High Order and Underresolution

Abstract

In this work, the accuracy of high order discontinuous Galerkin discretizations for underresolved problems is investigated. Whereas the superior behavior of high order methods for the well resolved case is undisputed, in case of underresolution, the answer is not as clear. The controversy originates from the fact that order of convergence is a concept for discretization parameters tending to zero, whereas underresolution is synonym for large discretization parameters. However, this work shows that even in the case of underresolution, high order discontinuous Galerkin approximations yield superior efficiency compared to their lower order variants due to the better dispersion and dissipation behavior. It is furthermore shown that a very high order accurate discretization (theoretically 16th order in this case) yields even better accuracy than state-of-the-art large eddy simulation methods for the same number of degrees of freedom for the considered example. This result is particularly surprising since those large eddy simulation methods are tuned specifically to capture coarsely resolved turbulence, whereas the considered high order method can be applied directly to a wide range of other multi-scale problems without additional parameter tuning.