Slope limiting for discontinuous Galerkin (DG) method

Question

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

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

Answer 1

The paper of Cockburn and Shu [1] 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 $$ u_{xi} = \frac{1}{\sqrt{3}} minmod(\sqrt{3}u_{xi}, u_i - u_{i-1}, u_{i+1}-u_i) $$

[1] https://doi.org/10.1006/jcph.1998.5892