Many books have been written on finite difference methods (FDM), but there are
good reasons to write still another one. The main reason is that even if higher order
methods have been known for a long time, the analysis of stability, accuracy and
effectiveness is missing to a large extent. For example, the definition of the formal
high order accuracy is based on the assumption that the true solution is smooth,
or expressed differently, that the grid is fine enough such that all variations in the
solution are well resolved. In many applications, this assumption is not fulfilled,
and then it is interesting to know if a high order method is still effective. Another
problem that needs thorough analysis is the construction of boundary conditions
such that both accuracy and stability is upheld. And finally, there has been quite a
strong development during the last years, in particular when it comes to very general
and stable difference operators for application on initial–boundary value problems.