Let us assume I have an A-stable numerical scheme. I believe that given any linear equation $y' = Ay$, it means that the numerical scheme applied to this equation is stable (and therefore convergent since it is consistent) if the eigenvalues of A have a negative real part.
My question is then, does this result extend to the non-linear case ? I am interested in particular in a system $y' = f(y,t)$ where the Jacobian of $f$ has negative real eigenvalues $\forall y$.
edit: Also I guess I could give a particular example here (but my question is not limited to that case): for example let us say the ODE describes a 2nd order chemical reaction network, i.e. $f \in R^n \rightarrow R^n$ is a polynomial function of the $y$ of order 2, where $n$ is the number of species. Considering the network I was able to prove that the eigenvalues of the Jacobian are real negative for any $y$, however I don't know if I can make the further argument that any A-stable method will converge (by the way, what about L-stable methods, B-stable methods, ... ?)