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

TL;DR

This paper introduces a novel temporally adaptive hybridized discontinuous Galerkin (HDG) method utilizing embedded DIRK schemes for efficient simulation of time-dependent compressible flows, enhancing accuracy and computational performance.

Contribution

The work extends HDG methods to time-dependent problems using embedded DIRK schemes with adaptive time-stepping, which is a new approach in this context.

Findings

Demonstrates effective performance on linear and nonlinear convection-diffusion equations.

Shows improved accuracy and efficiency with adaptive time-step control.

Validates the method through numerical experiments.

Abstract

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 difference formulae (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.