Spiegelman: Columbia University, February 18, 2002
Modeling Myths
• Modeling is Difficult
• Numerical models can do everything
• Numerical models will solve your problems
Modeling Facts
• Modeling is simple book-keeping (plus a bit of magic)
• There are no black boxes!
• Numerical models have a life of their own and make their own
problems
1
Spiegelman: Columbia University, February 18, 2002
Used carefully, numerical solutions are a very powerful tool for
gaining insight into possible physical processes. Used
indiscriminately, they become an intellectual black-hole. We need to
emphasize the two end-member approaches to modeling
1. The Kitchen Sink Approach: Throw everything into it and hope
something useful comes out (BAD IDEA)
2. The Model Problem Approach: Gain insight by developing
simple model problems that balance interesting behaviour with
comprehensibility (GOOD IDEA – but takes finesse).
2
Spiegelman: Columbia University, February 18, 2002
This course will stress the 2nd approach and will emphasize the
following axioms
• Be problem driven
• Understand your problem
• Keep it simple
• Never model more than you can understand (or observe?)
• Avoid the ‘reality trap’ (models 6= ‘reality’)
• Choose your techniques to mimic the underlying physics.
• The only successful model is an insightful model.
3
Spiegelman: Columbia University, February 18, 2002
Or more succinctly
• Think before you model
• Think while you model
• Think after you model
4
Spiegelman: Columbia University, February 18, 2002
5
Some Earth science problems you ought to know
(maybe)
1. Thermal convection (2-D Rayleigh-Benard)
2. The Lorenz Equations ( Chaos and ODE’s)
3. The shallow water equations
4. Seismic wave propagation
5. Flow in Porous Media
Spiegelman: Columbia University, February 18, 2002
6
Thermal Convection
(2-D Rayleigh Benard Convection)
∂T
+ V · ∇T = ∇2 T
∂t
∂T
1 ∂ω
2
+ V · ∇ω = ∇ ω − Ra
Pr ∂t
∂x
∇2 ψ = −ω
V
= ∇× ψk,
α∆T gd3
,
Ra =
νκ
ω = (∇× V) · k
ν
Pr =
κ
Spiegelman: Columbia University, February 18, 2002
The Lorenz Equations and chaos
Let
ψ(t, x) = W (t) sin(aπx) sin(πz)
T (t, x) = (1 − z) + T1 (t) cos(aπx) sin(πz) + T2 (t) sin(2πz)
Then RB convection becomes
dW
dt
dT1
dt
dT2
dt
= Pr(T1 − W )
= −W T2 + rW − T1
= W T1 − bT2
7
Spiegelman: Columbia University, February 18, 2002
8
Lorenz Equations Ra=28
(W,T1,T2)=(0,1,0)
60.0
W(t)
T1(t)
T2(t)
Variables
40.0
20.0
0.0
−20.0
−40.0
0.0
10.0
20.0
30.0
Time
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
Linearized Shallow water equations
equatorial β plane
∂v
+ βyk × v = −g∇η
∂t
∂η
+ ∇· (H v) = 0
∂t
With forcing and dissipation (ala Cane-Zebiak)
∂v
+ βyk × v = −gH0 ∇η + τ /ρ − rv
∂t
∂η
+ ∇· (H v) = −rη
∂t
9
Spiegelman: Columbia University, February 18, 2002
10
Sea Surface temperature anomalies
LDEO2 El Niño model
0˚
30˚S
Latitude
30˚N
Sea Surface temperature anomalies
150˚E
