The matrix-free macro-element hybridized Discontinuous Galerkin method for steady and unsteady compressible flows

TL;DR

The paper develops a matrix-free macro-element HDG method for steady and unsteady compressible flows, integrating second-layer static condensation to shrink local solvers and a multilevel FGMRES preconditioner for the global system. Time integration is performed with diagonally implicit Runge–Kutta schemes, enabling stable, high-CFL simulations. Numerical tests on Couette flow, flow past a sphere, and Taylor–Green vortex show that macro-element HDG reduces global degrees of freedom and iteration counts, shifts computational load toward local solves, and delivers competitive accuracy at moderate polynomial degrees. The approach exhibits strong scalability and memory efficiency, making it a promising option for large-scale compressible-flow simulations and future high-Re turbulence studies.

Abstract

The macro-element variant of the hybridized discontinuous Galerkin (HDG) method combines advantages of continuous and discontinuous finite element discretization. In this paper, we investigate the performance of the macro-element HDG method for the analysis of compressible flow problems at moderate Reynolds numbers. To efficiently handle the corresponding large systems of equations, we explore several strategies at the solver level. On the one hand, we devise a second-layer static condensation approach that reduces the size of the local system matrix in each macro-element and hence the factorization time of the local solver. On the other hand, we employ a multi-level preconditioner based on the FGMRES solver for the global system that integrates well within a matrix-free implementation. In addition, we integrate a standard diagonally implicit Runge-Kutta scheme for time integration. We test the matrix-free macro-element HDG method for compressible flow benchmarks, including Couette flow, flow past a sphere, and the Taylor-Green vortex. Our results show that unlike standard HDG, the macro-element HDG method can operate efficiently for moderate polynomial degrees, as the local computational load can be flexibly increased via mesh refinement within a macro-element. Our results also show that due to the balance of local and global operations, the reduction in degrees of freedom, and the reduction of the global problem size and the number of iterations for its solution, the macro-element HDG method can be a competitive option for the analysis of compressible flow problems.

Figures (10)

• Figure 1: Sample grid partitioned into non-overlapping subdomains.
• Figure 2: Compressible Couette Flow on a unit cube.
• Figure 3: Couette flow: convergence of the macro-element HDG method with $m = 2$.
• Figure 4: Flow past a sphere: cut halfway through the unstructured mesh, adaptively refined around the sphere in the center.
• Figure 5: Flow past a sphere at $Re = 100$ and $M_{\infty}=0.1$, computed on the mesh shown in Figure \ref{['fig:sphere mesh']} at $p=3$.

Theorems & Definitions (4)

• remark 1
• remark 2
• remark 3
• remark 4