Truncation errors in numerical integration are of two kinds:
local truncation errors – the error caused by one iteration, and global truncation errors – the cumulative error caused by many iterations.
Definitions Suppose we have a continuous differential equation
y ′ = f ( t , y ) , y ( t 0 ) = y 0 , t ≥ t 0 {\displaystyle y'=f(t,y),\qquad y(t_{0})=y_{0},\qquad t\geq t_{0}}
and we wish to compute an approximation y n {\displaystyle y_{n}} of the true solution y ( t n ) {\displaystyle y(t_{n})} at discrete time steps t 1 , t 2 , … , t N {\displaystyle t_{1},t_{2},\ldots ,t_{N}} . For simplicity, assume the time steps are equally spaced:
h = t n − t n − 1 , n = 1 , 2 , … , N . {\displaystyle h=t_{n}-t_{n-1},\qquad n=1,2,\ldots ,N.}
Suppose we compute the sequence y n {\displaystyle y_{n}} with a one-step method of the form
y n = y n − 1 + h A ( t n − 1 , y n − 1 , h , f ) . {\displaystyle y_{n}=y_{n-1}+hA(t_{n-1},y_{n-1},h,f).}
The function A {\displaystyle A} is called the increment function, and can be interpreted as an estimate of the slope y ( t n ) − y ( t n − 1 ) h {\displaystyle {\frac {y(t_{n})-y(t_{n-1})}{h}}} .
Local truncation error The local truncation error τ n {\displaystyle \tau _{n}} is the error that our increment function, A {\displaystyle A} , causes during a single iteration, assuming perfect knowledge of the true solution at the previous iteration. More formally, the local truncation error, τ n {\displaystyle \tau _{n}} , at step n {\displaystyle n} is computed from the difference between the left- and the right-hand side of the equation for the increment y n ≈ y n − 1 + h A ( t n − 1 , y n − 1 , h , f ) {\displaystyle y_{n}\approx y_{n-1}+hA(t_{n-1},y_{n-1},h,f)} :
τ n = y ( t n ) − y ( t n − 1 ) − h A ( t n − 1 , y ( t n − 1 ) , h , f ) . {\displaystyle \tau _{n}=y(t_{n})-y(t_{n-1})-hA(t_{n-1},y(t_{n-1}),h,f).}
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