On the performance of certain direct and iterative methods on equations arising on a two-dimensional in situ combustion simulator

A two-dimensional mathematical model of the in situ combustion process involves a set of nonlinear partial differential equations. These equations are discretized in implicit finite-difference form. The resulting set of nonlinear algebraic equations are solved for each time-step by use of a Newton-Raphson procedure. Each Newton iteration produces an equation of the form Ax = b, (*) where x is the Newton update, b is the current residual of the nonlinear equations and A is the Jacobian matrix. A is lar-e and has a non-symmetric, sparse structure. In this current work we wish to compare the performance of LU factorization, ORTHO-MIN(m) and more recent iterative methods, GMRES(m) and BI-CGSTAB to solve (*) on the model of the in situ combustion problem. To increase the convergence rate for the iterative methods a preconditioning and a scaling technique are used. (C) 2002 Elsevier Science Inc. All rights reserved.