二维三温热传导方程组的分数步隐式差分格式.
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2
4
2
n+1 n+1 n+1 n+1 n+1 n n n δx (kα δx Tαh )ij = h−2 kα,i + 1 ,j (Tαh,i+1,j − Tαh,ij ) − kα,i− 1 ,j (Tαh,ij − Tαh,i−1,j ) , n+1 n δy (kα δy Tαh )ij
(2.1) (2.2) (2.3)
n%;
+4
0 < C∗ ≤ Cvα ≤ C ∗ , ∂K (ρ, Tα ) ≤ D∗ , ∂Tα 0 < K∗ ≤ K (ρ, Tα ) ≤ K ∗ , α = e, i, r. (1.6) (1.7)
7 C ∗ , C∗ , K ∗ , K∗ , D∗ MeP*L +4tYM<!j/M~LK@BEM 2 qt 1 n n n , xi = ih, yj = jh; ∆t = T mh= N L , t = n∆t, Wij = W (xi , yj , t ). : α e, i, r YM!.)~ : Tαh ? Tα M'nDQ (
1
_ '
,
1
K GKJ (ICF) MbfLUw;Y **(p < UVd 8G Z d 8G d 8Gj/Mq2n a j/h [1], YA@!_}gT-4 M (k 8Gj/hM <M@](MjV r0wP [2,3] X4j}B7;\bfV{ X"a{MHT'n{< n^:nL"%H [4−6], 2`)~<bf(k 8Gj /hMnL"@ 'n{< =\r/n}j'n{<MCb qX LJk =7MZ+ H 1 hLz' &<!qX ^:nL"je Q, bftY e) ftY < -B9 &R Xx&R q tYML w A;Ip m bnj/hM3 UtY
S 27 L S 1 1 H 2004
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ACTA MATHEMATICAE APPLICATAE SINICA
Vol. 27 No. 1 Jan., 2004
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r u
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266071)
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o 5* =cg)l 9Hk0iNoM#5=A (o|= ="~hkf 5(o|=Nl>8 , H iM %{("rY x )l 9Hk0i oM#(o|= Dc rY
Cve 1 ∂Te = div (K (ρ, Te ) grad Te ) + ωei (Ti − Te ) + ωer (Tr − Te ), ∂t ρ 1 ∂Ti Cvi = div (K (ρ, Ti ) grad Ti ) − ωei (Ti − Te ), ∂t ρ 1 ∂Tr = div (K (ρ, Tr ) grad Tr ) − ωer (Tr − Te ), Cvr ∂t ρ (1.1) (1.2) (1.3)
n+1 Cvi,ij
∆t
=
1 n+1 n n δy (ki δy (Tih − Tih ))ij , ρ
1
cg)l 9Hk0iNoM#5=(o|= d 8Gj/ (1.3) MnL"4<'n{<e
n+1 Cvr,ij n 2 Trh,ij − Trh,ij n+ 1
J-Q
29
∆t 1 n+1 n n n n + δy (kr δy Trh )ij − ωer,ij (Trh − Teh )ij , ρ
n n n kα,i + 1 ,j = K (ρ, Tαh,i+1,j ) + K (ρ, Tαh,ij ) 2Fra Baidu bibliotek n n n Kα,i + 1 ,j = K (ρ, Tα,i+1,j ) + K (ρ, Tα,ij ) 2,
2 2
?
(
n n kα,i,j , Kα,i,j +1 +1
2 2
YQX/
Vd 8Gj/ (1.1) MnL"4<'n{<e
n+1 Cve,ij
1 1 n+ 1 n n n δx (ke δx Teh 2 )ij + δy (ke δy Teh )ij ∆t ρ ρ n+1 n+1 n n n n + ωei,ij (Tih − Teh )ij + ωer,ij (Trh − Teh )ij , 1 ≤ i ≤ N − 1, =
=h
−2
n+1 n kα,i,j (Tαh,i,j +1 +1 2
−
n+1 Tαh,ij )
−
n+1 n kα,i,j (Tαh,ij −1 2
−
n+1 Tαh,i,j −1 )
,
∇h ·
n+1 n (kα ∇h Tαh )ij
=
n+1 n δx (kα δx Tαh )ij
+
n+1 n δy (kα δy Tαh )ij .
