Advanced Mathematics for Engineers and Scientists/Finite Difference Method

Finite Difference MethodEdit

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 \Delta x 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^\text{th} derivative when divided by \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 + \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.

Last modified on 26 November 2013, at 22:01