Towards Robust Time-Accurate Anisotropically Adaptive Hybridized Discontinuous Galerkin Method

Abstract

Metric-based anisotropic mesh adaptation has proven effective for the solution of both steady and unsteady problems in terms of reduced computational time and accuracy gain. Especially for time-dependent problems, its generalization to implicit high-order space and time discretizations is, nevertheless, still a challenging task as it requires a great care to preserve consistency and stability of the numerical solution. In this regard, the objective of the present paper is twofold. First, to devise an accurate unsteady mesh adaptation algorithm, and second, to introduce a new solution transfer between anisotropic meshes, which preserves the local minima and maxima. Our findings are based on a hybridized discontinuous Galerkin (HDG) solver with diagonally implicit Runge-Kutta (DIRK) time integration, whereas the main focus is on problems for two-dimensional Euler equations including moving shocks.