I had a general query related to calculation of stiffness ratio evolution for a set of coupled odes over a certain time interval. My question is, while calculating the eigen values from the jacobian matrix do we need to know the exact solution at the different time points or can we use solutions approximated by an integration method?
Stiffness ratio evolution for odes
stiffness
I made a number of comments on stiffness in response to another question that are relevant here; I won't repeat them here in their entirety. The most important point is that stiffness ratio is only correlated with stiffness, and not a definition of stiffness. Consequently, it does not seem critical to calculate eigenvalues of a Jacobian matrix based on an exact solution; calculating approximate eigenvalues (for example, via a Krylov subspace method that provides eigenvalue estimates, like GMRES) of a reasonably accurate approximate solution, then calculating the stiffness ratio is probably fine for diagnostic purposes.Another criterion that could work, if you can calculate the necessary data, is to compare the time steps your integrator is taking at the end of each iterate to the maximum stable time step it could take. If stability is limiting time step size rather than accuracy, your system is probably stiff. (Again, this sort of behavior tends to occur in stiff systems, but it is not a definition.)