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I have been reading upon numerical techniques that are used to solve stiff ordinary differential equations.

From the description given here, I could follow the steps till equation (5).

I am finding it difficult to understand how equation 6 is obtained and the calculations involved in computing the coefficients of BDF.

Could someone recommend a reference in which these steps have been detailed?

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  • $\begingroup$ This is probably more suited to math.se $\endgroup$ – Nox Jan 6 at 17:22
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    $\begingroup$ I think there won’t be a reference because it’s just calculus: the lhs is a function of t, the rhs is a polynomial in t, so you just differentiate both to get a linear equation in $y_n$, solve for it and read off the coefficients of the other y’s. $\endgroup$ – Kirill Jan 6 at 17:50
  • $\begingroup$ Actually, there is no solve involved. One directly reads off the coefficients for all $y_{n-j}$s so that $f(t_n)=\dot y(t_n)$ is exactly approximated by the polynomial (up to order $k$). $\endgroup$ – Jan Jan 8 at 14:03
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In BDF schemes for $\dot y = f$, one uses $$ f(t_n)=\dot y(t_n) $$ and tries to approximate $\dot y(t_n)\approx \sum_{j=0}^k\alpha_k y_{n-j}$ by the current value $y_n$ (that is to be computed) and the $k$ previously computed approximations.

In the presented approach, in $(5)$, $y$ is approximated as a polynomial $p$ in $t$ fitted to $y_{n-j}$, so that the time derivative of the polynomial at $t_n$ approximates $\dot y(t_n)$ as desired. With that, for a given approximation order $k$, one can read of the coefficients $\alpha_j$ by evaluating $\dot p$ at $t_n$:

For $k=1$:

$$ \quad \dot y(t_n) \approx \dot p(t_n) = \frac{1}{h}(y_n - y_{n-1}) $$ which gives that $h\dot y(t_n)$ is approximated by $$1\cdot y_n + (-1)\cdot y_{n-1}.$$

For $k=2$ the terms read: $$ k=2: \quad \dot y(t_n) \approx \dot p(t_n) = \frac{1}{h}(y_n - y_{n-1})+\frac{1}{2h^2}[(t_n-t_n)\nabla^2y_n + (t_n-t_{n-1})\nabla^2y_n] $$ which, with $t_n-t_{n-1}=h$ and $\nabla^2y_n = y_n - 2y_{n-1} + y_{n-2}$ gives that $h\dot y(t_n)$ is approximated by $$\frac{3}{2}\cdot y_n + (-2)\cdot y_{n-1} + \frac{1}{2}y_{n-2}$$.

And so on...

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    $\begingroup$ If my understanding is right, the interpolating polynomial is in Newton's form, $p(t)=y[t_1] + y[t_1,t_2](t-t_1)+y[t_1,t_2,...,t_k](t-t_1)(t-t_2)..(t-t_{k-1})$ $\endgroup$ – Natasha Jan 9 at 4:19
  • $\begingroup$ yes, if you say it. I wasn't sure and I did not check the definition. $\endgroup$ – Jan Jan 9 at 7:22
  • $\begingroup$ How does one choose the intial data in the best way? $\endgroup$ – Emil Jan 9 at 19:19
  • $\begingroup$ Maybe my comment was off topic, found this anyway: math.stackexchange.com/a/421107/68036 (predictor-corrector) $\endgroup$ – Emil Jan 9 at 19:53

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