If you can't read please download the document
Upload
hadang
View
255
Download
8
Embed Size (px)
Citation preview
( 500080)
Partial Differential EquationsPDE mesh, grid, particle
2PDE
PDE 3
Finite-Difference Method, FDM 1 2 3
PDE
4
1/2
Partial Differential Equations, PDE
local governing equation boundary condition initial condition
Integral Equations global Green functionfundamental function
PDE
5
2/2
Partial Differential Equations, PDElocal
Integral Equationsglobal
conditional Variational Method
functional
PDE
6
1f(x)u
1.01.0
f(x)
u(x)
1.0
x
PDE
7
x:1 x(x)
1.0
1.0x
s
u(x)
u(x+x)
+x
f(x)s
0sin1sin1 sxfx x+x
0,0 xu
xdx
uddxdu
dxdu
dxdu
xxxx
xxx
xx
2
2
tansin
tansin
xxdxduxuxxuxs
222 1)( 02
2
xfdx
ud
PDE
8
xG(x,) x= f()
x
dfxGxu 1
0,
G(x,) x influence functionGreens function
PDE
9
functional
1
0
21
0
21
0
1
0 21111 dx
dxdudx
dxdudxds
1
0
2
1
0
1
0
2
21
21
dxxuxfdxdu
dxxuxfdxdxduuJ
1
0dxxuxf
stationarizeEuler
PDE 10
Finite-Difference Method, FDM 1 2 3
PDE
11
1/5 3 (x1,x2,x3) u(x1,x2,x3) 2 G=0 2Second order PDE
linearnon-linear
c=0 homogeneousinhomogeneous
0,,,,,,,,,,,, 32132131
2
21
2
23
2
22
2
21
2
xxxu
xu
xu
xu
xxu
xxu
xu
xu
xuG
0,,,,,, 3213
1321
3
1,
2
321
xxxcxuxxxb
xxuxxxa
i ii
ji jiij
PDE
12
2/5 u
aij aij=aji 3A
x(x1,x2,x3)A(x) 0xparabolic xhyperbolic xelliptic
ijji xxu
xxu
22
333231
232221
131211
aaaaaaaaa
A
PDE
13
3/5 2x,y
A
AC-B2 =00
CBBA
A
yxcuyu
xu
yuC
yxuB
xuA ,2 2
22
2
2
22 BACCACB
BA
IA
PDE
14
4/5
022
Yu
Xu
022
2
2
Y
uX
u
022
2
2
Y
uX
u
PDE
15
5/5 2
022
2
2
2
2
Q
zu
yu
xu
0222
2
2
2
2
uk
zu
yu
xu
Qzu
yu
xu
tuc
2
2
2
2
2
2
2
2
2
2
2
22
2
2
zu
yu
xuc
tu
Szu
yu
xuc
tu
PDE
16
conduction
convection
radiation
diffusion
convection
PDE
17
u heat fluxq
Fourier
c
2
2
xu
xu
xtuc
xuq
C mass fluxJ
Fick D
2
2
xCD
xCD
xtC
xCDJ
Fourier q(x,t)
x+x
PDE
18
1/2 1
u(x,t)
Ttemperature
x
q(x,t) q(x+x,t)
x
txutxq
,,
xx
txutxq
xx
txux
txux
txxutxxq
2
2
2
2
,,
,,,,
Q(x,t) heat source
PDE
19
2/2 x q
xx
txutxqtxxqq
2
2 ,,,
xt
txucxt
txucq
,,
t
txu
,
2
2
xu
tuc
Qxu
tuc
2
2
PDE
20
Initial/Boundary Conditions B.C.
