By Ricard Evans and Kirk Phillips
http://d.repec.org/n?u=RePEc:byu:byumcl:201502&r=dge
This paper presents an adjustment to commonly used approximation methods for dynamic stochastic general equilibrium (DSGE) models. Policy functions approximated around the steady state will be inaccurate away from the steady state. In some cases, this does not lead to substantial inaccuracies. In other cases, however, the model may not have a well-defined steady state, or the nature of the steady state may be at odds with its off-steady-state dynamics. We show how to simulate a DSGE model by approximating about the current state. Our method introduces an approximation error, but minimizes the error associated with a finite-order Taylor-series expansion of the model’s characterizing equations. This method is easily implemented using available simulation software and has the advantage of mimicking highly non-linear behavior. We illustrate this with a variety of simple models. We compare our technique with other simulation techniques and show that the approximation errors are approximately the same for stable, well-defined models. We also illustrate how this method can solve and simulate models that are not tractable with standard approximation methods.
(Log-)linearization clearly does not apply in some situations, and this paper details a relatively simple method that should help out those who want to stick with their standard solution tools. If you research question deals with large shocks or your model has an undefined steady-state, this paper is likely for you.
NEP-DGE Blog 
Christian, thanks for highlighting my work with Kerk Phillips. Here are a couple of comments.
1) Linear approximation is for every system of equations. It is just that the approximation error has been shown to be unacceptably large for many economic models. The nonlinearities in the dynamics matter.
2) Further, many models have multiple stationary equilibria or have instabilities such as unbalanced growth or a more fundamental instability such as unsustainable fiscal policy. Standard methods cannot even obtain a solution in these cases.
Linearization about the current state (or CSL, current state linearization) is an iteratively updating linear approximation about the current state. This method has the computational speed of linear approximation, while still accounting for the nonlinear dynamics of the underlying problem. There are some potential issues with this method.
A) We have not proven a theoretical guarantee that CSL will converge to the truth asymptotically. That said, CSL performs very well in all the applications we check in the paper.
B) When you are linearizing around a point other than the steady state, the method of undermined coefficients has a strange problem. It is not clear which eigenvalues to use in order to find the P matrix. Linearizing around the steady-state, you just choose the stable roots. However, away from the steady-state, it is likely that the linear approximation is not stable. We therefore have an issue about which roots to use. But again, in the paper, all the examples we checked used the stable roots and performed very well in terms of Euler errors and in terms of deviations from the true solution, both in impulse response functions and across simulations.
We think this approach is very valuable for simulating models that have the instabilities that are so important to many current research questions.