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Correction to: Journal of Scientific Computing https://doi.org/10.1007/s10915-022-01885-w
In this article, the Main results in ``Section 3'' contains error. It should read as given below.
For \(n=0, \ldots , N_T\), let \(\mathcal {M}_h^n\) be an approximate value of mass at \(t=t^n\) defined by
The value \(\mathcal {M}_h^n\) is an approximation of \(\int _\Omega \phi ^n dx\) due to the relations \(\frac{3}{2}\phi ^n-\frac{1}{2}\phi ^{n-1} (= \phi ^{n+1/2} + O(\Delta t^2)) = \phi ^n + O(\Delta t)\) and \(\frac{1}{2}(\phi ^0+\phi ^1) (= \phi ^{1/2} + O(\Delta t^2)) = \phi ^0 + O(\Delta t)\) for any smooth function \(\phi \).
FormalPara Theorem 1(conservation of mass) Suppose that Hypotheses 1 and 2 hold true. Let \(\phi _h = \{\phi _h^n\}_{n=1}^{{ N_T}}\) be a solution to scheme (9) for a given \(\phi _h^0\). Then, we have the following.
-
(i)
It holds that, for \(n={ 1},\ldots , N_T\),
$$\begin{aligned} \mathcal {M}_h^n = \mathcal {M}_h^0 + \Delta t \sum _{i=1}^n \Bigl ( \int _\Omega f^i dx + \int _\Gamma g^i ds \Bigr ). \end{aligned}$$(10) -
(ii)
Assume \(f=0\) and \(g=0\) additionally. Then, for the solution to scheme (9), it holds that, for \(n={ 1},\ldots , N_T\),
$$\begin{aligned} \int _\Omega \phi _h^n dx = \int _\Omega \phi _h^0 dx. \end{aligned}$$(11)
The identity (10) is equivalent to
(In Section 4, Proofs:)
1 Proof of Theorem 1
We first note that due to Proposition 1-(i)
hold for any \(\rho \in \Psi \) and \(n=1,\dots , N_{ T}\). We substitute \(1 \in \Psi _h\) into \(\psi _h\) in scheme (9) in the following.
We prove (i). Taking into account the case \(n = 1\):
we have, for any \(n \ge 2\),
which completes the proof of (i).
We prove (ii). Consider identity (10) with \(f=g=0\). Then, we have \(\int _\Omega \phi _h^1 dx = \int _\Omega \phi _h^0 dx \ ( = \mathcal {M}_h^0 )\) from \(n=1\), and \(\int _\Omega \phi _h^2 dx = \int _\Omega \phi _h^0 dx\) from \(n=2\). Using this argument repeatedly up to \(n=N_T\), we obtain (11). \(\square \)
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Futai, K., Kolbe, N., Notsu, H. et al. Correction to: A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems. J Sci Comput 102, 19 (2025). https://doi.org/10.1007/s10915-024-02720-0
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DOI: https://doi.org/10.1007/s10915-024-02720-0