Finite difference terms for boundaries

[hunt_mat]

Hi,

We all know that the finite difference formulae for the derivatives are given by:
$$\frac{dy}{dx}_{i}=\frac{y_{i}-y_{i-1}}{\delta x}$$
and
$$\frac{d^{2}y}{dx^{2}}=\frac{y_{i-1}-2y_{i}+y_{i+1}}{\delta x^{2}}$$
What would be the formulae for the boundary terms? when i=1? I think I can show that:
$$\frac{dy}{dx}_{0}=\frac{4y_{1}-y_{2}}{3}$$
Are there any other formulae? What about the second order derivative?

This is for a numerical code in matlab, I can use inbuilt functions but I want my code to run as fast as possible.

Mat

 

[gtoguy97]

lab has some in-built functions for computing derivatives, for example, the 'gradient' function. This function uses a five-point stencil scheme to compute the derivatives, which gives more accurate results. The formula for the second order derivative is:\frac{d^{2}y}{dx^{2}}=\frac{-y_{i-2}+16y_{i-1}-30y_{i}+16y_{i+1}-y_{i+2}}{12\delta x^{2}}For the boundary cases, you can use forward or backward difference formulae depending on what you are trying to do. For example, if you want to approximate the first derivative at the boundary, you can use the formula:\frac{dy}{dx}_{0}=\frac{y_{1}-y_{0}}{\delta x}For the second derivative, you can use the formula:\frac{d^{2}y}{dx^{2}}_{0}=\frac{y_{2}-2y_{1}+y_{0}}{\delta x^{2}}Hope this helps!