Why can't we use Finite Differences with non-parabolic PDEs?

Question

The title of the question says it all. Why can we only apply the method to parabolic PDEs like the heat equation, and not to ordinary PDEs?

Answer 1

Who gave you that idea?

You absolutely can use Finite Differences for other PDEs. They are routinely used to solve hyperbolic PDEs (wave equation, both first and second order) and elliptic PDEs (steady state diffusion/heat equation). You can even mix and match the equation types and create PDEs that have characteristic of both hyperbolic and parabolic equations, such as the Navier-Stokes equations.

If you're interested in learning how to implement solvers for these, most numerical fluid mechanics textbooks have a pretty thorough treatment on discretizing PDEs of many types with finite differences.

For reference:

First order wave equation: $\frac{\partial u}{\partial t} + \nabla \cdot(c\,u) = 0$

Second order wave equation: $\frac{\partial^2 u}{\partial^2 t} - c^2 \nabla^2 u = 0$

Elliptic diffusion: $\nabla^2 u + f = 0$

Navier Stokes:

$$ \rho\left(\frac{\partial\mathbf{u}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla{P} + \mu\nabla^2 \mathbf{u} $$