180˚
150˚W
120˚W
90˚W
Longitude
-0.8˚C
-0.6˚C
-0.4˚C
-0.2˚C
0˚C
0.2˚C
0.4˚C
0.6˚C
0.8˚C
1˚C
1.2˚C
1.4˚C
temperature anomaly
0˚
30˚S
Latitude
30˚N
Sep98
150˚E
180˚
150˚W
120˚W
90˚W
Longitude
Dec98
-1˚C
-0.5˚C
0˚C
0.5˚C
temperature anomaly
1˚C
1.5˚C
Spiegelman: Columbia University, February 18, 2002
Seismic Wave propagation
∂V
= ∇· σ + f
∂t
∂Vj
∂σij
∂Vi
+ λ∇· Vδij
=µ
+
∂t
∂xj
∂xi
ρ
Solutions by Pseudo-spectral techniques ala Gustavo Correa (LDEO)
11
Spiegelman: Columbia University, February 18, 2002
12
2.0
4.5
2.5
3.0
3.5
4.0
2.0
4.5
2.5
3.0
3.5
4.0
Spiegelman: Columbia University, February 18, 2002
13
2.0
4.5
2.5
3.0
3.5
4.0
2.0
4.5
2.5
3.0
3.5
4.0
Spiegelman: Columbia University, February 18, 2002
14
Flow in heterogeneous porous media
∇·
K
[∇P − ρf g] = 0
µ
elastic
-3.5 -3
-2
rigid
-2
-1
-1
0
0
1 1.5(rigid)
1
log Permeability
2 (elastic)
Spiegelman: Columbia University, February 18, 2002
15
Flow in deformable porous media (magma migration)
∂(ρf φ)
+ ∇· (ρf φv) = Γ
∂t
∂[ρs (1 − φ)]
+ ∇· [ρs (1 − φ)V] = −Γ
∂t
−kφ
[∇P − ρf g]
φ(v − V) =
µ
∇P = −∇× [η∇× V] + ∇ [(ζ + 4η/3)∇· V] + G − ρ̄g
a2 φn
kφ ∼
b
Spiegelman: Columbia University, February 18, 2002
16
Flow in deformable porous media
(2-D potential form, non-dimensional)
∂φ
+ V · ∇φ = (1 − φ0 φ)C + Γ
∂t
−∇· kφ ∇C + C = ∇· kφ [−∇× ωj − (1 − φ0 φ)k] + Γ
∇2 U s = C
g ∂ρ
2
∇ ω=−
η ∂x
∇2 ψ s = −ω
δρ
ρf
Spiegelman: Columbia University, February 18, 2002
17
Reactive Flow Localization
Non-linear Porosity waves
5
80
height z/d
40
3
2
1
t=0
distance
0
0
1
Earth Science Applications
2
width x/d
β=70.00°, U0= 8.00 cm/yr, slab age=150.00 Ma
3
4
Fluid Flow
Thermal Structure
1400
0
60
70
1200
50
80
1000
90
100
800
150
600
200
depth (km)
depth (km)
time
4
100
110
120
400
130
200
250
0
140
50
100
150
200
distance (km)
250
300
0
150
160
mid-ocean ridges
subduction zones
120
140
distance (km)
160
Spiegelman: Columbia University, February 18, 2002
18
Direction fields and solutions for
dc
= −c
dt
Concentration
1.5
1.0
0.5
0.0
0.0
0.5
1.0
Time
1.5
Spiegelman: Columbia University, February 18, 2002
19
Simple Stepping Schemes
Concentration
1.0
0.5
True Solution
Euler Step
h
(first order step)
0.0
0.0
0.5
1.0
1.5
Time
Concentration
1.0
1
2
0.5
Mid-point Step
3
True Solution
(second order step)
(euler)
h
0.0
0.0
0.5
1.0
Time
1.5
Spiegelman: Columbia University, February 18, 2002
20
4th Order Runge-Kutta Step
Concentration
1.0
1
3
2
4
Runge-Kutta step
0.5
(mid-point)
(4th order)
True Solution
(euler)
h
0.0
0.0
0.5
1.0
Time
1.5
Spiegelman: Columbia University, February 18, 2002
Bulirsch-Stoer Stepping using Richardson Extrapolation
21
Spiegelman: Columbia University, February 18, 2002
22
Transport by Characteristics: Particle based methods
t=0
2
4
6
t=10
8
concentration
2.5
2.0
1.5
1.0
0
10
20
distance
V(x)
= 0.2x
30
40
Spiegelman: Columbia University, February 18, 2002
23
FTCS Explodes!
