Parallel Conjugate Gradient with Schwarz Preconditioner Applied to Fluid Dynamics Problems

In the common implicit integration schemes of fluid-dynamics equations, the need to solve a linear system of equations—usually associated with a sparse matrix—derived from the discretization of the equations on a given mesh often arises. The original linear system is replaced by as many smaller systems as the number of subdomains, and the boundary conditions for each linear system are imposed at each iteration on the base of the solution obtained on the adjacent subdomains. However, its construction is immediate and does not require the generation of another mesh. The consequence is that the communications among processors, which involve delays because of latency and communication time, do not present a significant execution time overhead. The test cases discussed in this chapter are exemplary of many aspects of the Schwarz additive algorithm. The framework of Domain Decomposition, in which the Schwarz additive algorithm resides, evidences the intrinsically parallel nature of this preconditioner. The algorithm is well suited for running efficiently on distributed parallel systems composed of many processors because of these characteristics.