I was reading through a paper about the following PDE
$$ \frac{\partial u}{\partial t} = D_e \frac{\partial^2 u}{\partial z^2} - V_e \frac{\partial u}{\partial z}. $$
This PDE models the transport of gas tracers through unsaturated zones in soils.
Along with the paper I had a numerical solver for this PDE written by one of the paper's authors. In an attempt to understand the solver I wrote my own using the Crank-Nicolson method.
When I wrote my solver I approximated $ \frac{\partial u}{\partial t} $ using the forward difference approximation,
$$ \frac{\partial u}{\partial t} \approx \frac{u_{m + 1,\ j} - u_{m,\ j}}{\Delta t}, $$
where $u_{m,\ j}$ is the value of $u$ at the $m$th step in time and the $j$th step in position on the solution grid, and $\Delta t$ is the length of the time step which is constant.
This method obtained a solution close to the author's solver but after reading through the author's solver I noticed that he has used a different forward difference approximation for $ \frac{\partial u}{\partial t} $,
$$ \frac{\partial u}{\partial t} \approx \frac{u_{m + 1,\ j + 1} - u_{m,\ j + 1}}{6\Delta t} + 2\frac{u_{m + 1,\ j} - u_{m,\ j}}{3\Delta t} + \frac{u_{m + 1,\ j - 1} - u_{m,\ j - 1}}{6\Delta t}. $$
Is this approximation more accurate than the one I used? Should it always be used instead of the one I used?
It is not mentioned in the text I used as a reference for the Crank-Nicolson method. What is this weighted approximation called so I can look for a reference?
Why are the different forward difference approximations weighted as they are: one sixth for the positions before and after and two thirds for the current position?