The authors would like to report a mistake in (3.32) in the original paper. In the paper cited as citation number 12, i.e., Castonguay et al. [1], the summation over the elements is performed after transforming \(\varTheta _{adv}\) to the physical domain. When this transformation is carried out in [1], there is a \(J_k\) term that appears in \(\varTheta _{adv}\) which cancels out with the \(J_k\) in the LHS. This cancellation happens in our case as well through all the extra terms, which was missed. This can be seen by looking at (3.34), where \(J_k\) appears in all the terms of \(\varTheta _{extra}\). In the cited paper, the cancellation is important to obtain the result \(\varTheta _{adv} \le 0\) which is used by us. Thus, the correct approach would be to transform the quantities to physical domain, perform the cancellation and then sum over all the elements. This mistake affects the final result obtained in the paper and thereby some of the statements made in the Abstract and the Conclusions sections as well. All the derivations up until Lemma 3.4 are correct. Here, we give the corrected version of Theorem 3.5 and its proof. We apologize for any inconvenience caused.
Proof
Equations (3.28)–(3.31) are correct and remain as it is. However, instead of (3.32), let us just substitute (3.29), (3.30) and the other results similar to (3.31) into (3.28) without summing over all the elements to get
$$\begin{aligned}&\frac{1}{2} J_k \frac{d}{dt} \bigg (\int \limits _{\varOmega _k} (u^D_k)^2 d\varOmega _k + \frac{c}{2}\int \limits _{\varOmega _k} \bigg ( \bigg (\frac{\partial ^p u^D_k}{\partial \xi ^p} \bigg )^2 + \bigg (\frac{\partial ^p u^D_k}{\partial \eta ^p} \bigg )^2 \bigg ) d\varOmega _k \\&\qquad + \frac{c^2}{4} \int \limits _{\varOmega _k} \bigg (\frac{\partial ^{2p} u^D_k}{\partial \xi ^p \partial \eta ^p} \bigg )^2 d \varOmega _k \bigg ) \\&\quad = -\int \limits _{\varOmega _S} \hat{u}^D (\hat{\varvec{\nabla }} \cdot \hat{\varvec{f}}^D) d\varOmega _S - \int \limits _{\varGamma _{S}} \hat{u}^D (\hat{\varvec{f}}^C \cdot \hat{\varvec{n}}) d \varGamma _S \\&\qquad + \, c \Bigg ( \bigg [ \frac{1}{2} \frac{\partial ^p \hat{u}^D_R}{\partial \eta ^p} \frac{\partial ^p (\hat{\varvec{f}}^D \cdot \hat{\varvec{n}})_R}{\partial \eta ^p} - \frac{\partial ^p \hat{u}^D_R}{\partial \eta ^p} \frac{\partial ^p (\hat{\varvec{f}} \cdot \hat{\varvec{n}})^*_R}{\partial \eta ^p} \bigg ] \\&\qquad \qquad \quad + \bigg [ \frac{1}{2} \frac{\partial ^p \hat{u}^D_L}{\partial \eta ^p} \frac{\partial ^p (\hat{\varvec{f}}^D \cdot \hat{\varvec{n}})_L}{\partial \eta ^p} - \frac{\partial ^p \hat{u}^D_L}{\partial \eta ^p} \frac{\partial ^p (\hat{\varvec{f}} \cdot \hat{\varvec{n}})^*_L}{\partial \eta ^p} \bigg ] \\&\qquad \qquad \quad + \bigg [ \frac{1}{2} \frac{\partial ^p \hat{u}^D_T}{\partial \xi ^p} \frac{\partial ^p (\hat{\varvec{f}}^D \cdot \hat{\varvec{n}})_T}{\partial \xi ^p} - \frac{\partial ^p \hat{u}^D_T}{\partial \xi ^p} \frac{\partial ^p (\hat{\varvec{f}} \cdot \hat{\varvec{n}})^*_T}{\partial \xi ^p} \bigg ] \\&\qquad \qquad \quad + \bigg [ \frac{1}{2} \frac{\partial ^p \hat{u}^D_B}{\partial \xi ^p} \frac{\partial ^p (\hat{\varvec{f}}^D \cdot \hat{\varvec{n}})_B}{\partial \xi ^p} - \frac{\partial ^p \hat{u}^D_B}{\partial \xi ^p} \frac{\partial ^p (\hat{\varvec{f}} \cdot \hat{\varvec{n}})^*_B}{\partial \xi ^p} \bigg ] \Bigg ) \end{aligned}$$
Transforming the RHS of the above equation to the physical domain, we get,