t=0
1
2
2
3
4
3
t=4
5
6
concentration
2.0
1.5
1.0
0.5
0
1
distance
7
8
9
10
Spiegelman: Columbia University, February 18, 2002
24
Staggered Leapfrog works (α
< 1)
α = 0.9
t=0
1
2
2
3
4
3
t=4
5
6
concentration
2.0
1.5
1.0
0.5
0
1
distance
7
8
9
10
Spiegelman: Columbia University, February 18, 2002
25
α = 1.01
t=0
1
2
2
concentration
t=2.53
1
1
0
0
1
2
3
4
5
distance
6
7
8
9
10
Spiegelman: Columbia University, February 18, 2002
26
Staggered Leapfrog disperses (α
= 0.5)
3.0
t=0
distance
2.5
2.0
1.5
1.0
t=100
0.5
0
1
2
3
4
5
concentration
6
7
8
9
10
Spiegelman: Columbia University, February 18, 2002
27
3.0
t=0
distance
2.5
2.0
1.5
t=100
1.0
0.5
0
1
2
3
4
5
concentration
6
7
8
9
10
Spiegelman: Columbia University, February 18, 2002
28
Simple Upwind diffuses (badly) (α
= 0.5)
t=0
3
concentration
2
2
1
t=100
1
0
0
1
2
3
4
5
distance
6
7
8
9
10
Spiegelman: Columbia University, February 18, 2002
29
Better schemes
3.5
3.5
3
3
2.5
concentration
concentration
t=0
t=0−100
2
1.5
1
0.5
0
2
4
6
8
0
2
4
b
3
3
2.5
2.5
concentration
3.5
t=0−100
2
1.5
1
6
8
10
distance
3.5
0.5
t=100
1.5
0.5
10
distance
concentration
2
1
a
c
2.5
t=0−100 (all identical)
2
1.5
1
0
2
4
6
distance
8
0.5
10
d
0
2
4
6
distance
8
10
Spiegelman: Columbia University, February 18, 2002
30
Semi-Lagrangian Schemes: a recipe
true characteristic
u(n+1,j)
n+1
∆t
n+1/2
n
c(n)
xX
∆x
x
u(n+1/2)
j
Spiegelman: Columbia University, February 18, 2002
31
Behaviour with non-constant velocity
concentration
3
60
15
75
90
105
45
3
120
2
2
t=0
1
1
staggered−leapfrog
0.09s
concentration
30
1.32s
0
0
3
3
2
2
1
1
0
10
20
distance
90 15 105 30 120 45
75
t=0
semi−lagrangian
0.02s
0
60
mpdata (ncor=3, i3rd=1)
pseudo−spectral
15.85s
30
0
0
10
20
distance
30
Spiegelman: Columbia University, February 18, 2002
32
Comparison of Pseudo-spectral and Semi-Lagrangian schemes
semi−lagrangian (1024 pts, α=20, t=0.05s)
3
concentration
2
1
pseudo−spectral (256 pts,α=0.5,t=4.98s)
0
0
10
20
distance
30
Spiegelman: Columbia University, February 18, 2002
33
Advection-diffusion: FTCS
0.010
1
2
3
0.000
4
t=0
-0.005
-0.010
0.0
10.0
20.0
30.0
distance
40.0
50.0
3.0
t=0
True solution
2.5
Calculated solution
Temperature
relative error
0.005
2.0
1
2
1.5
1.0
0.0
10.0
3
4
20.0
30.0
distance
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
34
Advection-diffusion: Crank-Nicholson
0.0040
0.0030
1
2
3
0.0010
0.0000
4
t=0
-0.0010
-0.0020
-0.0030
-0.0040
0.0
10.0
20.0
30.0
distance
40.0
50.0
3.0
t=0
True solution
2.5
Calculated solution
Temperature
relative error
0.0020
2.0
1
2
1.5
1.0
0.0
10.0
3
4
20.0
30.0
distance
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
35
Advection-diffusion: Operator-Splitting MPDATA + CN
(no corrections. . . upwind scheme)
0.000
t=0
-0.010
-0.020
1
-0.030
0.0
10.0
2
3
4
20.0
30.0
distance
40.0
50.0
3.0
t=0
True solution
2.5
Temperature
relative error
0.010
Calculated Solution
2.0
1
2
1.5
