Understanding a Stiff ODE

[Master J]

A stiff ODE is defined as one for which the magnitude of the maximum eigenvalue of its Jacobian is much greater than that of the mininmum.

It is the real part of the eigenvalue which controls the error in an approximation when a numerical scheme is used to solve the ODE. If it is negative, the error decays away and the approximation approaches the true value for higher iterations.

My question is, is a positive real part also indicitive of a stiff ODE (in the definiton above)? If the error doesn't decay but grows instead, and some components dominate after a certain time, is this still a stiff ODE?

 

[jvicens]

Yes, a positive real part of the eigenvalue can also indicate a stiff ODE. In this case, the error in the numerical approximation will not decay but will instead grow. This can happen when certain components of the ODE dominate after a certain time, causing the error to increase rather than decrease. This behavior is still characteristic of a stiff ODE, as it is defined by the large difference between the maximum and minimum eigenvalues of the Jacobian.