$$\begin{aligned}&\frac{1}{2} J_k \frac{d}{dt} \bigg (\int \limits _{\varOmega _k} (u^D_k)^2 d\varOmega _k + \frac{c}{2}\int \limits _{\varOmega _k} \bigg ( \bigg (\frac{\partial ^p u^D_k}{\partial \xi ^p} \bigg )^2 + \bigg (\frac{\partial ^p u^D_k}{\partial \eta ^p} \bigg )^2 \bigg ) d\varOmega _k + \frac{c^2}{4} \int \limits _{\varOmega _k} \bigg (\frac{\partial ^{2p} u^D_k}{\partial \xi ^p \partial \eta ^p} \bigg )^2 d \varOmega _k \bigg ) \\&\quad = - J_k \int \limits _{\varOmega _k} u^D (\varvec{\nabla }\cdot \varvec{f}^D) d\varOmega _k - J_k \int \limits _{\varGamma _{k}} u^D (\varvec{f}^C \cdot \varvec{n}) d \varGamma _k \\&\qquad +\, c \bigg ( J_{y_k}^{2p+1} J_k \bigg [\frac{1}{2} \frac{\partial ^p u^D_R}{\partial y^p} \frac{\partial ^p F^D_R}{\partial y^p} - \frac{\partial ^p u^D_R}{\partial y^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_R}{\partial y^p} \bigg ]_k \\&\qquad \qquad \quad + \, J_{y_k}^{2p+1} J_k\bigg [ -\frac{1}{2} \frac{\partial ^p u^D_L}{\partial y^p} \frac{\partial ^p F^D_L}{\partial y^p} - \frac{\partial ^p u^D_L}{\partial y^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_L}{\partial y^p} \bigg ] \\&\qquad \qquad \quad + \, J_{x_k}^{2p+1} J_k \bigg [ \frac{1}{2} \frac{\partial ^p u^D_T}{\partial x^p} \frac{\partial ^p G^D_T}{\partial x^p} - \frac{\partial ^p u^D_T}{\partial x^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_T}{\partial x^p} \bigg ] \\&\qquad \qquad \quad + \,J_{x_k}^{2p+1} J_k\bigg [- \frac{1}{2} \frac{\partial ^p u^D_B}{\partial x^p} \frac{\partial ^p G^D_B}{\partial x^p} - \frac{\partial ^p u^D_B}{\partial x^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_B}{\partial x^p} \bigg ] \bigg ) \end{aligned}$$
It is clear from the above equation that \(J_k\) cancels across all the terms. After cancellation of \(J_k\), we sum over all the elements to get
$$\begin{aligned} \frac{d}{dt} \Vert u^D \Vert ^2 = \varTheta _{adv} + c \varTheta _{extra} \end{aligned}$$
where
$$\begin{aligned} \Vert u^D \Vert ^2 = \sum \limits _{k = 1}^N \; \bigg ( \int \limits _{\varOmega _k} \bigg [ (u^D_k)^2 + \frac{c}{2} \bigg ( \bigg (\frac{\partial ^p u^D_k}{\partial \xi ^p}\bigg )^2 + \bigg (\frac{\partial ^p u^D_k}{\partial \eta ^p}\bigg )^2 \bigg ) + \frac{c^2}{4} \bigg (\frac{\partial ^{2p} u^D_k}{\partial \xi ^p \partial \eta ^p}\bigg )^2 \bigg ] d \varOmega _k \bigg ) \end{aligned}$$
is a broken Sobolev norm of the solution in the entire domain,
$$\begin{aligned} \varTheta _{adv} = \sum \limits _{k=1}^{N} \bigg (- \int \limits _{\varOmega _k} u^D (\varvec{\nabla }\cdot \varvec{f}^D) d\varOmega _k - \int \limits _{\varGamma _{k}} u^D (\varvec{f}^C \cdot \varvec{n}) d \varGamma _k \bigg ) \end{aligned}$$
and
$$\begin{aligned} \varTheta _{extra}= & {} \sum \limits _{k=1}^N\bigg ( J_{y_k}^{2p+1} \bigg [\frac{1}{2} \frac{\partial ^p u^D_R}{\partial y^p} \frac{\partial ^p F^D_R}{\partial y^p} - \frac{\partial ^p u^D_R}{\partial y^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_R}{\partial y^p} \bigg ]_k \\&\quad +\, J_{y_k}^{2p+1} \bigg [ -\frac{1}{2} \frac{\partial ^p u^D_L}{\partial y^p} \frac{\partial ^p F^D_L}{\partial y^p} - \frac{\partial ^p u^D_L}{\partial y^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_L}{\partial y^p} \bigg ] \\&\quad + \, J_{x_k}^{2p+1} \bigg [ \frac{1}{2} \frac{\partial ^p u^D_T}{\partial x^p} \frac{\partial ^p G^D_T}{\partial x^p} - \frac{\partial ^p u^D_T}{\partial x^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_T}{\partial x^p} \bigg ] \\&\quad + \, J_{x_k}^{2p+1} \bigg [- \frac{1}{2} \frac{\partial ^p u^D_B}{\partial x^p} \frac{\partial ^p G^D_B}{\partial x^p} - \frac{\partial ^p u^D_B}{\partial x^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_B}{\partial x^p} \bigg ] \bigg ) \\ \end{aligned}$$
Note that the difference is that, now there is no \(J_k\) appearing in any of the above terms. Note the change in the definition of the broken Sobolev norm as well where the term \(J_k\) no longer appears.
