Advantages of static condensation in implicit compressible Navier-Stokes DGSEM solvers

We consider implicit time-marching schemes for the compressible Navier-Stokes equations, discretised using the Discontinuous Galerkin Spectral Element Method with Gauss-Lobatto nodal points (GL-DGSEM). We compare classic implicit strategies for the full Jacobian system to our recently developed static condensation technique for GL-DGSEM Rueda-Ramírez et al.(2019), A Statically Condensed Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations [1]. The Navier-Stokes system is linearised using a Newton-Raphson method and solved using an iterative preconditioned-GMRES solver. Both the full and statically condensed systems benefit from a Block-Jacobi preconditioner. We include theoretical estimates for the various costs involved (i.e. calculation of full and condensed Jacobians, factorising and inverting the preconditioners, GMRES steps and overall costs) to clarify the advantages of using static condensation in GL-DGSEM, for varying polynomial orders. These estimates are then examined for a steady three-dimensional manufactured solution problem and for an two-dimensional unsteady laminar flow over a NACA0012 airfoil. In all cases, we test the schemes for high polynomial orders, which range from 2 to 8 for a manufactured solution case and from 2 to 5 for the NACA0012 airfoil. The statically condensed system shows computational savings, which relate to the smaller system size and cheaper Block-Jacobi preconditioner with smaller blocks and better polynomial scaling, when compared to the preconditioned full Jacobian system (not condensed).