Advanced Mathematics for Engineers and Scientists/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:

$u^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{u(x+\mathrm{\Delta }x)-u(x)}{\mathrm{\Delta }x}$

The basic idea is that if $\mathrm{\Delta }x$ is "small", then

$u^{\prime }(x)\approx \frac{u(x+\mathrm{\Delta }x)-u(x)}{\mathrm{\Delta }x}$

Similarly,

$u^{″}(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{u(x+\mathrm{\Delta }x)-2u(x)+u(x-\mathrm{\Delta }x)}{\mathrm{\Delta }{x}^2}$

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

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 $n^{\text{th}}$ derivative when divided by $\mathrm{\Delta }{x}^n$.

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:

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

$\Downarrow$

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

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