How do they go from the left side to the approximation on the right? What does $j$ mean in this summation? $$ {\partial f_i^n \over \partial t} \approx \sum_j {\partial f_i^n \over \partial \hat x_j} v_j^{n+1} \\ \hat x_i = x_i + \Delta tv_i $$ Background:
They are computing this value on a discrete grid. $f_i$ is a force on the grid at position $i$ (where $i$ is a vector in 3-dimensional space). $v$ is velocity and $x$ is position, both vectors. $n$ is the current time, and $n+1$ is $t + \Delta t$.
I may be able to provide further information, if something is unclear. Original source with equation here.