time-marching unsteady, transienttime dependent
steady 1Dirichlet
2Neumann
0
3Robin 12
u
PDE
21
xT 2
T
T
x
s
u(x)
u(x+x)
+x
f(x)s
x x+x
xxfTTxtu
sinsin2
2
xfxuT
tu
2
2
2
2
Tc
xuc
tu 2
2
22
2
2
PDE 22
1 2 3
PDE
23
Discretization
Taylor
24PDE
Finite Difference Method
x x
i-1 i i+1
25PDE
x
xxxx
xxxdxd
x
0
lim
ii+1
x x
i-1 i i+1
xdxd ii
i
12/1
x0
i
211
11
2/12/12
2 2xx
xxx
dxd
dxd
dxd iii
iiii
ii
i
26PDE
Taylor1/3
x x
i-1 i i+1
iiiii x
xx
xx
x
333
2
22
1 !3!2
iiiii x
xx
xx
x
333
2
22
1 !3!2
27PDE
Taylor2/3
x x
i-1 i i+1
iiiii x
xx
xx
x
333
2
22
1 !3!2
iiiii x
xx
xx
x
333
2
22
1 !3!2
iii
ii
xx
xx
xx
3
32
2
21
!3!2
x
iii
ii
xx
xx
xx
3
32
2
21
!3!2
x
28PDE
Taylor3/3
x x
i-1 i i+1
iiiii x
xx
xx
x
333
2
22
1 !3!2
iiiii x
xx
xx
x
333
2
22
1 !3!2
ii
ii
xx
xx
3
3211
!32
2
(x)2
29PDE
x x
i-1 i i+1
2/13
33
2/12
22
2/12/11 !3
2/!22/2/
iii
ii xx
xx
xx
2/13
32
2/1
1
!32/2
ii
ii
xx
xx
(x)2
2/13
33
2/12
22
2/12/1 !3
2/!22/2/
iii
ii xx
xx
xx
x
30PDE
PDE
211
11
2/12/12
2 2xx
xxx
dxd
dxd
dxd iii
iiii
ii
i
222
11
1)(,2)(,1)(
)1()()()()(
xiA
xiA
xiA
NiiBFiAiAiA
RDL
iRiDiL
022
BFdxd
)1(0)(121
)1(0)(2
12212
211
NiiBFxxx
NiiBFx
iii
iii
31
Discretization
Taylor
Finite Element MethodFEM weak form
weak solution
32PDE
Handbook of Grid Generation
33FEM-intro
PDE 34
Finite-Difference Method, FDM 1 2 3
PDE
35
1Steady Convection-Diffusion Equation
1022
x
xu
xua 11,00 uu
a0
v
xPexa
BeABeAxu aaLPe
Pe: PecletL: =1
[L2T-1] [L1T-1]
PDE
36
2DNavier-Stokes + Continuity
0
1
1
2
2
2
2
2
2
2
2
yv
xu
yv
xv
yp
yvv
xvu
tv
yu
xu
xp
yuv
xuu
tu
PDE
37
Pex=0
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Pe=0.00Pe=0.10Pe=1.00Pe=2.00Pe=10.0
Pe
xPe
Pe
xPe
eePe
xu
eexu
1
11
PDE
38
21111 2
2 xuuu
xuua iiiii
Rc: Reynolds
xaR
uRuuR
c
iciic
0242 11
iii qcqcu 2211
0242 2 cc RqqRq1, q2: 0242 11 iciic qRqqR
: :
i
c
ci
c
c
RRccu
RRqq
22
22,1 2121
q2 0 |Rc|
PDE
39
Rc=1.00x= 0.10, a= 1.00, v= 0.10
Rc=2.50x= 0.10, a= 1.00, v= 0.04
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Central Difference
Exact
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Central Difference
Exact
PDE
40
Rc
a= 1.00, v= 0.04
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Dx= 0.10, Rc= 2.50Dx= 0.05, Rc= 1.25Exact
PDE
41
1/22
1111 22 x
uuuxuua iiiii
2
111 2x
uuuxuua iiiii
/1st order upwinding
iu 1iu1iu
a>0
xuua
xua ii
1
iu 1iu1iu
a
PDE
42
2/22
11111 222 x
uuuxaxuua
xuua iiiiiii
21111 2
22 xuuuxa
xuua iiiii
scheme
PDE
43
Rc=1.00x= 0.10, a= 1.00, v= 0.10
Rc=2.50x= 0.10, a= 1.00, v= 0.04
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Central DifferenceUpwindingExact
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Central DifferenceUpwindingExact
PDE
44
1/2implicit REAL*8(A-H,O-Z)real(kind=8), dimension(:), allocatable:: U1, U2real(kind=8), dimension(:,:), allocatable:: AMAT
!C!C-- INIT.
read (*,*) NE, VELO, DIFF
DX= 1.d0/dfloat(NE)N = NE + 1
LENGTH= 1.0d0allocate (U1(N), U2(N), AMAT(N,N))
REc= VELO*DX/DIFFPECLET= VELO/DIFF
COEFc2= VELO/(2.d0*DX)COEFc1= VELO/(1.d0*DX)COEFd= DIFF/(DX*DX)
!C!C-- Central Diff.
do j= 1, Ndo i= 1, NAMAT(i,j)= 0.d0
enddoenddo
do i= 1, NU1(i)= 0.d0
enddo
NE: N: =NE+1DT: tDX: xDIFF: vVELO: a
1022
x
xu
xua
11,00 uu
PDE
45
1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8
L=1.00
NENE=8
x=L/NENgrid pointN=NE+1
Fortran
C etc.
