The Deflated Relaxed Incomplete Cholesky CG method for use in a real-time ship simulator

Ship simulators are used for training purposes and therefore have to calculate realistic wave patterns around the moving ship in real time. We consider a wave model that is based on the variational Boussinesq formulation, which results in a set of partial differential equations. Discretization of these equations gives a large system of linear equations, that has to be solved each time-step. The requirement of real-time simulations necessitates a fast linear solver. In this paper we study the combination of the Relaxed Incomplete Cholesky preconditioner and subdomain deflation to accelerate the Conjugate Gradient method. We show that the success of this approach depends on the relaxation parameter. For low values of the relaxation parameter, e.g. the standard IC preconditioner, the deflation method is quite successfull. This is not the case for large values of the relaxation parameter, such as the Modified IC preconditioner. We give a theoretical explanation for this difference by considering the spectrum of the preconditioned and deflated matrices. Computational results for the wave model illustrate the expected convergence behavior of the Deflated Relaxed Incomplete Cholesky CG method. We also present promising results for the combination of the deflation method and the inherently parallel block-RIC preconditioner.