Preconditioning shifted linear systems arising in a wind model

Abstract

The numerical modelling of a wind field using a mass consistent model yields a linear system of equations with symmetric positive definite matrix for each set of wind velocities measured at meteorological stations. This type of linear system may be written as A ε x ε = b ε , where ε is related to the so-called stability parameter of the wind model, with A ε = M + ε N. Matrices M and N do not change along the process for a given discretisation level. The rest of parameters and the wind measures only affect the vector b. There are two extreme strategies for preconditioning such linear systems. On the one hand, we can construct an initial preconditioner which is applied without any modification in the resolution of all the linear systems. The performance of the preconditioner will become worse progressively as the process advances, depending on the great or small variation in the value of ". On the second hand, we can use a different preconditioner for each linear system independently. This strategy would be very expensive and slow. So, the main goal of this work is to develop a preconditioner that can be updated in function of ". We propose the construction of an intermediate approach that works better than the former strategy and worse than the latter in terms of Conjugate Gradient iterations, but evidently at a lower computational cost.