A stable unstructured finite volume method with multigrid for parallel large-scale incompressible viscous fluid flow computations

This manuscript briefly describes a parallel unstructured finite volume method for large-scale simulation of viscous fluid flows in a fully coupled form. The numerical method based on side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face while the pressure is defined at the element centroid. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance pressure coupling. The continuity equation is satisfied within each element exactly and the summation of the continuity equations can be exactly reduced to the domain boundary, which is important for the global mass conservation. The resulting system of algebraic linear equations is solved using the non-nested geometric multi-grid method with the flexible GMRES(m) algorithm. However, the standard multigrid methods with classical smoothing techniques can not applied for the coupled iterative solution of the momentum and continuity equations because of the zero-block in the saddle point problem. In order to avoid the zero-block in the saddle point problem, two different approaches are proposed. In the first approach, we use an upper triangular right preconditioner which results in a scaled discrete Laplacian instead of a zero block in the original system. In the second approach, we replace the original system with an equivalent larger system by introducing a new variable which is equal to the pressure. Therefore, a zero block in the original system can be replaced with an identity matrix. The implementation of the preconditioned iterative solvers is based on the PETSc library for improving the efficiency of the parallel code. The present method is validated for the Kovasznay flow, the 2D/3D lid-driven cavity flow problem and the 3D flow past a confined sphere in a circular tube. The parallel efficiency of the code is tested on an SGI-Altix 3000. The numerical results indicate substantial improvement in the computation time.