Finite Difference - Higher-order Differences

Higher-order Differences

In an analogous way one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for and and applying a central difference formula for the derivative of at x, we obtain the central difference approximation of the second derivative of f:

2nd Order Central

Similarly we can apply other differencing formulas in a recursive manner.

2nd Order Forward

More generally, the nth-order forward, backward, and central differences are respectively given by:

$\Delta^n_h(x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x + (n - i) h),$

$\nabla^n_h(x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x - ih),$

$\delta^n_h(x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f\left(x + \left(\frac{n}{2} - i\right) h\right).$

Note that the central difference will, for odd, have multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of and .

The relationship of these higher-order differences with the respective derivatives is very straightforward:

Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order h. However, the combination

approximates f'(x) up to a term of order h2. This can be proven by expanding the above expression in Taylor series, or by using the calculus of finite differences, explained below.

If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.

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