Incomplete Gaussian Elimination as a Preconditioning for Generalized Conjugate Gradient Acceleration
J. R. Wallis · 1983
ABSTRACT This paper discusses the application of preconditioned generalized conjugate gradient acceleration to fully implicit thermal simulation. The preconditioning step utilizes incomplete Gaussian elimination (IGE) to form an approximate factorization of the Jacobian matrix. The implementation allows any finite difference approximationany grid block orderingassociated well constraint equationsany level of incomplete factorizationreduced system preconditioningconstrained residual preconditioning Acceleration procedures include ORTHOMINORTHORES IGE preconditioning and its implementation is discussed with respect to five, seven, nine or eleven-point finite difference approximations and optional well constraint equations. Numerical results were obtained using a thermal model allowing any number Nc of components. The model's implicit formulation requires the solution of a linear system of equations in which each grid block has Nc + 1 unknowns. Test problems involved combustion, steam drive and cyclic steam stimulation processes, some of which exhibited ill-conditioning, negative transmissabilities and high transmissability ratios. Numerical results indicate how different grid block orderings, different levels of incomplete factorization and different acceleration procedures affect convergence, storage and work requirements.