PDE
46
1/2implicit REAL*8(A-H,O-Z)real(kind=8), dimension(:), allocatable:: U1, U2real(kind=8), dimension(:,:), allocatable:: AMAT
!C!C-- INIT.
read (*,*) NE, VELO, DIFF
DX= 1.d0/dfloat(NE)N = NE + 1
LENGTH= 1.0d0allocate (U1(N), U2(N), AMAT(N,N))
REc= VELO*DX/DIFFPECLET= VELO/DIFF
COEFc2= VELO/(2.d0*DX)COEFc1= VELO/(1.d0*DX)COEFd= DIFF/(DX*DX)
!C!C-- Central Diff.
do j= 1, Ndo i= 1, NAMAT(i,j)= 0.d0
enddoenddo
do i= 1, NU1(i)= 0.d0
enddo
21111 2
2 xuuu
xuua iiiii
PECLET: =aL/v=a/vREc: =ax/v
COEFc1: a/xCOEFc2: a/(2*x)COEFd: v/x2
PDE
47
2/2do i= 2, N-1AMAT(i,i )= 2.d0 * COEFdAMAT(i,i-1)= -COEFc2 - COEFdAMAT(i,i+1)= COEFc2 - COEFd
enddo!C!C-- Boundary Conditions
U1(1)= 0.d0U1(N)= 1.d0AMAT(1,1 )= 1.d0AMAT(N,N )= 1.d0
!C!C-- Gaussian Elimination
call GAUSS (AMAT, U1, N, N)
stopend
xaRwhereuRuuR
uxx
aux
uxx
ax
uuuxuua
ciciic
iii
iiiii
0242
02
22
22
11
12212
21111
COEFc1: a/xCOEFc2: a/(2*x)COEFd: v/x2
1 2 3 4 5 6 7 8 9
i=2~N-1
PDE
48
2/2do i= 2, N-1AMAT(i,i )= 2.d0 * COEFdAMAT(i,i-1)= -COEFc2 - COEFdAMAT(i,i+1)= COEFc2 - COEFd
enddo!C!C-- Boundary Conditions
U1(1)= 0.d0U1(N)= 1.d0AMAT(1,1 )= 1.d0AMAT(N,N )= 1.d0
!C!C-- Gaussian Elimination
call GAUSS (AMAT, U1, N, N)
stopend
101
Nuu
1 2 3 4 5 6 7 8 9
1022
x
xu
xua
11,00 uu
PDE 49
Finite-Difference Method, FDM 1 2 3
PDE
50
21st order wave equation
a
-25.0
0.0
25.0
50.0
75.0
100.0
125.0
0 50 100 150 200 250 300
t=0.00 t=0.45
0),300(),0(
11050:60
50sin1000,
300110,500:00,
300
0
tutu
xxxu
xxxu
a
axua
tu
PDE
51
xuua
tu ii
211
explicitforward Euler
ninininin
in
in
in
i uuxtauu
xuua
tuu
11111
1
22
implicitbackward Euler
ni
ni
ni
ni
ni
ni
ni
ni uu
xtauu
xta
xuua
tuu
1
111
1
11
11
1
222n
iun-
PDE
52
FTCSForward-Time/Center-Space
xtacuucuu
uuxtauu
xuua
tuu
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
,2
2
2
111
111
111
Courant1
x=5.0, t= 0.01666, c~1.00 x=5.0, t= 0.015, c=0.90 x=5.0, t= 0.0075, c=0.45
PDE
53
t=0.45
t/x
-3.00E+04
-2.00E+04
-1.00E+04
0.00E+00
1.00E+04
2.00E+04
3.00E+04
0 50 100 150 200 250 300
C=1.00C=0.90C=0.45Exact
PDE
54
Von Neumann Stability Analysis
IinxiIknn
i egegu gngnIk= k x
1sin1
sin1sin2
22
22
111
cg
IcgIee
ueecuguuucuu
II
ni
IIni
ni
ni
ni
ni
ni
ni
IIinIiInni
ni
IIinIiInni
ni
IinIinni
ueegeegu
ueegeegu
gueggegu
11
11
11
gt|g|11
FTCSc=0
PDE
55
nininini
ni
ni
ni
ni
uucuuxuua
tuu
11
11
C=1.00
211 2
2 xuuuxa iii
-25.0
0.0
25.0
50.0
75.0
100.0
125.0
0 50 100 150 200 250 300
C=1.00C=0.90C=0.45Exact
PDE
56
Von Neumann Stability Analysis
ccccccgIcccIcccecg
uceucguuucuuI
ni
Ini
ni
ni
ni
ni
ni
