The movies below are my simulations of the heat equation and of the
wave equation, both with the homogeneous Dirichlet data, and the
same initial condition. Wave equation also has homogeneous initial
velocity. The movies show a tremendous difference between
qualitative properties of these two equations: dissipative and
conservative nature.

The last movie is a simulation of the advection equation with linear
and nonlinear (convex increasing) flux, both with the same initial
data. The simulation shows a tremendous difference between
qualitative properties of the linear and nonlinear equations
(preserving smoothness versus developing shocks even from smooth
data).