Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 6 / c

The Forward Euler method uses the current slope:Here $x_n=x_0+nh$ and the initial approximation normally equals the given initial value. Each step takes a tangent-line increment of width $h$.

Define the one-step local truncation error as the defect obtained by inserting the exact solution into an update:For a smooth solution, Taylor's theorem and $y^{\prime }=f(x,y)$ giveThus $m=2$ for the one-step defect convention. The leading derivative can also be written $y^{\prime \prime }=f_x+f_{y}f$ by the chain rule.

Some conventions instead call ${\tau }_{n+1}={\delta }_{n+1}/h$ the truncation error. That normalized local truncation error is $O(h)$, so $m=1$ under the normalized convention. Both describe the same Euler one-step defect; neither should be confused with an unqualified global error claim.