11cos212cos2cos221
sincos1sincos11
1
22
11
(cos-1)0(1-c)0|g|1
ccg 11cos21
1
xtac
CFLCourant-Friedrichs-Lewy
PDE
57
Von Neumann Stability Analysis
I
ni
ni
Ini
ni
ni
ni
ni
ni
ni
ni
eccg
uuecgguc
ucuucuucuu
11
1
1 1111
111
(1-cos) 01|g|1
unconditional stable
cos1121
cos2cos2221
sincos1122
cc
cccc
Icccecc I
cos11211
ccg
PDE
58
Lax-Wendroff2
322
22
1
2
22
2
2
32
2
21
2
!2
tOxuatt
xuauu
xua
tu
xa
xu
ta
tu
tOttut
tuuu
xua
tu
ni
ni
ni
ni
nininininini
ni
ni
ni
ni
nin
in
i
uuucuucu
xuuuta
xuutauu
112
11
21122111
221
21
221
2
PDE
59
Lax-Wendroff
-25.0
0.0
25.0
50.0
75.0
100.0
125.0
0 50 100 150 200 250 300
C=1.00C=0.90C=0.45Exact
PDE
60
sin2
sin21
sincos12
1sincos12
221
22
221
22
22
2
22
2
22
2
Icc
IcccIcc
eccceccg
ueccucueccgu
II
ni
Ini
ni
Ini
(1-c2)0|g|1
2sin141 422 ccg
ninininininini uuucuucuu 112111 221
21
1
xtac
CFLCourant-Friedrichs-Lewy
PDE
61
Modified Equation1/2
xtacuucuu
xua
tu n
in
in
in
i
,1
1
Taylor
Taylor
1
43
3
32
2
2
1
43
3
32
2
21
!3!2
!3!2
tOttut
tut
tuuu
tOttut
tut
tuuu
ni
ni
ni
ni
2
3
231
332
3
33
3
3
2
22
2
2
,6622
xtOxxuat
tux
xuat
tu
xua
tu
4
PDE
62
Modified Equation2/25647
4
5
7122
7
xtOxua
tu
xtxtOxuxata
xua
tu
,
,,
3
33
3
3
223
3232
2
2
3223
3
32
2
2
2
,,,
1326
12
xxtxttOxuxcxa
xucxa
xua
tu
K.A. Hoffmann & S.T. Chiang, Computational Fluid Dynamics for Engineers Volume I, Chapter 4, Section 4.6 Modified Equation, Engineering Education System (EES), 1993.
PDE
63
Artificial/Numerical Viscosity
7u
eArtificial/Numerical Viscositydissipate
7 3223
3
32
2
2
2
,,,
1326
12
xxtxttOxuxcxa
xucxa
xua
tu
22
121
xucxae
e=0c=1Courant2tCourantLax-Wendroff
PDE 64
Finite-Difference Method, FDM 1 2 3
PDE
65
31/3Unsteady Diffusion Equation
1022
x
xu
tu
1,1,0,00)0,(
0.1
tutuxu
FTCSExplicitForward Euler
2111
122121
211
1
21
21
2
xtrruurruu
ux
tux
tux
tu
xuuu
tuu
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
PDE
66
32/3Unsteady Diffusion Equation
ImplicitBackward Euler
nininini
ni
ni
ni
ni
ni
ni
ni
ni
ni
uruurru
uux
tux
tux
tx
uuut
uu
11
111
112
12
112
2
11
111
1
21
21
2
1022
x
xu
tu
1,1,0,00)0,(
0.1
tutuxu
PDE
67
33/3Unsteady Diffusion Equation
-Crank-Nicolson
nininininini
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
urururururur
ux
tux
tux
tux
t
xuuu
xuuu
tuu
111
111
1
21
121
21
12
211
2
11
111
1
21
221
2
21
2221
-
PDE
68
ExplicitForward Euler
niu
niu 1
niu 1
1niu
niu
1niu
11
niu
11
niu
niu
niu 1
niu 1
1niu
11
niu
11
niu
ImplicitBackward Euler
-Crank-Nicolson
PDE
69
21
2
x
tr
)1(cos2121cos2
21
21
21 111
rrrgreerg
ureururegu
ruurruu
II
ni
Ini
ni
Ini
ni
ni
ni
ni
