Erratum to: On the Stability of the Flux Reconstruction Schemes on Quadrilateral Elements for the Linear Advection Equation

Excerpt

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.

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