High performance parallel algorithms for incompressible flows

Abstract

The generalized Stokes problem, which arises frequently in the simulation of Navier-Stokes equations for incompressible fluid flow, gives rise to symmetric linear system of equations. These systems are indefinite because of the incompressibility constraint on the velocity, causing difficulty for most preconditioners and iterative methods. Solenoidal basis methods are a class of projection methods where the velocity is projected into a space in which it is incompressible. This thesis presents innovative algorithms using solenoidal basis methods to solve the generalized Stokes problem for 3D MAC (Marker and Cell) and 2D unstructured P1-isoP1 finite element grids. It details a localized algebraic approach to construct solenoidal basis. An efficient Object-Oriented design for the algorithms and its parallel implementation in multi-threading and multi-processing environments is presented. Inexpensive parallel matrix-vector products using bounded buffers for inter-processor communication are suggested. Experimental results show that the number of iterations needed for convergence is stable across wide range of flow parameters, such as the Reynolds number, time step and the mesh width. Scalability of the algorithms is suggested by the experiments on SGI Origin 2000.