Towards a 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 great care to preserve consistency and stability of the numerical solution. In this regard, the objective of the present work is two-fold. First, we devise an accurate unsteady mesh adaptation algorithm, and second, we 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.