Block-selective algebraic multigrid for implicitly coupled pressure-velocity system

Abstract

The experience with the implicitly coupled pressure–velocity solver has proven that multigrid methods are the most robust and efficient in solving the system. Until now the usual choice of the linear solver for the block–variable has been the Aggregative Algebraic Multigrid which has a relatively simple implementation: Coarsening is achieved by joining the cells into groups, coarse level operators are calculated by summing the coefficients from the fine level matrix, and restriction and prolongation is done by injection. However, our experience has shown that this type of AMG was inefficient for the implicitly coupled solver: In a single non–linear iteration, the residual would stall, not reaching the prescribed convergence tolerance.To remedy the problem, we have implemented a Block–Selective Algebraic Multigrid in foam-extend which is a community driven fork of OpenFOAM, an open–source CFD software library for arbitrary polyhedral Finite Volume Method. Coarsening is done by selecting a set of equations which will be solved on the coarse level, based on a heuristic algorithm. Prolongation operator is calculated from the approximation of the algebraically smooth error. Restriction and coarse level matrix are formed based on a variational principle for a symmetric positive–definite matrix. To apply the algorithm on the matrix arising from the discretisation of the implicitly coupled pressure–velocity system, a primary matrix must be chosen by calculating a norm of the block–coefficient.