3
^0e
0 Tα (x, y, 0) = Tα (x, y ),
α = e, i, r, ωei
(1.4) ωer
∗
Y Te , Ti, Tr AiS L n ?Vdk] Zdk] dk] ?VdCZd VdC dM}g4 ~L ρ e@XMt] p 2001 6 I 4 #EK 2003 5 I 14 #EK uz )e P A (19871043 ) . 7 e P A (Q98A07115) c` y
n+1 Teh,ij − Teh,ij n+ 1 2
n 2 Teh,ij − Teh,ij
n+ 1
(2.4a)
n+1 Cve,ij
∆t
=
1 n+1 n n δy (ke δy (Teh − Teh ))ij , ρ
1 ≤ j ≤ N − 1. (2.4b)
Zd 8Gj/ (1.2) MnL"4<'n{<e
n
28
T
ej eP*L
8 M 27 L / 4 < De Ω = Ωxy × [0, T ], Ωxy = (x, y) | 0 ≤ x ≤ 1; 0 ≤ y ≤ 1 , nBOpR)Y < ?^0M T $
Tα = 0, α = e, i, r, (x, y, t) ∈ ∂ Ωxy × [0, T ]. (1.5)
n+1 Cvi,ij n 2 Tih,ij − Tih,ij n+ 1
∆t 1 n+1 n n n n + δy (ki δy Tih )ij − ωei,ij (Tih − Teh )ij , ρ
n+1 2 Tih,ij − Tih,ij n+ 1
=
1 n+ 1 n δx (ki δx Tih 2 )ij ρ 1 ≤ i ≤ N − 1, 1 ≤ j ≤ N − 1. (2.5b) (2.5a)
4
2
n+1 n+1 n+1 n+1 n+1 n n n δx (kα δx Tαh )ij = h−2 kα,i + 1 ,j (Tαh,i+1,j − Tαh,ij ) − kα,i− 1 ,j (Tαh,ij − Tαh,i−1,j ) , n+1 n δy (kα δy Tαh )ij
(2.1) (2.2) (2.3)
n%;
+4
0 < C∗ ≤ Cvα ≤ C ∗ , ∂K (ρ, Tα ) ≤ D∗ , ∂Tα 0 < K∗ ≤ K (ρ, Tα ) ≤ K ∗ , α = e, i, r. (1.6) (1.7)
7 C ∗ , C∗ , K ∗ , K∗ , D∗ MeP*L +4tYM<!j/M~LK@BEM 2 qt 1 n n n , xi = ih, yj = jh; ∆t = T mh= N L , t = n∆t, Wij = W (xi , yj , t ). : α e, i, r YM!.)~ : Tαh ? Tα M'nDQ (
1
_ '
,
1
K GKJ (ICF) MbfLUw;Y **(p < UVd 8G Z d 8G d 8Gj/Mq2n a j/h [1], YA@!_}gT-4 M (k 8Gj/hM <M@](MjV r0wP [2,3] X4j}B7;\bfV{ X"a{MHT'n{< n^:nL"%H [4−6], 2`)~<bf(k 8Gj /hMnL"@ 'n{< =\r/n}j'n{<MCb qX LJk =7MZ+ H 1 hLz' &<!qX ^:nL"je Q, bftY e) ftY < -B9 &R Xx&R q tYML w A;Ip m bnj/hM3 UtY
S 27 L S 1 1 H 2004
9<
N
ACTA MATHEMATICAE APPLICATAE SINICA
Vol. 27 No. 1 Jan., 2004
}
;
; 876 532 9 4 Z '@ F
r u
(
wy~v
z
∗
O
266071)
o
o 5* =cg)l 9Hk0iNoM#5=A (o|= ="~hkf 5(o|=Nl>8 , H iM %{("rY x )l 9Hk0i oM#(o|= Dc rY