21
2sin2
1
12
sin4111
2sin41)1(cos21
2
2
2
r
rg
rrg
)cos1(211
21
21 1111
1
rg
uuegrurguegr
uruurru
ni
ni
Ini
ni
I
ni
ni
ni
ni
1cos1210cos11
rg
Crank-Nicolson
PDE
70
t=2.00
r= 0.515x= 0.10, t= 0.00515
r= 0.500x= 0.10, t= 0.00500
-0.50
0.00
0.50
1.00
1.50
0.00 0.20 0.40 0.60 0.80 1.00
r= 0.515
r= 0.500
Exact
PDE
71
1/3implicit REAL*8(A-H,O-Z)real(kind=8), dimension(:), allocatable:: U, U0, Breal(kind=8), dimension(:,:), allocatable:: AMATreal(kind=8) :: DX, DT, TIME, C, LENGTHinteger :: N
!C!C +-------+!C | INIT. |!C +-------+!C===
write (*,*) 'NE, DT, OMEGA'read (*,*) NE, DT, OMG
DIFF= 1.d0
DX= 1.d0/dfloat(NE)N = NE + 1EPS= 1.d-08
allocate (B(N), U(N), U0(N), AMAT(N,N))COEF= DIFF*DT/(DX*DX)
do j= 1, Ndo i= 1, NAMAT(i,j)= 0.d0
enddoenddo
do i= 1, NU (i)= 0.d0U0(i)= 0.d0
enddo
NE: N: =NE+1DT: tDX: xOMG: SOR1
PDE
72
2/3do i= 2, N-1AMAT(i,i )= 2.d0*COEF + 1.d0AMAT(i,i-1)= -COEFAMAT(i,i+1)= -COEFAMAT(i,i )= 1.d0/AMAT(i,i)
enddo!C!C-- Boundary Conditions
U(1)= 0.d0U(N)= 1.d0
!C===
!C!C +------------------+!C | TIME integration |!C +------------------+!C===
TIME= 0.d0doTIME= TIME + DT
BNRM2= 0.d0do i= 2, N-1
B(i) = U0(i)BNRM2= BNRM2 + B(i)**2
enddoif (BNRM2.eq.0.d0) BNRM2= 1.d0
nininini uruurru 11111 21
1,1,0,00)0,(
tutuxu
n
iib
1
2
2b
AMAT(i,i)
PDE
73
3/3!C!C-- SOR
do iter= 1, 100*NDNRM2= 0.d0do i= 2, N-1
UU= U(i)RESID= ((B(i) - AMAT(i,i-1)*U(i-1)
& - AMAT(i,i+1)*U(i+1))*AMAT(i,i)-UU)*OMGU(i)= UU + RESIDDNRM2= DNRM2 + RESID**2
enddoif (dsqrt(DNRM2/BNRM2).lt.EPS) exit
enddo
do i= 1, NU0(i)= U(i)
enddoif (TIME.ge.2.d0) exit
enddo!C===
!C!C-- Result
write (*,'(a,1pe16.6)') '### TIME', TIMEdo i= 1, N
XX = dfloat(i-1)*DXwrite (*,'(3(1pe16.6))') XX, U(i)
enddo
stopend
2
2
)()1(
b
xx kkui*ui n+1
**111,111,,
1
111,
1,
111,
1ii
niii
niiii
ii
ni
in
iiin
iiin
iii
uuuAuABA
u
BuAuAuA
PDE 74
Finite-Difference Method, FDM 1 2 3
PDE
75
1/3 1 3-
21 LU
PDE
76
2/3 31 LU
S.V. PatankarExponential Method
Patankar, S.V., A Calculation Procedure for Two-Dimensional Elliptic Situations, Numerical Heat Transfer, Vol.4, p.409, 1981
1111
115
15
1
211111
,0,10,1.01,1.0110
,,,0
iiiicic
iiiccicccic
ccciiiiii
mmmmFmRifmmmRDmFRDmRif
xaF
xDxaRumumum
PDE
77
Exponential MethodRc
Rc=1.00x= 0.10, a= 1.00, v= 0.10
Rc=2.50x= 0.10, a= 1.00, v= 0.04
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Central DifferenceUpwindingExponentialExact
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00
Central DifferenceUpwindingExponentialExact
PDE
78
3/3 23ScilabMatlabC/C++Fortran
201421913:00
1
1 23A4 8
8
2318