Understanding the MC flux limiter

[nicksauce]

I'm reading through these notes on numerical hydrodynamics. On pg17, they introduce the monotonized central difference flux limiter, which looks something like

$$\frac{\partial~a}{\partial~x}|_i = min\left({\frac{|a_{i+1}-a_{i-1}|}{2\Delta~x},2\frac{|a_{i+1}-a_{i}|}{\Delta~x},2\frac{|a_i-a_{i-1}|}{\Delta~x}}\right)sign\left(a_{i+1}-a_{i-1}\right)$$

Now I am a bit confused about what's happening here. The first term in the min looks like a derivative, but the next two terms look like twice the derivative. Can anyone explain what exactly is going on here?


Thanks

 

[Vailanor]

!The monotonized central difference flux limiter is a numerical technique used to approximate the true value of a partial derivative at a point. The first term in the min refers to the standard central difference formula, which is an approximation of the true derivative. The next two terms are modified versions of the central difference formula that take into account other nearby points, and therefore provide a more accurate approximation. The min operator allows the limiter to choose the most accurate approximation at each point. The sign operator ensures that the limiter only produces an estimate of the derivative that is consistent with the direction of the gradient.