Question

A particular finite difference approximation for the first derivative of a function is

f′(xᵢ) = -f(xᵢ+3)+9f(xᵢ+1)-8f(xᵢ)/6h
where the points are all equally spaced with a step size h. What is the order of the truncation error? Hint: use the Taylor series expansions for f (xᵢ+3) f (xᵢ+1).

Answer 1

The order of the truncation error for the given finite difference approximation is second order.

Step-by-step explanation:

The order of the truncation error for this particular finite difference approximation can be determined using Taylor series expansions for f(xi+3) and f(xi+1).

Let's expand f(xi+3) and f(xi+1) in Taylor series around xi:

f(xi+3) = f(xi) + 3hi)> + \frac{9h^2}{2}i)> + O(h^3)

f(xi+1) = f(xi) + hi)> + \frac{h^2}{2}i)> + O(h^3)

Now substitute these expansions into the finite difference approximation formula:

f'(xi) = -f(xi+3) + 9f(xi+1) - 8f(xi) / 6h

f'(xi) = -f(xi) - 3hi)> - \frac{9h^2}{2}i)> + 9f(xi) + 9hi)> + \frac{9h^2}{2}i)> - 8f(xi) / 6h

This can be simplified to:

f'(xi) = -\frac{10}{6}f(xi) + \frac{9}{6}f(xi+1) - \frac{1}{6}f(xi+3) - \frac{h}{3}i)> + O(h^2)

Therefore, the order of the truncation error for this finite difference approximation is second order.