Parallelizable block diagonal preconditioners for the compressible Navier-Stokes equations

Abstract

Abstract We propose a new block diagonal preconditioner for solving the Navier-Stokes equations for compressible 2D flows, using finite element methods on unstructured grids and discuss the parallelization of this method. The core of the Navier-Stokes solver is the coupled solution of the nonlinear equations at each time step by a preconditioned nonlinear GMRES algorithm. In this work, we suggest to partition the Jacobian matrix A into n × n blocks, n being the number of available processors. In practice, our goal is to obtain good results on commonly available parallel workstations, with from 4 to 16 processors. Special attention is paid to the partitioning strategy, which is easily generalizable to 3D problems. We approximate A by A [n] taking only the non-zero coefficients lying in the diagonal blocks of A . We show that, with an appropriate partitioning strategy and a convenient node reordering, the ILU(0) factorization of A [n] is an effective preconditioner of the nonlinear GMRES algorithm. Because of its diagonal block structure both the factorization and the triangular system solution steps associated with the preconditioning matrix can then be parallelized. Numerical results are presented for several transonic and supersonic calculations around a NACA 0012 aerofoil.