An Optimal Domain Decomposition Method for the C-Grid Navier-Stokes Jacobian

Abstract

The Arakawa C-grid scheme is perhaps the most well-known discretization of the incompressible Navier-Stokes equations. The different variables (velocity components and pressure) are placed on the faces and in the center of the grid cells, respectively, to achieve good conservation and stability properties. In [De Niet and Wubs, IMA J. Num Anal. 2009] an optimal ordering of the variables for the sequential LU-decomposition of the resulting Jacobian was developed based on observations how fill is generated for such matrices during Gaussian elimination. In [Wubs and Thies, S!MAX 2011] the method was used in a domain decomposition approach and extended by a robust dropping strategy that leads to a preconditioner achieving a grid-independent convergence rate of GMRES. Structure-preserving properties of this incomplete LU (ILU) factorization allow recursive application and achieving optimal complexity of O(N log N) for scalar problems already. In this talk we show that this goal can be achieved for the 3D C-grid Navier-Stokes equations as well by using a special choice of space-filling subdomain shapes (parallelepipeda). We demonstrate results with this novel partitioning approach for both direct factorization and the multi-level ILU method on several thousand CPU cores. Possible applications include fully implicit time integration and bifurcation analysis of fluid dynamics problems.