the order of accuracy

The order of accuracy describes how fast an approximation's error shrinks as you make a step size h smaller — captured by big-O notation in its accuracy sense: error = O(h^p) means the error is bounded by some constant times h^p for small h. The exponent p is the order. A first-order method (p = 1) has error roughly proportional to h; a second-order method (p = 2) has error proportional to h^2, which falls far faster.

The practical punch of the order is what it predicts when you refine. With an O(h^p) method, halving the step multiplies the error by (1/2)^p: a first-order error halves, a second-order error drops to a quarter, a fourth-order error to a sixteenth. So order tells you the payoff of extra work. The forward difference (f(x+h) - f(x))/h is O(h), first order; the central difference (f(x+h) - f(x-h))/(2h) is O(h^2), second order; the trapezoidal rule is O(h^2) and Simpson's rule O(h^4). Higher order buys dramatically more accuracy per step, though usually at more work per step.

Be careful: O(h^p) is an asymptotic statement about the limit h to 0; it hides the constant and says nothing for large h, where a nominally lower-order method can win. It also assumes the function is smooth enough — a fourth-order method on a non-smooth function silently drops to a lower observed order. The standard way to measure the true order in practice is an order-of-accuracy (convergence) study: halve h repeatedly, look at the ratio of successive errors, and read off p from how fast they fall. This is distinct from the OTHER use of big-O, which measures an algorithm's cost rather than its error.