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Variable degree Schwarz methods for the implicit solution of unsteady compressible Navier-Stokes equations on two-dimensional unstructured meshes

We report our experiences on using a new variant of the Schwarz preconditioned GMRES methods in the implicit solution of the unsteady compressible Navier-Stokes equations discretized on two-dimensional unstructured meshes. We first partition the global mesh with the recursive spectral bisection method into submeshes, and then we introduce a family of Schwarz methods, referred to as the Variable Degree Schwarz methods (VDS) on the overlapping submeshes. In VDS, the subdomain preconditioner is constructed by using a polynomial in two matrix variables, namely the matrix, in its un-factorized form, of the current time step k and another matrix, in its factorized form, obtained at a previous time step j. The degree of the matrix polynomial in each subdomain is determined automatically so that extra preconditioning is performed only in subdomains whose associated local matrices have large condition numbers. The extra preconditioning occurs often near the body of the airfoil. We show numerically that VDS is very effective. Unlike the well-known elliptic theory, we observe that the convergence rate of VDS preconditioned GMRES degenerates very mildly without a coarse space for reasonably large number of subdomains. We also study the effects of the overlapping size, the number of subdomains and the level of inexactness of the subdomain solvers. The other purpose of the study is to understand the robustness of the Schwarz methods with respect to flow parameters, such as the CFL, the free stream Mach number and the Reynolds number. Numerical results for both subsonic and transonic problems are reported.