Multiple Step G-Contractivity with Applications to Slowly Varying Linear Systems

摘要

The possibility of constructing contractivity sets which correspond to the notion of stiff stability is explored. The contractivity set K should have three properties: (1) zero belongs to K, i.e., zero-contractivity; (2) A(zero)-contractivity; and (3) q belongs to a set of complex numbers: Re q or = negative beta is contained in K, beta or = zero. Only fixed-h G-contractivity is considered, restricting attention to stiffly stable one-leg methods. A k-step method (rho, sigma) is said to be G-contractive with respect to inner product norm sub G if the solution to the linear test equation satisfies certain criteria. It is recalled that K sub G which is a subset of the stability set, is a circular domain. As a consequence, K sub G cannot contain the set of all negative numbers unless (rho, sigma) is A-stable. By constructing a transformed method (rho*, sigma*), the step number of which is an integer multiple of k, a G norm with a noncircular constractivity set, corresponding to the notion of stiff stability, is constructed.