This expression can be rewritten as a sum over all the edges instead of all the elements. Consider one such summation along an interior vertical edge. Let \(-\) and \(+\) subscripts denote the element on the left and right. For the element on the left, this edge is its right boundary and for the right element, it is the left boundary. Also, note that for a Cartesian mesh with no mortar elements, the \(J_y\) for these left and right elements are the same, since it is the edge length of their common boundary. Adding the 2 terms coming from each element from this edge, we get
$$\begin{aligned} J_{y}^{2p+1} \bigg [\frac{1}{2} \frac{\partial ^p u^D_-}{\partial y^p} \frac{\partial ^p F^D_-}{\partial y^p} - \frac{\partial ^p u^D_-}{\partial y^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_-}{\partial y^p} - \frac{1}{2} \frac{\partial ^p u^D_+}{\partial y^p} \frac{\partial ^p F^D_+}{\partial y^p} - \frac{\partial ^p u^D_+}{\partial y^p} \frac{\partial ^p (\varvec{f} \cdot \varvec{n})^*_+}{\partial y^p} \bigg ] \end{aligned}$$
(3.35)
The above equation is the corrected version of Eq. (3.35). Since the \(J_k\) terms got cancelled out, \(J_+\) and \(J_-\) do not appear in the equation above.
Now, we use the fact that f is a linear advective flux, i.e., \(F^D = a u^D\) and \(G^D = b u^D\). Also, from the definition of the Lax–Friedrichs flux, we have,
$$\begin{aligned} (\varvec{f} \cdot \varvec{n})^*_- = \frac{1}{2}a \left( u^D_- + u^D_+\right) + \frac{\lambda }{2}|a|\left( u^D_- - u^D_+\right) \end{aligned}$$
(3.36)
$$\begin{aligned} (\varvec{f} \cdot \varvec{n})^*_+ = -\frac{1}{2}a \left( u^D_- + u^D_+\right) + \frac{\lambda }{2}|a|\left( u^D_+ - u^D_-\right) \end{aligned}$$
(3.37)
Substituting these results in (3.35), we get
$$\begin{aligned} J_{y}^{2p+1} \bigg [ -\frac{\lambda }{2}|a| \bigg ( \frac{\partial ^p u^D_-}{\partial y^p} \frac{\partial ^p u^D_-}{\partial y^p} - \frac{\partial ^p u^D_-}{\partial y^p} \frac{\partial ^p u^D_+}{\partial y^p} \bigg ) -\frac{\lambda }{2}|a| \bigg ( \frac{\partial ^p u^D_+}{\partial y^p} \frac{\partial ^p u^D_+}{\partial y^p} - \frac{\partial ^p u^D_-}{\partial y^p} \frac{\partial ^p u^D_+}{\partial y^p} \bigg ) \bigg ] \end{aligned}$$
(3.38)
which can be further simplified as
$$\begin{aligned} J_{y}^{2p+1} \bigg [ -\frac{\lambda }{2}|a| \bigg ( \frac{\partial ^p u^D_-}{\partial y^p} - \frac{\partial ^p u^D_+}{\partial y^p} \bigg )^2 \bigg ] \end{aligned}$$
(3.39)
From this equation, we can see that the terms contributing to \(\varTheta _{extra}\) are non-positive. The same is true for \(\varTheta _{adv}\) and therefore, for any \(c \ge 0\), we have
$$\begin{aligned} \frac{d}{dt} \Vert u^D \Vert ^2 = \varTheta _{adv} + c \varTheta _{extra} \le 0 \end{aligned}$$
which proves Theorem 3.5.
