Two Unconventional Classes of Methods for Stiff Systems
In several areas of numerical analysis there is a current feeling that what is needed is greater insight into the working of existing methods, rather than the development of yet more new methods. This point of view can be convincingly argued in the case of the numerical solution of initial value problems for systems of ordinary differential equations, in the absence of stiffness. However, when stiffness is present, the performance of conventional methods is not very impressive. The basic reasons for this are well-known; conventional methods can have adequate stability to cope with stiff systems only if they are implicit, and stiffness precludes the solution of the resulting implicit difference schemes by direct iteration, necessitating the use of some form of Newton iteration, with the attendant need to compute inverses of matrices. The real computational difficulty with stiff systems is centred round this need repeatedly to compute matrix inverses. This paper will describe two attempts to break out of this situation by considering unconventional classes of methods. One class, that of linear multistep methods with variable matrix coefficients, can be shown theoretically to possess adequate stability, and the methods appear to be computationally sound. However, with methods of this class, it is still necessary to invert a matrix, but only once per step. The second class, that of nonlinear methods, lacks the same level of theoretical backing; computationally, methods of this class appear to possess adequate stability, but there are difficulties, as yet unresolved, in controlling the level of local accuracy. Their advantage is that they require no matrix inversions.
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