1.0
0.0
10.0
3
4
20.0
30.0
distance
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
36
Advection-diffusion: Operator-Splitting MPDATA + CN
(one correction)
0.0004
relative error
0.0002
0
t=0
-0.0002
2
3
4
1
-0.0004
0.0
10.0
20.0
30.0
distance
40.0
50.0
3.0
t=0
True solution
Temperature
2.5
Calculated Solution
2.0
1
2
1.5
1.0
0.0
10.0
3
4
20.0
30.0
distance
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
37
Advection-diffusion: Operator-Splitting MPDATA + CN
Da Woiks! (ncor=3, i3rd=1)
0.0002
1
2
0.0000
3
t=0
4
-0.0001
-0.0002
0.0
10.0
20.0
30.0
distance
40.0
50.0
3.0
t=0
True solution
2.5
Temperature
relative error
0.0001
Calculated Solution
2.0
1
2
1.5
1.0
0.0
10.0
3
4
20.0
30.0
distance
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
38
Advection-diffusion: Operator-Splitting Semi-Lagrangian + CN
α = 2.5
0.0002
1
0.0000
2
3
4
t=0
-0.0001
-0.0002
0.0
10.0
20.0
30.0
distance
40.0
50.0
3.0
t=0
True solution
2.5
Temperature
relative error
0.0001
Calculated Solution
2.0
1
2
1.5
1.0
0.0
10.0
3
4
20.0
30.0
distance
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
39
Advection-diffusion: All-in-one Semi-LagrangianCN
α = 2.5
0.0002
1
2
3
0.0000
4
t=0
−0.0001
−0.0002
0.0
10.0
20.0
30.0
40.0
50.0
distance
3.0
t=0
True solution
2.5
Temperature
relative error
0.0001
Calculated Solution
2.0
1
2
3
1.5
1.0
0.0
10.0
20.0
4
30.0
distance
40.0
50.0
Spiegelman: Columbia University, February 18, 2002
40
2-D control volume
i,j+1
Fy(i,j+1/2)
Fx(i+1/2,j)
i-1,j
i,j
i,j-1
i+1,j
Spiegelman: Columbia University, February 18, 2002
41
Boundary condition pointers
side 2
iout(2,1)
iout(1,2)
side 1
dir 2 (j)
side 1
side 2
iout(1,1)
iout(2,2)
dir 1 (i)
Spiegelman: Columbia University, February 18, 2002
42
Bicubic interpolation
j+2
x
j+1
x
+ (ri,rj)
x
j
(i,j)
x
j-1
i-1
i
i+1
i+2
Spiegelman: Columbia University, February 18, 2002
43
Some useful 2-D advection fields
Rigid body rotation vs. a shear cell
a
1
b
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spiegelman: Columbia University, February 18, 2002
Rigid Body rotation test: a methods sampler
44
Spiegelman: Columbia University, February 18, 2002
45
1.0
1.0
11
1
0.5
0.5
1
1
1
1
a
0.0
0.0
0.5
1.0
b
1.0
0.0
0.0
0.5
1
1
1
0.5
0.5
1
1
1
1
1
c
1
0.0
0.0
0.5
1.0
d
1.0
0.0
0.0
0.5
1.0
1.0
11
11
0.5
0.5
1
1
1
0.0
0.0
0.5
1
1
1
e
1.0
1.0
1.0
f
0.0
0.0
0.5
1.0
Spiegelman: Columbia University, February 18, 2002
Rigid Body rotation test: a methods sampler
46
Spiegelman: Columbia University, February 18, 2002
47
1.0
1.0
11
1
0.5
0.5
1
1
1
1
stag-leap 65
2 0.0
0.0
0.5
1.0
upwind 65
1.0
2 0.0
0.0
0.5
1.0
1
1
1
0.5
0.5
1
1
1
1
1
mpdata 1 65
2 0.0
0.0
1
0.5
1.0
mpdata 2 65
1.0
2 0.0
0.0
0.5
11
0.5
0.5
1
1
1
0.0
0.5
1
1
1
2 0.0
1.0
1.0
11
mpdata 33 65
1.0
1.0
stag-leap 129
2 0.0
0.0
0.5
1.0
Spiegelman: Columbia University, February 18, 2002
48
Shear cell test
Staggered Leapfrog vs. Upwind
1.0
1.
