The most popular iterative linear solvers in Computational Fluid Dynamics (CFD) calculations are restarted GMRES and BiCGStab. At the beginning of most incompressible flow calculations, the computation time and the number of iterations to converge for the pressure Poisson equation are quite high, since the initial guess is far from the solution. In this case, the BiCGStab algorithm, with relati...

In this report we describe the implementations of the GMRES algorithm for both real and complex, single and double precision arithmetics suitable for serial, shared memory and distributed memory computers. For the sake of simplicity, exibility and eeciency the GMRES solvers have been implemented using the reverse communication mechanism for the matrix-vector product, the preconditioning and the...

‎The global FOM and GMRES algorithms are among the effective‎ ‎methods to solve Sylvester matrix equations‎. ‎In this paper‎, ‎we‎ ‎study these algorithms in the case that the coefficient matrices‎ ‎are real symmetric (real symmetric positive definite) and extract‎ ‎two CG-type algorithms for solving generalized Sylvester matrix‎ ‎equations‎. ‎The proposed methods are iterative projection metho...

In the present work, we propose a new projection method for solving the matrix equation AXB = F . For implementing our new method, generalized forms of block Krylov subspace and global Arnoldi process are presented. The new method can be considered as an extended form of the well-known global generalized minimum residual (Gl-GMRES) method for solving multiple linear systems and it will be calle...

When solving large nonsymmetric systems of linear equations with the restarted GMRES algorithm, one is inclined to select a relatively large restart parameter in the hope of mimicking the full GMRES process. Surprisingly, cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values. Here, two simple examples are presented where GMRES(1) conver...

GMRES(k) is widely used for solving nonsymmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating ...

This paper presents, a preconditioned version of global FOM and GMRES methods for solving Lyapunov matrix equations AX + XA = −BTB. These preconditioned methods are based on the global full orthogonalization and generalized minimal residual methods. For constructing effective preconditioners, we will use ADI spiliting of above lyapunov matrix equations. Numerical experiments show that the solut...

The Generalized Minimum Residual (GMRES) iterative method and variations of it are frequently used for solving systems of linear equations of the form Ax = b, where A is a large sparse nonsingular nonsymmetric matrix. We discuss ways to reorganize the algorithm to improve its efficiency.

To date, the most efficient solver used in the weather sciences for the resolution of linear system in numerical weather prediction is the generalized minimal residual method called GMRES. However, difficulties still appear in matrix resolution when the GMRES iterative method is used without an appropriate preconditioner. For improving the computation speed in numerically solving weather equati...