A Higher-order Hybridisable Discontinuous Galerkin IMEX method for the incompressible Euler equations

Abstract

The incompressible Euler equations are an important model system in computational fluid dynamics. Fast high-order methods for the solution of this time-dependent system of partial differential equations are of particular interest: due to their exponential convergence in the polynomial degree they can make efficient use of computational resources. To address this challenge we describe a novel timestepping method which combines a hybridised Discontinuous Galerkin method for the spatial discretisation with IMEX timestepping schemes, thus achieving high-order accuracy in both space and time. The computational bottleneck is the solution of a (block-) sparse linear system to compute updates to pressure and velocity at each stage of the IMEX integrator. Following Chorin's projection approach, this update of the velocity and pressure fields is split into two stages. As a result, the hybridised equation for the implicit pressure-velocity problem is reduced to the well-known system which arises in hybridised mixed formulations of the Poisson- or diffusion problem and for which efficient multigrid preconditioners have been developed. Splitting errors can be reduced systematically by embedding this update into a preconditioned Richardson iteration. The accuracy and efficiency of the new method is demonstrated numerically for two time-dependent testcases that have been previously studied in the literature.

Figures (11)

• Figure 1: Two neighbouring cells $K^+$, $K^-$ with corresponding outward normals $n^+$, $n^-$.
• Figure 2: Advection of passive tracer in the Taylor Green vortex at different times on a $32\times 32$ grid with polynomial degree $k=2$. The results were obtained with the IMEX-HDG discretisation and a SSP2(3,3,2) timestepper with timestep size $\Delta t=0.005$. The decay constant in the forcing function in \ref{['eqn:taylor_green_Q0_f']} is set to $\kappa=0.1$.
• Figure 3: Vorticity for double layer shear flow at time $t=6$ on two different grids. The two upper plots show results obtained with the original method in Guzman2016.
• Figure 4: Vorticity for double layer shear flow at time $t=8$ on two different grids. The two upper plots show results obtained with the original method in Guzman2016.
• Figure 5: Spatial distribution of the error in velocity (left) and pressure (right) on a $32\times 32$ grid. Results are obtained with a SSP3(4,3,3) timestepper and a polynomial degree of $k=3$.