If we take a 1st order polynomial approximation in each cell, we can find the (2x2) -mass matrix, differentiation matrix, and flux matrix through the integration of Lagrange polynomials.
However, I am not able to comprehend the simplified equation which we obtain on using centered numerical flux. It seems that we get the result same as up-winding/ down-winding if we ensure continuity of solution.
According to me, if we multiply the element equation by its mass inverse we should be getting the final form, but it doesn't seem to be the case. I would be glad if someone points out my mistake in understanding.
I am attaching the content, which I am referring to: -