User:DanielCarrera

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

$$ \begin{bmatrix} 2 &  &   &   &   &   &   \\ 1 & 4 & 1 &   &   &   &   \\  & 1 & 4 & 1 &   &   &   \\  &   & 1 & 4 & 1 &   &   \\  &   &   & 1 & 4 & 1 &   \\  &   &   &   & 1 & 4 & 1 \\  &   &   &   &   &   & 2 \\ \end{bmatrix} \begin{bmatrix} f_1' \\ f_2' \\ f_3' \\ f_4' \\ f_5' \\ f_6' \\ f_7' \\ \end{bmatrix} = \frac{1}{h} \begin{bmatrix} -3& 4 &-1 &  &   &   &   \\ -3&   & 3 &   &   &   &   \\  &-3 &   & 3 &   &   &   \\  &   &-3 &   & 3 &   &   \\  &   &   &-3 &   & 3 &   \\  &   &   &   &-3 &   & 3 \\  &   &   &   & 1 &-4 & 3 \\ \end{bmatrix} \begin{bmatrix} f_1 \\ f_2 \\ f_3 \\ f_4 \\ f_5 \\ f_6 \\ f_7 \\ \end{bmatrix} $$

$$ f_{i+1} = f_i + f'_i h   {\color{red} + \frac{f''_i}{2!}h^2} {\color{blue} + \frac{f^{(3)}_i}{3!}h^3 + \frac{f^{(4)}_i}{4!}h^4 + \frac{f^{(5)}_i}{5!}h^5 + \frac{f^{(6)}_i}{6!}h^6 + \cdots} $$

$$ f_{i+1}' = f'_i {\color{red} + f''_i h}   {\color{blue} + \frac{f^{(3)}_i}{2!}h^2 + \frac{f^{(4)}_i}{3!}h^3 + \frac{f^{(5)}_i}{4!}h^4 + \frac{f^{(6)}_i}{5!}h^5 + \cdots} $$

$$ f_1' + 3 f_2' = \frac{- 17f_1 + 9f_2 + 9f_3 - f_4}{6h} + {\color{blue}\mathcal{O}( h^4 )} $$

$$ f_N' + 3 f_{N-1}' = \frac{17f_N - 9f_{N-1} - 9f_{N-2} + f_{N-3}}{6h} + {\color{blue}\mathcal{O}( h^4 )} $$

$$ \frac{1}{3}f'_{i-1} + f'_{i} + \frac{1}{3}f'_{i+1} = \frac{14}{9} \frac{f_{i+1}-f_{i-1}}{2h} + \frac{1}{9} \frac{f_{i+2}-f_{i-2}}{4h} + {\color{blue}\mathcal{O}( h^6 )} $$

$$ f'_{i} = - \frac{49f_i}{20h} + \frac{ 6f_{i+1}}{h} - \frac{15f_{i+2}}{2h} + \frac{20f_{i+3}}{3h} - \frac{15f_{i+4}}{4h} + \frac{ 6f_{i+5}}{5h} - \frac{ f_{i+6}}{6h} + {\color{blue}\mathcal{O}( h^6 )} $$

$$ f'_{i} = + \frac{49f_i}{20h} - \frac{ 6f_{i-1}}{h} + \frac{15f_{i-2}}{2h} - \frac{20f_{i-3}}{3h} + \frac{15f_{i-4}}{4h} - \frac{ 6f_{i-5}}{5h} + \frac{ f_{i-6}}{6h} + {\color{blue}\mathcal{O}( h^6 )} $$