Forward Difference Approximation

Throughout numerical weather prediction, you often need to calculate the gradient of a function at a number of points. As we rarely know the equation which defines the function, we need to calculate numerically an estimate of the gradient. This method requires the use of finite difference schemes. In this module we will be considering finite difference schemes to approximate the gradient of a function in one variable. The problem can be stated more specifically in these terms: how to estimate the gradient of a function in f(x), at a point .

We can take a point say, and define as. We know the values of at and and we can use this to give the following approximation to the gradient

What the above expression actually calculates is the gradient of the line which intersects the points and as you can see from the figure above. In the figure, the function is shown with a red line. The blue line represents the approximation to the gradient and the black line is the actual gradient of the function at .

This approximation to the gradient is called the forward difference approximation.