A multilevel Schwarz shooting method for the solution of the Poisson equation in two dimensional incompressible flow simulations

摘要

This work presents a Multilevel Schwarz Shooting method for the numerical solution of the Poisson equation, using a five point finite difference molecule, and subject to Dirichlet boundary conditions, which arises in two dimensional incompressible flow simulations. The iterative simple shooting algorithm utilized has optimal complexity, but becomes unstable when the problem size is increased. In this work, this limitations is overcome using a domain decomposition technique, more specifically, the alternating Schwarz method, which is used to generalize the iterative multiple shooting approach in the partial differential equation (PDE) context. In order to accelerate the convergence of the algorithm a multilevel approach is used. The resulting iterative algorithm has optimal complexity and is scalable in terms of problem size. Results of computational experiments are presented for the Poisson equation and for the two dimensional incompressible Navier–Stokes equation using the stream function-vorticity formulation.