# Order of local Truncation error and order of method

I am getting confused with how to find the order of local truncation error and order of method. I solved a multistep ODE method and my local truncation error is coming out to be:

$$local truncation error =-\frac{y^{(5)}(t_i)h^5}{16}+{O(h^6)}$$ If my local truncation error is coming out as above, then what should be my order of method and order of local truncation. Thank you for great help.

Edit I am editing the question to make it more general. It is not specific to any ODE method.I found one solved example in the book where is calculates the order of multistep method. Reading through, it gives a hint that order of local truncation error is one more than order of method. My question is this can this rule be applied to all multistep ODE methods to find order and the order of local truncation error is 1 more than order of method? Thank you for great help.

Book - Numerical Analysis: Mathematics of Scientific Computing By Kincaid and Cheney, 3rd edition

• RK4 is of order 4, the remaining terms should be $+O(h^6)$. See math.stackexchange.com/a/3303621/115115 on my view on the different definitions, and math.stackexchange.com/a/1921748/115115 or math.stackexchange.com/a/2715978/115115 on how these local terms fit together in a global error. Commented Dec 9, 2022 at 8:23
• Thank you for your response @LutzLehmann. I edited the question a little. Somehow I think I found a general method to calculate order. Earlier I though order of method is found from the local truncation error. But I think I was wrong. For finding order of method, we only need the ODE method equation. Please review.
– Manu
Commented Dec 10, 2022 at 0:25
• Yes, for multi-step method the order as quadrature method gives the order of the integration method. For Runge-Kutta methods the quadrature method, solving $y'(x)=f(x)$, can be higher than the order as ODE integration method, solving $y'(x)=f(x,y(x))$. Commented Dec 10, 2022 at 6:47