EPODE

Stability and solution unicity

For the unicity of the exact solution we put the following conditions:

the system function f is continous on
D={(t,y) | t in [a,b], ||y(t)-y|| <= d}, where d in R₊;
f satisfies the Lipschitz's inequality
||f(t,u)-f(t,v)|| <= L||u-v||, a <= t <= b, for all (t,u),(t,v) in D

For almost all differential equation systems, the value L(b-a) is of hundreds order. For such systems the classical step by step methods, like Runge-Kutta's processes or Adams's formulae, are satisfactorily (in the sense of a small error).

Unfortunately, there are some exceptions, for instance, the case when the function variation is very strong. When the value L(b-a) exceeds the thousands order, a variety of restrictions are imposed to the classical methods, especially on the stepsize, so there are useless. Such a system we classified to be of stiff kind one.

Let y'=l y, l in R. The solutions family of this equation, can be represented by {y | y(t)=ce^{{l t}, c in R}. Note that, when l>0, for some large t values, the difference between two arbitrary solutions increases exponentially, even if the difference between the initial values is very small. In such a case, applying an approximate method, the results are unforeseeables. If l<0, the difference between two arbitrary exact solutions decreases when t -> infinity. These remarks can be extended for the case

y'=l y, y in C (discussion on Re l>0 or Re l<0). The last equation is referred to as the scalar test equation.

The scalar test equation is used in the numerical stability study of the step by step methods, especially for the multistep formulae.

Let f:[t₀,infinity) x R^N -> R^N, y(.,t₀,y₀) the exact solution of the IVP with y(t₀)=y₀, and j an arbitrary solution for the differential system.

The solution j is stable in Liapunov's sense if, for any e>0, there is a value d(e,t₀) such that

||j(t)-y(t,t₀,y₀)||<e, for all t in [t₀,infinity), holds for any y₀ satisfying ||y₀-j(t₀)|| <= d(e,t₀). The solution j is asympotically stable if, for any e>0, there is a value d(t₀) such that the last relationship holds for any y₀ satisfying ||y₀-j(t₀)||<d(t₀).

In the particular case of a linear system, y'=Ay, if the solution y=0 is stable (asymptoticallly stable), all the system solutions are stable (asymptotically stable). The solution y=0 is asymptotically stable if all the eigenvalues of the matrix A have some negative real parts, that means that the characteristic polynomial of A is an Hurwitz polynomial.

The stability of the exact solutions is out of the present study subject.

The following supposition is esentially in our study.

The exact solutions of the numerical solved problems with initial values are asymptotically stable

By this supposition we eliminate the possibility of a numerical instability due to the instability of the exact solutions.