ABSTRACT
As we have seen in the previous chapters, when we simply apply the discrete variational derivative method (DVDM) to a nonlinear equation, the resulting scheme naturally inherits the nonlinearity, unless some linearization technique, such as the one discussed in Chapter 6, is utilized. The issue of \either nonlinear or linear" seems to be quite a big problem,
and no simple answer seems to be able to be found. From the perspective of speed, linear (linearly implicit) schemes are much more e±cient; instead, they can lose stability and/or qualitative good behavior that nonlinear schemes generally have. In this respect, it seems that there are still many situations where nonlinear schemes should be willingly employed. The biggest problem in nonlinear schemes is, of course, how we should solve
them. We need some iterative solver, among which the most popular choice is the Newton method. As far as we can supply a good initial guess, the Newton method rapidly converges (if we observe the convergence per iteration), and quite reliable. The method has, however, several drawbacks when it comes to nonlinear equations arising from numerical schemes for nonlinear PDEs.
