Iterative methods for nonsymmetric systems in DAEs and stiff ODEs codes

Abstract

Past research demonstrated that the Newton method coupled with Krylov subspace iterative methods is more efficient than the Newton method coupled with Gaussian elimination when used to solve the corrector equation in stiff systems of Ordinary Differential Equations (ODEs)-with large and sparse Jacobian. Some Krylov subspace methods have been incorporated in the mathematical software codes (such as LSODE), which solve numerical stiff systems of ODEs. In this article we review the past results on this topic and three iterative methods for solving nonsymmetric linear systems of equations. We then incorporate these iterative methods in DASSL, which is a program for numerical integration of systems of stiff ODEs and Differential-Algebraic Equations (DAEs). We present numerical tests (with systems of stiff ODEs) and timing comparisons on the CRAY-2 supercomputer.