Ref to problem 11 in chapter 5.11 in Numerical Analysis 9th ed. R. Burden, J. Faires.
The question asks to "Discuss the stability, consistency and convergence of the implicit trapezoidal method." I feel like this question is vague and will split my concerns in three questions.
(1) For stability, are they asking to compute a stability region or to prove stability through satisfying Lipschitz constant? A-stability was easy to find but the Lipschitz constant I am still finding a hard time understanding how that proves that the method is unconditionally stable for all $h > 0$. Can someone walk me through this?
(2) For consistency, I know the method is consistent if $$\lim_{h->0} |\tau_n / h| = 0$$ where $\tau_n$ is the local truncation error and the order of consistency is given by: $$\frac{y(t_{n+1}) - y(t_{n})}{h} - \phi(t_n,y(t_n),h)$$ now I got that $\tau_n = \frac{h^3}{3!}y'''(t_n)$ which is $0$ for $h$ -> $0$ which means that the method is consistent. For the order of consistency I am getting the following: $$\frac{y(t_{n+1}) - y(t_{n})}{h} - \phi(t_n,y(t_n),h) = \frac{h^2}{2}y''(t_n) + O(h^3)$$ which suggests the order of consistency is of order $O(h^2)$. My issue is that one can confirms consistency by having $$\phi(t,y,h) = \frac{1}{2}[f(t,y) + f(t+h,y+h)]$$ $$\phi(t,y,0) = \frac{1}{2}[f(t,y) + f(t+0,y+0)] = f(t,y)$$ and since $\phi(t,w,0) = f(t,y)$ the method is considered consistent. This seems too trivial, can someone explain this further and maybe show me how it's applied to the trapezoidal rule?
(3) Lastly, it's said that if the method is stable and consistent then it will converge. Can someone confirm this and show me how to calculate the order of convergence?
Any help is greatly appreciated!
(2) The simplified approach is a theorem about multi-step methods. The trapezoidal method has that form.
(3) The order of convergence is the order of consistency, if convergence is established. The order $2$ you found is correct, even if the second consistency/local truncation error formula you did it with is wrong.