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Frederic Thomas
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An elementary question about an integration formula (nomenclature probably archaic)

so I am working on this research paper: https://docdro.id/sZsZiYL

Basically, the authors use a so-called Yee grid in order to discretize Maxwell's equations for computational purposes. In the paper, there is a mention of an integration formula that will be applied to the Maxwell-Faraday equation, namely:

$\oint_{\Gamma_S} \vec{E} \cdot \vec{dl} = -\iint_{S_\Gamma} \frac{\partial \vec{B}}{\partial t} \cdot \vec{ds}$

In the paper you find the following grid setup for the integration of the E field:

Integration contour

Now the discretized integral of the left-hand side of our Maxwell-Faraday is given by:

Formula

which they describe in the paper as first-order integration formula

enter image description here

My questions to the respectable members in here are:

  1. what do they mean by the first-order integration formula? does it have a more contemporary name?
  2. What is the deal with the big Os? what do they represent?

Many thanks for considering my request.

PS:

  1. I have already went throw the internet to have a reliable definition of the formula, but it was in vain.
  2. The ds in the paper designates a contour and not an area, I just went with l since it makes more sense to me.