Finite Differences

Finite differences are discrete approximations to derivatives, obtained by replacing the limiting process $dx\to 0$ with a small but finite step $dx$.

Forward Difference Approximation (FDA)

$$\frac{df}{dx}\approx \frac{f(x+dx)-f(x)}{dx}$$

Backward Difference Approximation (BDA)

$$\frac{df}{dx}\approx \frac{f(x)-f(x-dx)}{dx}$$

Second Derivative (Centred)

$$\frac{d^{2}f}{d{x}^2}\approx \frac{f(x+dx)-2f(x)+f(x-dx)}{d{x}^2}$$

This centred second-derivative formula is used in the FTCS scheme and BTCS scheme for the spatial part of the Heat equation, and in the Laplacian difference equation for the Laplace equation.

The forward difference is the basis for the Explicit Euler method; the backward difference gives the Implicit Euler method.

Explicit Euler method | Implicit Euler method | FTCS scheme | BTCS scheme | Laplacian difference equation