The linear solver in a typical reservoir simulator consumes around 60 to 70 % of the total simulation time. To speed up the solution of the linear systems we will use a two stage preconditioner, where the first stage is the ILU preconditioner and the second stage is obtained with a technique of Reduction of Order Modelling (ROM) called Proper Orthogonal Decomposition (POD). The benefits of using this method is that it is relatively easy to implement because it doesn't require big modifications in existing code. There is already research in this area with positive results, but the method has been tested with the Richardson Algorithm. This is not a very realistic scenario given that there are better solvers widely available. The objective of this thesis is to test the method with the Generalized Minimal Residual Method (GMRES). The results validate the findings that two stage preconditioner improves the performance of the Richardson algorithm, however it doesn't improve the performance of the GMRES algorithm. The drawback is that although the two stage preconditioner increases convergence it is too costly to compute. And the time saved by the decrease in the number of iterations is offset by the increase in time in preconditioner computation.