Cve 1 ∂Te = div (K (ρ, Te ) grad Te ) + ωei (Ti − Te ) + ωer (Tr − Te ), ∂t ρ 1 ∂Ti Cvi = div (K (ρ, Ti ) grad Ti ) − ωei (Ti − Te ), ∂t ρ 1 ∂Tr = div (K (ρ, Tr ) grad Tr ) − ωer (Tr − Te ), Cvr ∂t ρ (1.1) (1.2) (1.3)
n+1 Cvi,ij
∆t
=
1 n+1 n n δy (ki δy (Tih − Tih ))ij , ρ
1
cg)l 9Hk0iNoM#5=(o|= d 8Gj/ (1.3) MnL"4<'n{<e
n+1 Cvr,ij n 2 Trh,ij − Trh,ij n+ 1
J-Q
29
∆t 1 n+1 n n n n + δy (kr δy Trh )ij − ωer,ij (Trh − Teh )ij , ρ
n n n kα,i + 1 ,j = K (ρ, Tαh,i+1,j ) + K (ρ, Tαh,ij ) 2Fra Baidu bibliotek n n n Kα,i + 1 ,j = K (ρ, Tα,i+1,j ) + K (ρ, Tα,ij ) 2,
2 2
?
(
n n kα,i,j , Kα,i,j +1 +1
2 2
YQX/
Vd 8Gj/ (1.1) MnL"4<'n{<e
n+1 Cve,ij
1 1 n+ 1 n n n δx (ke δx Teh 2 )ij + δy (ke δy Teh )ij ∆t ρ ρ n+1 n+1 n n n n + ωei,ij (Tih − Teh )ij + ωer,ij (Trh − Teh )ij , 1 ≤ i ≤ N − 1, =
=h
−2
n+1 n kα,i,j (Tαh,i,j +1 +1 2
−
n+1 Tαh,ij )
−
n+1 n kα,i,j (Tαh,ij −1 2
−
n+1 Tαh,i,j −1 )
,
∇h ·
n+1 n (kα ∇h Tαh )ij
=
n+1 n δx (kα δx Tαh )ij
+
n+1 n δy (kα δy Tαh )ij .
3
^0e
0 Tα (x, y, 0) = Tα (x, y ),
α = e, i, r, ωei
(1.4) ωer
∗
Y Te , Ti, Tr AiS L n ?Vdk] Zdk] dk] ?VdCZd VdC dM}g4 ~L ρ e@XMt] p 2001 6 I 4 #EK 2003 5 I 14 #EK uz )e P A (19871043 ) . 7 e P A (Q98A07115) c` y
n+1 Teh,ij − Teh,ij n+ 1 2
n 2 Teh,ij − Teh,ij
n+ 1
(2.4a)
n+1 Cve,ij
∆t
=
1 n+1 n n δy (ke δy (Teh − Teh ))ij , ρ
1 ≤ j ≤ N − 1. (2.4b)
Zd 8Gj/ (1.2) MnL"4<'n{<e
n
28
T
ej eP*L
8 M 27 L / 4 < De Ω = Ωxy × [0, T ], Ωxy = (x, y) | 0 ≤ x ≤ 1; 0 ≤ y ≤ 1 , nBOpR)Y < ?^0M T $
Tα = 0, α = e, i, r, (x, y, t) ∈ ∂ Ωxy × [0, T ]. (1.5)
n+1 Cvi,ij n 2 Tih,ij − Tih,ij n+ 1
∆t 1 n+1 n n n n + δy (ki δy Tih )ij − ωei,ij (Tih − Teh )ij , ρ
n+1 2 Tih,ij − Tih,ij n+ 1
=
1 n+ 1 n δx (ki δx Tih 2 )ij ρ 1 ≤ i ≤ N − 1, 1 ≤ j ≤ N − 1. (2.5b) (2.5a)