A SOFTWARE PACKAGE FOR SOLVING LARGE SYSTEMS OF NONLINEAR EQUATIONS

WE PROPOSE TO BUILD A SOFTWARE PACKAGE FOR SOLVING GENERAL LARGE SYSTEMS OF NONLINEAR EQUATIONS SUCH AS THOSE ARISING FROM NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS. PHASE I WILL FOCUS ON TESTING A RANGE OF METHODS BASED ON EFFICIENT VARIANTS OF NEWTON'S METHOD AND PRECONDITIONED CONJUGATE GRADIENT TYPE TECHNIQUES AS WELL AS ON A CLASS OF JACOBIAN-FREE ITERATIVE METHODS. THESE METHODS HAVE ATTRACTED MUCH INTEREST RECENTLY IN PARTICULAR AMONG THE COMPUTATIONAL FLUID DYNAMICS COMMUNITY, BECAUSE OF THEIR SIMPLICITY OF USAGE AND THEIR GENERALITY. THE PACKAGE WILL CONSIST OF TWO PARTS. ITS FIRST PART WILL USE EXPLICIT JACOBIANS AND WILL INTERFACE NEWTON'S METHOD WITH PRECONDITIONED CONJUGATE GRADIENT TYPE TECHNIQUES. ITS SECOND PART WILL BE BASED ON JACOBIAN-FREE METHODS WHICH AREPARTICULARLY DESIRABLE FOR ACCELERATING STANDARD NONLINEAR ITERATIONS OF THE FORM UN+1=M(UN) THAT ARISE IN MANY SPECIALIZED INDUSTRIAL CODES. THUS, THE PACKAGE WILL BE FLEXIBLE ENOUGH TO ALLOW USERS TO TAKE ADVANTAGE OF THEIR OWN AVAILABLE SOFTWARE, IF ANY. THE CODE WILL EMBODY THE BEST KNOWN METHODS TO HANDLE STRONG NONLINEARITIES BY TRUST REGION AND POSSIBLY HOMOTOPY TECHNIQUES AND TO TAKE ADVANTAGE OF SPARSITY. THE AIM OF THE PACKAGE IS TO COMBINE EFFICIENCY, FLEXIBILITY, ROBUSTNESS AND WIDE RANGE OF APPLICABILITY.