Difference between Orders of Error, Accuracy & Convergence

Key Differences

Degree of Accuracy $(K)$

Used for quadrature formulas - the maximum degree of polynomials that are integrated exactly.

Order of Consistency $(q)$

Used for ODE methods - The order of consistency tells us how quickly the local error of a numerical method decreases as we make the step size smaller. It is the power of $h$ in the local truncation error: $\tau =O(h^q)$.

Convergence Order

The power of $h$ in the global error bound (multiple steps)- often equals consistency order for stable methods.

Summary Table

Term	Context	Definition	Example
Degree of Accuracy $(K)$	Numerical Integration	Max polynomial degree integrated exactly	Trapezoid rule: $K=1$
Consistency Order $(q)$	ODE Methods	Power in local error: $\tau =O(h^q)$	Euler method: $q=1$
Convergence Order	ODE Methods	Power in global error: $\Vert y(t_i)-y_i\Vert =O(h^q)$	Euler method: order $1$

Quick Examples

• Simpson’s rule: Degree of accuracy K = 3 (integrates cubics exactly)
• Euler method: Consistency order q = 1 (local error ~ h²)
• RK4: Consistency order q = 4 (local error ~ h⁵)