Finite Differences Apply to Hyperbolic and Elliptic PDEs

Summary

Finite difference methods are not limited to parabolic partial differential equations such as the heat equation. The answer explains that they are also routinely used for hyperbolic equations, including first- and second-order wave equations, and for elliptic equations such as steady-state diffusion. The method approximates derivatives on a grid; the PDE’s classification does not by itself rule out that approach.

The document also points to equations with mixed characteristics, using Navier–Stokes as an example, and suggests numerical fluid mechanics texts for implementation details. It provides representative equations but no discretization scheme, stability analysis, convergence results, or comparison of methods. In practice, choosing a finite difference formulation still depends on the equation, boundary and initial conditions, and numerical properties of the scheme; those implementation questions are outside the scope of this brief answer.

Key ideas

• Finite differences can be used for hyperbolic and elliptic PDEs as well as parabolic PDEs.
• The wave equation is an example of a hyperbolic PDE that can be discretized with finite differences.
• Steady-state diffusion provides an elliptic PDE example.
• Some systems, including Navier–Stokes, combine characteristics associated with different PDE types.
• The document gives examples but does not discuss solver design, stability, or accuracy.

Tags

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


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












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 by Tyler Olsen (score 3, accepted)

https://quant.stackexchange.com/a/20757

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:

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.