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).