  Recently, various evolutionary partial differential equations (PDEs) with a
mixed derivative have been emerged and drawn much attention. Nonetheless, their
PDE-theoretical and numerical studies are still in their early stage. In this
paper, we aim at the unified framework of numerical methods for such PDEs.
However, due to the presence of the mixed derivative, we cannot discuss
numerical methods without some appropriate reformulation, which is
mathematically challenging itself. Therefore, we first propose a novel
procedure for the reformulation of target PDEs into a standard form of
evolutionary equations. This contribution may become an important basis not
only of numerical analysis, but also of PDE-theory. In order to illustrate this
point, we establish the global well-posedness of the sine-Gordon equation.
After that, we classify and discuss the spatial discretizations based on the
proposed reformulation technique. As a result, we show the average-difference
method is suitable for the discretization of the mixed derivative.
