IN the past few years numerical analysts have come to realize that linear stability theory cannot be applied to nonlinear systems. One cannot say that the Jacobian represents the local behaviour of the solutions except at a fixed point. This had not previously been appreciated in numerical analysis, and there was a tendency to believe that looking at the Jacobian at one point as a constant, the solutions nearby would behave like the linearized system produced from this `frozen' Jacobian. Numerical analysts have now recognized the failings of linear stability theory when applied to nonlinear systems, and have constructed a new theory of nonlinear stability.
The theory looks at systems that have a property termed contractivity; if and are any two solutions of the system satisfying different initial conditions, then if
for all , where , the system is said to be contractive. An analogous definition may be framed for a discrete system; if
the discrete system is said to be contractive. Now we must define another property of the system which numerical analysts have termed dissipativity. Since we have already used this word in dynamics, we shall call this new property NA-dissipativity (numerico-analytic dissipativity). The system is said to be NA-dissipative in if
where is an inner product, holds for all y, in and for , where is the domain of regarded as a function of y. NA-dissipativity can be shown to imply contractivity. We needed to define NA-dissipativity because we can obtain a usable test for it; a system is NA-dissipative if
where is the logarithmic norm of the Jacobian . If , are the eigenvalues of , Eq.(75) can be shown to be equivalent to
We now have a practical sufficient condition for contractivity.
We further define: if a Runge--Kutta method applied with any step length to a NA-dissipative autonomous system is contractive, then the method is said to be B-stable. If we have the same situation with a nonautonomous system, then the method is said to be BN-stable. A sufficient condition for both of these properties is given by algebraic stability. A Runge--Kutta method is said to be algebraically stable if and are both nonnegative definite. and here come from Eq.(11), the Butcher array of the Runge--Kutta method. Algebraic stability implies A-stability, but the reverse is not true.
Let us look at this theory from a dynamics viewpoint. The problem is that contractivity is a very severe requirement to impose; in fact it precludes the possibility of chaos occurring in the system. Is is easy to see that this must be so, since chaos demands that neighbouring trajectories be divergent, whereas contractivity demands that they be convergent. We show in Appendix A.2 that contractivity is sufficient to give nonpositive Lyapunov exponents; positive Lyapunov exponents are a necessary condition for chaos to occur. Unfortunately then, one can only investigate the stability of contractive, nonchaotic systems with the nonlinear stability theory as it stands. However, as we have remarked previously, practical numerical codes do not use any stability theory to evaluate their accuracy, so this is a theoretical, rather than a practical, problem. It should be pointed out that what numerical analysts call dissipativity, what we have termed NA-dissipativity, is a much stronger requirement even than strict dissipativity () in dynamics. The latter merely requires that , whereas the former insists that ( being the Lyapunov exponents of the system).