Order of Consistency

Intuition

The order of consistency tells us how quickly the error of a numerical method decreases as we make the step size smaller.

Examples from ODE Methods:

1. Euler Method (Order 1)

• $\mathrm{T}(\mathrm{t},\mathrm{h})=\mathrm{O}(\mathrm{h})$
• Halving $\mathrm{h}\to$ error reduces by factor 2
• Simple but not very accurate

2. Euler-Heun Method (Order 2)

• $\mathrm{T}(\mathrm{t},\mathrm{h})=\mathrm{O}\left({\mathrm{h}}^2\right)$
• Halving $\mathrm{h}\to$ error reduces by factor 4
• Much more accurate than Euler

3. Classical Runge-Kutta (Order 4)

• $\mathrm{T}(\mathrm{t},\mathrm{h})=\mathrm{O}\left({\mathrm{h}}^4\right)$
• Halving $\mathrm{h}\to$ error reduces by factor 16
• Very high accuracy

Definition to be corrected