the one step method defined by $x_{k+1} = x_k + hγ(t_k,x_k)$
for the ODE $dx(t)/dt = f(t,x(t))$
with $γ(t_k,x_k) = f(t_k+h/2,x_k+(h/2)*f(t_k,x_k))$.
what is the conditions for the convergence of one-step methods considered above
And, show that if for some constant M one has $|f(t,x)-f(t,y)| \le M|x-y|$ then the one-step method with the above γ is stable.
Also, show that if in addition to this condition one also has $|f(t,x)-f(s,x)|\le C|t-s|$ then the method is consistent.
