You may want to look at this excellent book on continuation methods. It includes pseudo-code for several robust Newton-type predictor-corrector methods, including bifurcation detection. Most of the book is not specifically transient, but it should be helpful anyway.
@book{allgower2003inc,
title={{Introduction to Numerical Continuation Methods}},
author={Allgower, EL and Georg, K.},
year={2003},
publisher={Society for Industrial and Applied Mathematics
Philadelphia, PA, USA}
}
Also, this paper on pseudotransient continuation is a good introduction,
you can apply the same framework to stiff problems by limiting the
maximum time-step and using the identity as the scaling matrix (instead
of one scaled to local properties to accelerate convergence to steady
state). Specifically, in (1.1) your V=I.
@article{coffey2003ptc,
author = {Todd S. Coffey and C. T. Kelley and David E. Keyes},
collaboration = {},
title = {Pseudotransient Continuation and Differential-Algebraic
Equations},
publisher = {SIAM},
year = {2003},
journal = {SIAM Journal on Scientific Computing},
volume = {25},
number = {2},
pages = {553-569},
keywords = {pseudotransient continuation; nonlinear equations;
steady-state solutions; global convergence; differential-algebraic
equations; multirate systems},
url = {http://link.aip.org/link/?SCE/25/553/1},
doi = {10.1137/S106482750241044X}
}
Jed
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