A Statically Condensed Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations

Abstract

We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers.The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advectiondiffusion partial differential equations in conservation form. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to 200 when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system.The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summationby-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

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

“…The selected fluxes yield a compact mesh stencil and are differentiated to obtain an analytical Jacobian. Further details on how the Jacobian can be obtained along with the peculiarities and sparsity patterns resulting from using Gauss-Lobatto nodal points, can be found in our previous works [1,44].…”

“…When treated implicitly, the nonlinear operator F , in equation ( 1) is evaluated for the unknown solutions, Q s+1 . Considering this, equation (1) can then be rewritten as…”

“…We have recently shown an additional advantage of GL-DGSEM [1]: it is well suited for the static condensation approach, whilst the classic Gauss point version is not. In this work we exploit the statically condensed system, to accelerate implicit time advancement with and iterative GMRES solver, and compare the accelerations to the traditional full Jacobian system.…”

“…The remaining references were developed for HDG formulations, where the method decouples the degrees of freedom belonging to the mesh elements from the mesh skeleton, enabling static condensation. However, HDG requires specific numerical fluxes [1,26,29], restricting the use of well known Riemann approximations, such as Roe's. Our static condensation for GL-DGSEM allows any flux.…”

“…In our previous work [1], we showed the detailed implementation of the static condensation approach in GL-DGSEM, and applied the method to solve steady cases using direct solvers and an implicit GMRES with a point-Jacobi preconditioner. In this work, we extend that analysis further by comparing the performance of statically condensed and full Jacobian (noncondensed) systems for Block-Jacobi preconditioner in steady and unsteady problems, and show that the statically condensed system can lead to faster iterative GMRES solves.…”