I am currently learning nodal DG methods, primarily through the book by Warburton, and am a bit confused on how to handle surface integrals using straight edged elements. On page 187 (and on page 214) of Warburton's book, we have
$$ \int_{\partial E_{k}} \mathbf{n} \cdot f_{h} \ell_{i}^{k}(\mathbf{x}) d\mathbf{x}. $$
Recall that $f_{h} = \sum_{j=1}^{N} f_{j}(x,t)\ell_{j}(\mathbf{x})$ and $\ell_{i}(\mathbf{x})$ is our basis function, and so,
$$ \int_{\partial E_{k}} \mathbf{n} \cdot f_{h} \ell_{i}^{k}(\mathbf{x}) d\mathbf{x} = \int_{\partial E_{k}} \mathbf{n} \cdot \left[ \sum_{j=1}^{N} f_{j}(x,t)\ell_{j}(\mathbf{x}) \right]\ell_{i}^{k}(\mathbf{x}) d\mathbf{x}\\ = \sum_{j=1}^{N} \mathbf{n} \cdot f_{j}(x,t) \int_{\partial E_{k}} \ell{j}(\mathbf{x}) \ell_{i}(\mathbf{x}) d \mathbf{x}\\ = \sum_{j=1}^{N} \mathbf{n} \cdot f_{j}(x,t) M^{k}_{ij}. $$ Now, to discretize the surface mass matrix, we must use a quadrature rule on each face (since I am working in 2D it's a line integral along each edge). With that said, since we have an orthonormal basis, we can compute the mass matrix analytically. For volume integrals, Warburton shows that $M^{k}_{ij} = (\mathcal{V} \mathcal{V}^{T})^{-1} J,$ where $J$ is the metric Jacobian. So intuitively, the same would be for the mass matrix, however, the Vandermonde matrix, $\mathcal{V}$ would be constructed so its on the local face points $(r,s)$, where $(r,s) \in [-1,1]$. Now, looking at their code (https://github.com/tcew/nodal-dg/blob/master/Codes1.1/CFD2D/CurvedEulerRHS2D.m), we see that the surface integral term has no surface mass matrix. There is an inverse mass matrix applied to the numerical flux but it is the one from the time-derivative. I was curious if someone could point out why there is no surface mass matrix in their formulation.