Finite difference approximation of derivative

jf22901 · **Posted:** Mon, 22 Jan 2018 23:27:40 UTC

Hi all,

I'm reading the methods section of a Nature Astronomy research article, and am confused by an approximation to a derivative. The equation (mass conservation) is:

$\frac{\partial F}{\partial z} = -\frac{\partial}{\partial t}(\rho q)$ ,

where F is the flux (kg m^-2 s^-1), rho is the atmospheric density, q is a mixing ratio (in this case dust), t is time and z is altitude. They have approximated the RHS as:

$\frac{\partial}{\partial t}(\rho q) = \frac{\rho_2 q_2}{t_2} - \frac{\rho_1 q_1}{t_1}$ ,

where the subscripts 1 and 2 refer to times 1 and 2. What I can't understand is why it isn't:

$\frac{\partial}{\partial t}(\rho q) = \frac{\rho_2 q_2 - \rho_1 q_1}{t_2 - t_1}$ .

I assume I've forgotten something obvious relating to approximating derivatives, but I can't think at the moment what it is! In case anyone can access these things, the article is at https://www.nature.com/articles/s41550-017-0353-4, and the approximation is Eq. 2 in the methods.

Thanks!

S.O.S. Mathematics CyberBoard

Finite difference approximation of derivative

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