The Use of Iterative Linear-Equation Solvers in Codes for Large Systems of Stiff IVPs for ODEs

Systems of linear algebraic equations must be solved at each integration step in all commonly used methods for the numerical solution of systems of stiff IVPs for ODES. Frequently, a substantial portion of the total computational-work and storage required to solve stiff IVPs is devoted to solving these linear algebraic systems, particularly if the systems are large. Over the past decade, several efficient iterative methods have been developed to solve large sparse (nonsymmetric) systems of linear algebraic equations. We study the use of a class of these iterative methods in codes for stiff IVPs. Our theoretical estimates and preliminary numerical results show that the use of iterative linear-equation solvers in stiff-ODE codes improves the efficiency—in terms of both computational—work and storage-with which a significant class of stiff IVPs having large sparse Jacobians can be solved.