Past exam of the numerical linear algebra and numerical solution of differential equations course of the University of Oxford / 2020 / 1 / a / Solution

Write the increment function of the one-step method as $\Phi (x,v,h)=f(x+\beta h,v+\beta hf(x,v))$. Its consistency error isAs $h\to 0$, the first term tends to $y^{\prime }(x_n)$ and the second tends to $f(x_n,y(x_n))=y^{\prime }(x_n)$ by continuity. Both limits are uniform along the exact-solution curve, a compact set: its small stage perturbations lie in a fixed compact neighborhood, so uniform continuity applies there. Thus ${\max }_{n<N}|T_n|\to 0$, proving consistency for every $\beta \in [0,1]$.