but also to transitional and turbulent flow problems. In this context, we developed a robust and efficient DG discretization for under-resolved turbulent flows that might be classified as an implicit large eddy simulation (LES) approach and that has been shown to be competitive in accuracy to the most accurate LES approaches currently available.

Below, we show some examples for two-dimensional, high Reynolds number vortex dynamics.

2D driven cavity flow for a Reynolds number of Re=1e5

2D Kelvin-Helmholtz instability for a Reynolds number of Re=1e4

 

A detailed description for turbulent flow problems can be found here.