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So I managed to fix my problem. First, off I was missing a factor of 0.5 when calculating the surface integral. Additionally, I am using the following normalization $\phi(x) = \sqrt{2m + 1} L(\xi)$. I have one last question (I apologize if this is off topic). A problem I identified in my code was that my way of doing tensor products was not the same as the original paper by Cockburn or a paper by Praveen et al (https://arxiv.org/pdf/1506.06140.pdf). The standard way of computing tensor products is $$ u_{h}(x,y) = \sum_{k = 0}^{N} \sum_{l = 0}^{M} u_{ij}(t) \phi_{k}(x) \phi_{l}(y) (1), $$ The tensor product formula adapted in the last two papers are $$ U_{h}(x,y) = \sum_{k=0}^{\frac{1}{2}(k + 1)(k + 2)} u_{ij}^{k}(t) \phi_{k}(x,y) (2) $$ Is there a reason to use formula (1) over formula (2)? Mathematically, they seem both correct.

The standard way of computing tensor products is $$ u_{h}(x,y) = \sum_{k = 0}^{N} \sum_{l = 0}^{M} u_{ij}(t) \phi_{k}(x) \phi_{l}(y) (1), $$ The tensor product formula adapted in the last two papers are $$ U_{h}(x,y) = \sum_{k=0}^{\frac{1}{2}(k + 1)(k + 2)} u_{ij}^{k}(t) \phi_{k}(x,y) (2) $$ Is there a reason to use formula (1) over formula (2)? Mathematically, they seem both correct.

Above is a snippet in how I perform the tensor product quadrature. I'm a little confused on why removing the normalizing of the Gauss quadrature is causing such severe results. Below are images of when I remove normalization of the weights and Legendre polynomials (but normalize the basis). As we can see, when I remove the normalization of the weights my solution does not evolve in time (it just stays in the same place as when T = 0).

Above is a snippet in how I perform the tensor product quadrature. I'm a little confused on why removing the normalizing of the Gauss quadrature is causing such severe results.
Solution:

So I managed to fix my problem. First, off I was missing a factor of 0.5 when calculating the surface integral. Additionally, I am using the following normalization $\phi(x) = \sqrt{2m + 1} L(\xi)$. I have one last question (I apologize if this is off topic). A problem I identified in my code was that my way of doing tensor products was not the same as the original paper by Cockburn or a paper by Praveen et al (https://arxiv.org/pdf/1506.06140.pdf). The standard way of computing tensor products is $$ u_{h}(x,y) = \sum_{k = 0}^{N} \sum_{l = 0}^{M} u_{ij}(t) \phi_{k}(x) \phi_{l}(y) (1), $$ The tensor product formula adapted in the last two papers are $$ U_{h}(x,y) = \sum_{k=0}^{\frac{1}{2}(k + 1)(k + 2)} u_{ij}^{k}(t) \phi_{k}(x,y) (2) $$ Is there a reason to use formula (1) over formula (2)? Mathematically, they seem both correct.

Above is a snippet in how I perform the tensor product quadrature. I'm a little confused on why removing the normalizing of the Gauss quadrature is causing such severe results. Below are images of when I remove normalization of the weights and Legendre polynomials (but normalize the basis).

Above is a snippet in how I perform the tensor product quadrature. I'm a little confused on why removing the normalizing of the Gauss quadrature is causing such severe results. Below are images of when I remove normalization of the weights and Legendre polynomials (but normalize the basis). As we can see, when I remove the normalization of the weights my solution does not evolve in time (it just stays in the same place as when T = 0).