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

Andrés M. Rueda-Ramírez and Esteban Ferrer and David A. Kopriva and Gonzalo Rubio and Eusebio Valero
2019 arXiv   pre-print
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
more » ... 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 advection-diffusion 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 summation-by-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.
arXiv:1911.02366v2 fatcat:vlpoqeeia5gatfxsx36islttly