Partial Differential Equations/Finite Difference Method

Finite Difference Method

The finite difference method is a basic numeric method which is based on the approximation of a derivative as a difference quotient. We all know that, by definition:

$u'(x) = \lim_{\Delta x \to 0}\frac{u(x + \Delta x) - u(x)}{\Delta x}$

The basic idea is that if Δx is "small", then

$u'(x) \approx \frac{u(x + \Delta x) - u(x)}{\Delta x}$

Similarly,

$u''(x) = \lim_{\Delta x \to 0}\frac{u(x + \Delta x) - 2 u(x) + u(x - \Delta x)}{\Delta x^2}$

$u''(x) \approx \frac{u(x + \Delta x) - 2 u(x) + u(x - \Delta x)}{\Delta x^2}$

It's a step backwards from calculus. Instead of taking the limit and getting the exact rate of change, we approximate the derivative as a difference quotient. Generally, the "difference" showing up in the difference quotient (ie, the quantity in the numeriator) is called a finite difference which is a discrete analog of the derivative and approximates the n^th derivative when divided by Δxⁿ.

Replacing all of the derivatives in a differential equation ditches differentiation and results in algebraic equations, which may be coupled depending on how the discretization is applied.

For example, the equation

$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$

may be discretized into:

$\frac{u(x, t + \Delta t) - u(x, t)}{\Delta t} = \frac{u(x + \Delta x, t) - 2 u(x, t) + u(x - \Delta x, t)}{\Delta x^2}$

$\Big\Downarrow$

$u(x, t + \Delta t) = u(x, t) + \frac{\Delta t}{\Delta x^2} (u(x + \Delta x, t) - 2 u(x, t) + u(x - \Delta x, t))$

This discretization is nice because the "next" value (temporally) may be expressed in terms of "older" values at different positions.