It is then possible to replace the original discretized
equation ( 5 ) by one written in the local coordinate of (휉,휂):
휕
휕푥
(퐾
휕푢푖,푗
휕푥
)+
휕
휕푦
(퐾
휕푢푖,푗
휕푦
)
=
휕
휕휉
(퐾
휕푢푖,푗
휕휉
)+
휕
휕휂
(퐾
휕푢푖,푗
휕휂
)
=
푛푠
∑
푚=1훾푚푢푖+푖푚,푗+푗푚+퐶푖,푗.
(10)
Further, in respect of local coordinates system, the solution
푢푖+푖푚,푗+푗푚 atagridpointcouldbewrittenasaTaylor
expansion at (푥∗,푦∗)sothat
푢(푥푖+푖푚,푦푗+푗푚)=푢(휉푚,휂푚)
=푢±+휉푚푢±휉+휂푚푢±휂+
1
2
휉^2 푚푢±휉휉
+
1
2
휂^2 푚푢±휂휂+휉푚휂푚푢±휉휂+
1
2
휉푚휂^2 푚푢±휉휂휂+푂(ℎ^3 ).
(11)
The reasons of keeping third-order derivative of solution푢휉휂휂
in Taylor’s development ( 11 ) are related to their presence on
the interface conditions relations as it will be clear later (see
( 15 )). Additionally, the values푢±,푢±휉,푢±휂,푢±휉휉,푢±휂휂,푢±휉휂,and푢±휉휂휂
in ( 11 ) represent the solution and its derivatives determined at
projection point (푥∗,푦∗) of the interface. The (+) or (−)sign
is chosen depending on whether (휉푚,휂푚) belongs to the (+)
or (−) side of the interface.
By replacing in ( 10 )eachcomponent푢푖+푖푚,푗+푗푚as given
by ( 11 ) and rearranging terms, the following expression of the
truncation error at an irregular point is obtained (푖,푗):
휅푖,푗=푎 1 푢−+푎 2 푢++푎 3 푢휉−+푎 4 푢+휉
+푎 5 푢−휂+푎 6 푢+휂+푎 7 푢−휉휉+푎 8 푢+휉휉+푎 9 푢−휂휂
+푎 10 푢+휂휂+푎 11 푢휉휂−+푎 12 푢+휉휂+푎 13 푢−휉휂휂
+푎 14 푢+휉휂휂−퐶푖,푗+푂(max푚儨儨
儨儨훾푚
儨儨
儨儨ℎ^3 ).
(12)
The coefficients{푎 1 ,푎 2 ,...,푎 14 }of ( 12 ) depend on the
position of the stencil point relative to the interface:
푎 1 = ∑
(푚∈푀−)
훾푚;푎 2 = ∑
(푚∈푀+)
훾푚;
푎 3 = ∑
(푚∈푀−)
훾푚휉푚;푎 4 = ∑
(푚∈푀+)
훾푚휉푚;
푎 5 = ∑
(푚∈푀−)
훾푚휂푚;푎 6 = ∑
(푚∈푀+)
훾푚휂푚;
푎 7 =
1
2
∑
(푚∈푀−)
훾푚휉푚^2 ;푎 8 =
1
2
∑
(푚∈푀+)
훾푚휉^2 푚;
푎 9 =
1
2
∑
(푚∈푀−)
훾푚휂^2 푚;푎 10 =
1
2
∑
(푚∈푀+)
훾푚휂푚^2 ;
푎 11 = ∑
(푚∈푀−)
훾푚휉푚휂푚;푎 12 = ∑
(푚∈푀+)
훾푚휉푚휂푚;
푎 13 =
1
2
∑
(푚∈푀−)
훾푚휉푚휂푚^2 ;푎 14 =
1
2
∑
(푚∈푀+)
훾푚휉푚휂^2 푚,
(13)
where the sets푀±are defined as푀±={푚:(휉푚,휂푚)∈Ω±}. (14)At this end it is important to specify the relations of the
solution and its derivatives at both sides of the interface. After
some algebraic manipulation, from the contact conditions ( 3 )
the following relations are deduced:푢+=푢−+훼퐾−푢−휉,푢+휉=휌푢−휉,푢+휂=(1−훼휒耠耠퐾−)푢−휂+훼퐾−휂푢−휉+훼퐾−푢−휉휂,
푢+휉휉=휌푢−휉휉
−
[퐾+휂푢휂+−퐾−휂푢−휂+퐾휉+푢+휉−퐾−휉푢−휉+퐾+푢+휂휂−퐾−푢−휂휂]
퐾+
;
푢+휂휂=푢−휂휂−휒耠耠(푢휉+−푢−휉)+훼(휒耠耠퐾휉−+퐾−휂휂)푢−휉
+훼퐾휂−(푢−휉휂−휒耠耠푢−휂)+훼퐾−휂(푢휉휂−−휒耠耠푢−휂)
+훼퐾−(휒耠耠푢−휉휉+푢휉휂휂−2휒耠耠푢−휂휂)−훼(휒耠耠)
2
퐾−푢−휉,푢+휉휂=
(퐾−휂푢−휉−퐾+휂푢+휉)
퐾+
+휌(푢−휉휂−휒耠耠푢휂−)+휒耠耠푢+휂,
푢+휉휂휂=휌푢−휉휂휂+휒耠耠(휌푢−휉휉−푢+휉휉)
+
[푢−휉(휒耠耠퐾휉−+퐾−휂휂)−푢+휉(휒耠耠퐾+휉+퐾휂휂+)]
퐾+
+2
[퐾−휂푢−휉휂−퐾+휂푢+휉휂−휒耠耠(퐾−휂푢−휂−퐾+휂푢+휂)]
퐾+
−2휒耠耠(휌푢−휂휂−푢휂휂+),
(15)
with휌=퐾−/퐾+.Thecurvature휒耠耠of the interface at (푥∗,
푦∗) is the second order derivative of function휉=휒(휂)with
respect to휂.Wenotealsothatat(푥∗,푦∗)wehave휒(0) = 0
and for a smooth interface (as assumed here)휒耠(0) = 0.
The interface relations ( 15 ) are used to recast ( 12 )ina
compact form using only quantities from one side of the
interface (theΩ−side, e.g.):휅푖,푗=퐵 1 푢−+퐵 2 푢−휉+퐵 3 푢−휂+퐵 4 푢−휉휉+퐵 5 푢−휂휂+퐵 6 푢휉휂−+퐵 7 푢−휉휂휂+푂(max푚儨儨
儨儨훾푚
儨儨
儨儨ℎ^3 ),
(16)
where 퐵푖 (푖=1,7) are expressions of coefficients
{푎 1 ,푎 2 ,...,푎 14 }and other quantities used in ( 15 ). Since we