User:DanielCarrera

I am an astronomer. I am a postdoc at Iowa State University. I use computer simulations to test models of planet formation.

$$$$

\begin{bmatrix}

2 & & & & & & \\

1 & 4 & 1 & & & & \\

& 1 & 4 & 1 & & & \\

& & 1 & 4 & 1 & & \\

& & & 1 & 4 & 1 & \\

& & & & 1 & 4 & 1 \\

& & & & & & 2 \\

\end{bmatrix}

\begin{bmatrix}

f_1' \\

f_2' \\

f_3' \\

f_4' \\

f_5' \\

f_6' \\

f_7' \\

\end{bmatrix}

= \frac{1}{h}

\begin{bmatrix}

-3& 4 &-1 & & & & \\

-3& & 3 & & & & \\

&-3 & & 3 & & & \\

& &-3 & & 3 & & \\

& & &-3 & & 3 & \\

& & & &-3 & & 3 \\

& & & & 1 &-4 & 3 \\

\end{bmatrix}

\begin{bmatrix}

f_1 \\

f_2 \\

f_3 \\

f_4 \\

f_5 \\

f_6 \\

f_7 \\

\end{bmatrix}

$$

f_{i+1} = f_i + f'_i h

{\color{red} + \frac{f''_i}{2!}h^2}

{\color{blue}

+ \frac{f^{(3)}_i}{3!}h^3

+ \frac{f^{(4)}_i}{4!}h^4

+ \frac{f^{(5)}_i}{5!}h^5

+ \frac{f^{(6)}_i}{6!}h^6

+ \cdots}

$$

f_{i+1}' = f'_i

{\color{red} + f''_i h}

{\color{blue}

+ \frac{f^{(3)}_i}{2!}h^2

+ \frac{f^{(4)}_i}{3!}h^3

+ \frac{f^{(5)}_i}{4!}h^4

+ \frac{f^{(6)}_i}{5!}h^5

+ \cdots}

$$

f_1' + 3 f_2' = \frac{- 17f_1 + 9f_2 + 9f_3 - f_4}{6h}

+ {\color{blue}\mathcal{O}( h^4 )}

$$

f_N' + 3 f_{N-1}' = \frac{17f_N - 9f_{N-1} - 9f_{N-2} + f_{N-3}}{6h}

+ {\color{blue}\mathcal{O}( h^4 )}

$$

\frac{1}{3}f'_{i-1}

+ f'_{i} + \frac{1}{3}f'_{i+1}

= \frac{14}{9} \frac{f_{i+1}-f_{i-1}}{2h}

+ \frac{1}{9} \frac{f_{i+2}-f_{i-2}}{4h}

+ {\color{blue}\mathcal{O}( h^6 )}

$$

f'_{i} =

- \frac{49f_i}{20h}

+ \frac{ 6f_{i+1}}{h}

- \frac{15f_{i+2}}{2h}

+ \frac{20f_{i+3}}{3h}

- \frac{15f_{i+4}}{4h}

+ \frac{ 6f_{i+5}}{5h}

- \frac{ f_{i+6}}{6h}

+ {\color{blue}\mathcal{O}( h^6 )}

$$

f'_{i} =

+ \frac{49f_i}{20h}

- \frac{ 6f_{i-1}}{h}

+ \frac{15f_{i-2}}{2h}

- \frac{20f_{i-3}}{3h}

+ \frac{15f_{i-4}}{4h}

- \frac{ 6f_{i-5}}{5h}

+ \frac{ f_{i-6}}{6h}

+ {\color{blue}\mathcal{O}( h^6 )}