I've asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well.

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the stability of the schemes using the Von Neumann's stability analysis. However, as the authors realize, this is only applicable to linear PDE's. So the authors work around this by "freezing" the non-linear term, i.e. they replace the $uu_x$ term with $Uu_x$, where $U$ is "considered to represent locally constant values of $u$."

So my question is two-fold:

1: how to interpret this method and why does it (not) work?

2: could we also replace the $uu_x$ term with the $uU_x$ term, where $U_x$ is "considered to represent locally constant values of $u_x$"?

**References**

- Eilbeck, J. C., and G. R. McGuire. "Numerical study of the regularized long-wave equation I: numerical methods." Journal of Computational Physics 19.1 (1975): 43-57.