Refer to problem 11 in chapter 5.11 of Numericical Analysis, 9 th ed. R. Burden, J. Faires.

The question asks "Discuss the stability, consistency and convergence of the implicit trapezoidal method." I have the impression that this question is vague and will divide my concerns into three questions.

(1)

For stability, do they ask to calculate a region of stability or to prove stability by satisfying the Lipschitz constant? A-stability was easy to find but Lipschitz's constant I still find it difficult to understand how this proves that the method is unconditionally stable for all $ h> $ 0. Can someone guide me through this?

(2) For the sake of consistency, I know that the method is consistent if $$ lim_ {h-> 0} | tau_n | = 0 $$

or $ tau_n $ is the local truncation error and the order of consistency is given by:

$$ frac {y (t_ {n + 1}) – y (t_ {n})} {h} – f (t_n, y (t_n)) $$

now I have it $ tau_n = frac {h ^ 3} {3!} y & # 39; & # 39; & # 39; (t_n) $ Which one is $ 0 for $ h $ -> $ infty $ which means that the method is consistent. For the sake of consistency, I get the following:

$$ frac {y (t_ {n + 1}) – y (t_ {n})} {h} – f (t_n, y (t_n)) = frac {h ^ 2} {2} y & # 39; & # 39; (t_n) + O (h ^ 3) $$ which suggests that the order of consistency is of order $ O (h ^ 2) $. My problem is that we can confirm consistency by having

$$ phi (t, w, h) = frac {1} {2}[f(t,y) + f(t+h,y+h)]$$

$$ phi (t, w, 0) = frac {1} {2}[f(t,y) + f(t+0,y+0)] = f (t, y) $$

and since $ phi (t, w, 0) = f (t, y) $ the method is considered consistent. This sounds too trivial, can anyone explain this further and perhaps show me how this applies to the trapezoidal rule?

(3) Finally, we say that if the method is stable and consistent, it will converge. Can someone confirm it to me and show me how to calculate the order of convergence?

Any help is greatly appreciated!