A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids

• A third-order, parallel, implicit reconstructed discontinuous Galerkin method.
• Significant lower memory requirement versus third-order implicit DG (P2).
• Jacobian formulation based on automatic differentiation is easy and robust.
• For the compressible Euler and NavierStokes equations on 3D hybrid grids.
• Fast convergence for a wide rage of flow problems including complex configurations.

A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P2). Also, three approaches — analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flow solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.