# Slope limiting for discontinuous Galerkin (DG) method

I had a question regarding the implementation of the TVB limiter for the RKDG method by Cockburn. I have seen that some implementations of the DG method use normalized Legendre polynomials such that the basis functions take the form

$$\phi_{i}^{k}(x) = \sqrt{2k + 1}P^{k}(x).$$

With this formulation, we have $$u_{h}(x) = \bar{u} + \sqrt{3} u_{x} \phi_{i}(x)$$

Therefore we have a limited slope of the form

$$s_{i}^{m} = \frac{1}{\sqrt{3}} \text{minmod} \left(\sqrt{3} u_{x}, \bar{u}_{i} - \bar{u}_{i-1}, \bar{u}_{i+1} - \bar{u}_{i}\right)$$ However, I am using a non-normalized Legendre modal formulation. As such, should'nt my limited slope just be

$$s_{i}^{m} = \text{minmod} \left(u_{x}, \bar{u}_{i} - \bar{u}_{i-1}, \bar{u}_{i+1} - \bar{u}_{i}\right)?$$

If I don't have the normalization constants my results are not correct. I am currently validating my 1D solver on the famous Sod-Shock tube. As further details my system of equations with my DG formulation is of the form

$$\frac{\Delta x}{2m + 1} \frac{d}{d t} c_{i,m}(t) - \int_{\Omega_{i}} f\left( \sum_{m=0}^{k} c_{i,m}(t) \phi_{i,m}(x) \right) \frac{\partial \phi_{i,l}(x)}{\partial x} dx + \left[\hat{f} \left( \sum_{m=0}^{k} c_{i,m}(t) \phi_{i,m}(x)\right) \phi_{i,l}(x)\right]^{x_{i}}_{x_{i-1}} = 0, m = 0, \dots, k$$

The paper of Cockburn and Shu  explains this. If the solution is $$u_h(x) = u_i + u_{xi} \phi(x), \qquad \phi(x) = \frac{x - x_i}{\Delta x/2}$$ Then the limiter is $$u_{xi} = minmod(u_{xi}, u_i - u_{i-1}, u_{i+1}-u_i)$$ Does your solution representation look as in the first equation above ?
Also, the solution with normalised Legendre basis will look like this $$u_h(x) = u_i + u_{xi} \sqrt{3} \phi(x), \qquad \phi(x) = \frac{x - x_i}{\Delta x/2}$$ Then the limiter is $$u_{xi} = \frac{1}{\sqrt{3}} minmod(\sqrt{3}u_{xi}, u_i - u_{i-1}, u_{i+1}-u_i)$$