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

$${\displaystyle {\begin{bmatrix}2&&&&&&\\1&4&1&&&&\\&1&4&1&&&\\&&1&4&1&&\\&&&1&4&1&\\&&&&1&4&1\\&&&&&&2\\\end{bmatrix}}{\begin{bmatrix}f_{1}'\\f_{2}'\\f_{3}'\\f_{4}'\\f_{5}'\\f_{6}'\\f_{7}'\\\end{bmatrix}}={\frac {1}{h}}{\begin{bmatrix}-3&4&-1&&&&\\-3&&3&&&&\\&-3&&3&&&\\&&-3&&3&&\\&&&-3&&3&\\&&&&-3&&3\\&&&&1&-4&3\\\end{bmatrix}}{\begin{bmatrix}f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}\\\end{bmatrix}}}$$

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

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

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

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

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

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

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