A linear multigrid preconditioner for the solution of the Navier-Stokes equations using a discontinuous Galerkin discretization

• Main Author: Diosady, Laslo Tibor
• Other Authors: David Darmofal.
• Format: Others
• Language: English
• Published: Massachusetts Institute of Technology 2008
• Subjects: Computation for Design and Optimization Program.
• Online Access: http://dspace.mit.edu/handle/1721.1/39326; http://hdl.handle.net/1721.1/39326

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007. === This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. === Includes bibliographical references (p. 69-72). === A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element Line-Jacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(O)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling used for the element Line-Jacobi preconditioner. This reordering is shown to be far superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and a novel algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. A linear p-multigrid algorithm using element Line-Jacobi, and Block-ILU(O) smoothing is presented as a preconditioner to GMRES. === (cont.) The coarse level Jacobians are obtained using a simple Galerkin projection which is shown to closely approximate the linearization of the restricted problem except for perturbations due to artificial dissipation terms introduced for shock capturing. The linear multigrid preconditioner is shown to significantly improve convergence in terms of the number of linear iterations as well as to reduce the total CPU time required to obtain a converged solution. A parallel implementation of the linear multi-grid preconditioner is presented and a grid repartitioning strategy is developed to ensure scalable parallel performance. === by Laslo Tibor Diosady. === S.M.