  We analyze numerically a forward-backward diffusion equation with a
cubic-like diffusion function, -emerging in the framework of phase transitions
modeling- and its "entropy" formulation determined by considering it as the
singular limit of a third-order pseudo-parabolic equation. Precisely, we
propose schemes for both the second and the third order equations, we discuss
the analytical properties of their semi-discrete counter-parts and we compare
the numerical results in the case of initial data of Riemann type, showing
strengths and flaws of the two approaches, the main emphasis being given to the
propagation of transition interfaces.
