Journal of Computational Physics 360 (2018) 247
Contents lists available atScienceDirect
Journal
of
Computational
Physics
www.elsevier.com/locate/jcp
Corrigendum
Corrigendum
to
“On
the
relation
between
conservation
and
dual
consistency
for
summation-by-parts
schemes”
[J.
Comput.
Phys.
344
(2017)
437–439]
Jan Nordström
∗
,
Fatemeh Ghasemi
DepartmentofMathematics,ComputationalMathematics,LinköpingUniversity,SE-58183Linköping,Sweden
Afewnotationalerrorswererecentlydiscoveredintheabovepublication.
•
Thenotationusedinthenoteisvalidforfluxesoftheform fL(
u)
=
ALu,
fR(
v)
=
ARv whereAL=
AR ism×
m constantsymmetricmatrix.
•
ThematricesL and
R givenin(10)shouldbetransposed.
•
Weshow thatProposition 1inthenote isvalidevenif AL and AR arevariable,non-symmetric aswell asequalandinvertibleattheinterface.
Thedualproblemwithinterfaceconditions(neglectingboundaryconditions)is
θ
τ−
ATθ
x=
0,
−
1≤
x≤
1,
τ
>
0,
φ (
0,
t)
= ψ(
0,
t),
x=
0,
τ
>
0,
(1)where
θ,
A= φ,
AL(
x,
t)
∈ [−
1≤
x≤
0]
andθ,
A= ψ,
AR(
x,
t)
∈ [
0≤
x≤
1]
.Thesemi-discreteprimalproblemwithvariableAL
,
AR iswt
+
Lw=
0,
L=
P−1(
Q+ )
A,
A=
diag(
AL,
AR),
(2)whereP
,
Q,
aregiveninthenoteandAL
,
AR areblockdiagonalmatricesapproximating AL,
AR atpointwisepositionsinx respectively.
Thesemi-discretedualproblemrelatedto(2) is
θ
τ+
P−1LTPθ
=
0.SubstitutingL from(2) leadstoθ
τ−
ATP−1(
Q− ˜)θ =
0,
(3)where
˜
isgiveninthenote.ThevectorATP−1Qθ
approximates ATθ
x andATP−1
˜θ
imposethedualinterfaceconditionsifandonlyiftheconservationcondition(7)inthenoteissatisfied.Hence(3) isadualconsistentapproximationof(1),and Proposition1 holdsalsointhiscase.
TheauthorswouldliketoapologiseforanyinconveniencecausedandthankDrSofiaErikssonforspottingtheerrors.
DOIoforiginalarticle:https://doi.org/10.1016/j.jcp.2017.04.072.
*
Correspondingauthor.E-mailaddress:jan.nordstrom@liu.se(J. Nordström). https://doi.org/10.1016/j.jcp.2018.02.046
