the order of a method
When you halve the step size of a numerical solver, how much does the error shrink? For some methods it merely halves; for others it drops to a quarter, or a sixteenth. The ORDER of a method is the single number that answers this: it is the power p such that the error behaves like h^p, telling you how richly accuracy is rewarded as you refine the step.
Precisely, a method has order p if its GLOBAL error scales as O(h^p) as h -> 0. Equivalently, its per-step local truncation error is O(h^(p+1)) (one power higher, because the errors of about 1/h steps accumulate and lose a power of h). So forward Euler, with local error O(h^2) and global error O(h), is order 1; the midpoint and Heun methods are order 2; the classic RK4 is order 4. The practical meaning of order p: halving h multiplies the error by (1/2)^p, so a fourth-order method gains 4 correct binary digits per halving versus 1 for a first-order method. A method achieves order p by matching the Taylor expansion of the true solution through the h^p term — higher-order methods sample the slope at more places to cancel more terms.
Order is the headline figure on a solver's spec sheet, but read the fine print. Higher order is not free: each step costs more function evaluations (RK4 needs four), so the win is in accuracy-per-unit-work, which favors high order when you want many correct digits and the solution is smooth. Crucially, the order is an ASYMPTOTIC statement — it describes behavior as h -> 0 and may not kick in until h is small enough; for a non-smooth f, a singularity, or a stiff problem, the observed convergence can be slower than the nominal order, or absent entirely until stability is satisfied. A higher-order method is also no help if round-off, not truncation, dominates.
Solving a smooth ODE and halving h: a first-order method's error goes 0.1 -> 0.05 -> 0.025 (factor 2); a second-order method's goes 0.1 -> 0.025 -> 0.00625 (factor 4); RK4's goes 0.1 -> 0.00625 -> 0.00039 (factor 16). Plotting log(error) versus log(h) gives a straight line whose SLOPE is the order p — the standard way to measure a method's order empirically.
Order p = the slope of log(error) vs log(h); halving h shrinks error by 2^p.
Order is asymptotic and assumes smoothness and stability. A nominal fourth-order method can show first-order convergence near a kink or singularity, and for stiff problems an explicit method shows NO convergence until the step satisfies the stability limit — order means nothing across that line.