Question regarding notation in numerical methods

Question

I am sitting with a problem where I have to show that Heun's method is consistent w.r.t. properties which is not related to my question. Anyway.

I am given two different definitions and the above statement is easy to prove with

(1) $\phi (t,y;h) = \frac{1}{2}(f(t,y)+f(t+h,y+hf(t,y)))$ (Heuns method)

and for the second definition I am given I have that

(2) $\int_{x_{j}}^{x_{j+1}} f(t,y(t))dt \approx \frac{1}{2}h_{j}(f(x_{j},y_{j}) + f(x_{j+1},y_{j+1})) \\ \approx \frac{1}{2}h_{j}(f(x_{j},y_{j}) + f(x_{j+1},y_{j}+h_{j}f(x_{j},y_{j})))$

But I am confused about how these two different ways of expressing Heuns method are related, especially since notation (1) is given after the exercise where I have to use it

If you use $x$ instead of $t$ in (1) do you see some similarites to (2)? — gammatester, Oct 06 '18 at 13:06
No I am struggling with seeing how I can go from (2) to (1), and why we can do it — Chengdu, Oct 06 '18 at 13:12
(2) is just a composite quadrature. (1) is just one application of the rule. — Sean Roberson, Oct 06 '18 at 14:06
You want to compute the function value $y_{j+1}$ from $y_j,$ but in the integral rule (2) it is implicit in the first line, to make it explicit you simply do a linear approximation $y_{j+1} = y_{j} + h_jf(x_j, y_j)$ and get the second line of (2). The rest is substitution. — gammatester, Oct 06 '18 at 14:12

score 1 · Accepted Answer · answered Oct 08 '18 at 14:27

The ODE is equivalent to the integral equation $$ y(x+h)=y(x)+\int_0^h f(x+s, y(x+s))\,ds. $$ First we approximate the integral with the trapezoidal formula $$ y(x+h)=y(x)+\frac h2[f(x,y(x))+f(x+h, y(x+h))] + O(h^3) $$ which now is an implicit equation for $y(x+h)$. Reflect that we now compute approximations by using $y_k$ for the approximation of $y(x_k)$, $x_k=x_0+kh$, $$ y_{k+1}=y_k+\frac h2[f(x_k,y_k)+f(x_{k+1}, y_{k+1})]. $$ This is the implicit trapezoidal method. Now we can replace the $y_{k+1}$ on the right side with an $O(h^2)$ approximation $\tilde y_{k+1}$ without changing the error order $O(h^3)$ of the step. The immediate candidate for that first approximation is the value of the Euler step $\tilde y_{k+1}=y_k+hf(x_k,y_k)$. This then constitutes the explicit trapezoidal or modified Euler or Heun's method.

Question regarding notation in numerical methods

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