Correcting for Staggered Leapfrog

In practice the program uses a staggered leapfrog algorithm so in solving for $a''(t)$ the value of $a'$ is known at $t-d/2$ where $d$ is the time step. See section 6.4 for more details. The solution to this problem is to use the two equations

$$a_+' \approx a_-' + d a'';\;a' \approx {1 \over 2}\left(a_+' + a_-'\right)$$ (6.27)

where $a_+'$ and $a_-'$ refer to the values of $a'$ at $t+d/2$ and $t-d/2$ respectively and all other variables are evaluated at time $t$. Take the evolution equation to be

$$a'' = -C_1 {a'^2 \over a} + C_2.$$ (6.28)

Plugging this form into equation (6.27) and eliminating $a'$ gives

$$a_+' \approx a_-' + d \left(-{C_1 \over 4 a} \left(a_+' + a_... ...\over 2 a} a_+' a_-' - {d C_1 \over 4 a} a_-'^2 + d C_2 + a_-'$$ (6.29)

$$a_+' \approx -{2 a \over d C_1} \left({d C_1 \over 2 a} a_-' + 1 \pm \sqrt{{d^... ..._1^2 \over 4 a^2} a_-'^2 + {d^2 C_1 C_2 \over a} + {d C_1 \over a} a_-'}\right)$$ (6.30)

$$= -a_-' - {2 a \over d C_1} \pm {2 a \over d C_1}\sqrt{1 + {2 d C_1 \over a} a_-' + {d^2 C_1 C_2 \over a}}.$$ (6.31)

To determine whether to use the plus or minus sign in equation (6.31) consider the limit as $d \rightarrow 0$. In this limit

$$a_+' \approx -a_-' - {2 a \over d C_1} \pm {2 a \over d C_1}... ...{2 a \over d C_1} \pm \left({2 a \over d C_1} + 2 a_-'\right).$$ (6.32)

This suggests that the plus sign must be used in order to reduce to the limit $a_+' \approx a_-'$. Hence

$$a_+' \approx -a_-' - {2 a \over d C_1}\left(1 - \sqrt{1 + {2 d C_1 \over a} a_-' + {d^2 C_1 C_2 \over a}}\right).$$ (6.33)

In the program it's useful to calculate $a''$, which is roughly $(a_+' - a_-')/d$, so

$$a'' \approx {1 \over d} \left[-2 a_-' - {2 a \over d C_1}\le... ...{2 d C_1 a_-' \over a} + {d^2 C_1 C_2 \over a}}\right)\right].$$ (6.34)

Thus equation (6.26) becomes

$$a'' \approx {1 \over d}\left\{-2 a_-' - {2 a \over d C_1} \l... ...a_{pr} f_{i,pr}\vert^2 + a^{C_4} V_{pr}\right)}\right]\right\}$$ (6.35)

where

$$C_1 = s+2;\;C_3 = 2s+2r+2;\;C_4 = 2s+2$$ (6.36)