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

Question

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:

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

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

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.

Answer 1

I would say that it is a standard nomenclature still in use. Certainly, you have heard about the second-order or fourth-order Runge-Kutta algorithms.

In your problem, this naming is applied to an algorithm to evaluate a surface integral, but the reason for the name is the same. It has to do with the way the global error goes to zero once the formula for a single element is summed over a finite set of elements required to evaluate the integral (or integrate a differential equation) over a finite domain.

Let me explain the mechanism underlying the name with a simple 1D example.

Trapezoidal rule gets its name from the fact that the integral of a function $f$ over an interval $ [ x_i, x_{i}+h]$ can be evaluated as $$ \int_{x_i}^{x_i+h}f(x)dx = \frac{h}{2}(f(x_i)+f(x_{i+1}))-\frac{h^3}{12}f''(\xi_i) $$ where $\xi_i$ is a point in the interval $(x_i,x_i+h)$, and $f''$ is the second derivative of $f$. When this single-interval rule is applied additively to the $N$ equispaced intervals required to get the integral over the finite interval $[a,b]$, we have $$ \begin{align} &h = \frac{b-a}{N}\\ &x_0=a\\ &x_N=b\\ &\int_{a}^{b}f(x)dx= \frac{h}{2}\sum_{i=0}^{N-1}(f(x_i)+f(x_{i+1}))-\frac{h^3}{12}\sum_{i=0}^{N-1}f''(\xi_i). \end{align} $$ The sum in the error term grows as $N$. It is then convenient to rewrite it as $$ -\frac{h^3}{12}\sum_{i=0}^{N-1}f''(\xi_i)=-N\frac{h^3}{12}\left(\frac{1}{N}\sum_{i=0}^{N-1}f''(\xi_i) \right). $$ Assuming the continuity of $f''$, a point $\xi \in (a,b)$ exists such that $$ \frac{1}{N}\sum_{i=0}^{N-1}f''(\xi_i) =f''(\xi) $$ and we arrive to the expression of the global error as $$ -h^2\frac{(b-a)}{12}f''(\xi). $$ Thus, the global error is second order in $h$, i.e., one order smaller than the local error.

This phenomenon is completely general: a local error of the order $n$ becomes a global error of order $n-1$, providing an answer to why a discretization method with an error $O(h^2)$ is called a first-order method.