On Liapunov Stability of Stiff Non-Linear Multistep Difference Equations,

摘要

The concepts of G-stability and (G,mu)-stability recently introduced by Dahlquist are useful for discussing Liapunov stability of solutions to systems of non-linear difference equations, generated by applying linear multistep formulas to monotone, dissipative, arbitrarily stiff systems of non-linear differential equations. In this paper, the theory of G-stability and (G,mu)-stability is reviewed and a construction is proposed which facilitates the finding of a quadratic Liapunov function. By this construction it is proved that, for the four-parameter family of all three-step formulas which are second-order accurate, A-stability is necessary and sufficient for G-stability. Some results on (G,mu)-stability are also obtained by this construction.