A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows

• We combine the hybridized discontinuous Galerkin method with embedded DIRK schemes.
• We investigate the use of time-step control in this setting.
• The most suitable DIRK method is identified.

The potential of the hybridized discontinuous Galerkin (HDG) method has been recognized for the computation of stationary flows. Extending the method to time-dependent problems can, e.g., be done by backward differentiation formula (BDF) or diagonally implicit Runge–Kutta (DIRK) methods. In this work, we investigate the use of embedded DIRK methods in an HDG solver, including the use of adaptive time-step control. Numerical results demonstrate the performance of the method for both linear and nonlinear (systems of) time-dependent convection–diffusion equations.