orderprecision

Order precision, or order of accuracy, is a concept in numerical analysis used to describe how the error of a numerical method decreases as the discretization parameter is refined. If the global error satisfies |error| ≤ C h^p for small h, where h represents a typical discretization size (such as grid spacing or timestep) and C is a constant, the method is said to have order p. The exponent p is the order of precision. Higher p means faster convergence as the discretization is refined. In many contexts, the local truncation error is O(h^{p+1}) and the global error is O(h^p), given suitable smoothness of the exact solution.

In floating-point computation, precision also refers to the number of significant digits used to represent numbers. The machine epsilon is a measure of the maximum relative rounding error when representing numbers in the floating-point system and influences the practical accuracy and stability of computations. Rounding, cancellation, and error accumulation can affect the realized order of convergence in finite-precision arithmetic, sometimes reducing or temporarily masking the theoretical order.

Common examples illustrate typical orders of accuracy: the forward Euler method for ordinary differential equations is first-order accurate; the classical Runge–Kutta method (RK4) is fourth-order accurate in the global sense for smooth problems; central difference schemes for derivatives are often second-order accurate, assuming sufficient smoothness.

Practical assessment of order involves computing a solution with different step sizes (e.g., h and h/2) and estimating the observed order p from the error ratios, typically using a log-log slope. Deviations from the expected order can indicate stability issues, stiffness, or insufficient regularity in the problem data.