5
1.0
0.
5
0.5
1
1
0.5
0.5
0.5
1.5
1
0.0
0.0
1.0
5
0.0
1.0 0.0
1.0
0.5
0.5
1.0
0.5
1.0
1
0.5
0.
5
1.
5
a
-0.
5
0.
0.5
0.5
b
0.0
0.0
0.5
0.0
1.0 0.0
Spiegelman: Columbia University, February 18, 2002
49
Shear cell test
Mpdata (with everything) vs. High res Staggered Leapfrog
1.0
0.
5
1.
5
1.0
1
0.5
1
0.5
0.5
1.5
1
0.0
0.0
1.0
0.0
1.0 0.0
1.0
0.5
0.5
1.0
1
0.5
0.5
0.
5
0.5
1.5
0.5
1.5
-0.5
1
0.
5
1.5
c
0.5
1
d
0.0
0.0
0.5
0.0
1.0 0.0
0.5
1.0
Spiegelman: Columbia University, February 18, 2002
50
The winner!
Semi-Lagrangian behaviour
Spiegelman: Columbia University, February 18, 2002
51
1.0
1.0
1
1
0.5
0.5
1
1
1
1
1
0.0
0.5
b
1.0
0.0
0.5
1.0
5
1.0
0.0
0.
1
5
1.
1
0.5
1 1.5
c
0.5
0.0
0.0
0.5
0.5
0.5
0.0
1.0 0.0
1.0
0.5
1.0
0.5
1
1.5
1
0.
5
1.0
0.5
0.5
0.5
5
1.
0.
0.0
1.5
a
1
5
1
0.5
d
0.0
0.0
0.5
0.0
1.0 0.0
0.5
1.0
1.0
Spiegelman: Columbia University, February 18, 2002
52
2-D Diffusion (booooring. . . )
2-D FTCS scheme
1.0
1.0
0.5
1
5
0.
0.5
0.0
0.0
1
5
1.
0.5
0.0
1.0 0.0
1.0
0.5
1.0
0.5
0.5
1.0
0.5
0.5
0.5
0.0
0.5
1.0
0.5
0.0
1.0 0.0
1.0
0.5
0.5
1.0
0.5
0.0
0.5
0.0
0.0
0.5
0.0
1.0 0.0
0.5
1.0
Spiegelman: Columbia University, February 18, 2002
53
2-D Diffusion
Errors for 2-D FTCS scheme
1.0
0
5e
-50e
-0
-05
5
0.5
-5e-05
5e-
05
0
5e-05
-5e-05
-5e-05
5e
-05
0
0.0
0.0
0.5
1.0
1.0
15
0.
02
00
00
0.0
0.
00
01
5
01
55e-05
5 5e-0
2
00
0.0
00
.0
5
1
00
-0
0
-5e
-05
15
0
5
00
5
001
0.0
001
-05
-5e
0.
-5e-05 0.0
0
0
5
5e-0
0.0
0.0
0.
-0.0
5 05
001e-0.0 -5
5e-05
0.5
5
5e-05
5
-0
5e
-5e-05
2
-0.
15
000
01
00
00
-0.
5
00
0.0
05
e-5
5
01
00
5
5e-05
1
00
.0
-0
.0
0
0.
0.5
-0
5
-0
5e
-5e-05
0.0001
0
015
15
00
.0
-0
0.00
15
00
02
0
0
1.0
0.0
1.0 0.0
1.0
0.5
0.5
1.0
0
0.5
0.0
1.0 0.0
15
00
001 5
-0
-5e
5e-05
0
-5e
-05
5
01
5
.0
0
-0
-0.0
05
0.0
01
5e
0
-05
01
5
-0 5e-0
5
.0
00
15
0
15
-0.0001
5
001
00
-05
00
01
00
0.0
5e-
0.0
0.0
0.
0.0
5
5e-05
-05
-0.0001-5e
-0.
.00
0.5
5
15
000
-0
001 e-05
-5
-0.
-5e-05
00
1
-05.0
001
5e
15
.0
0.0
-05
00
-0
-05
-5e
-5e
0.0
05
e-5
0
15
15
00
00
015
.0
0.0
0.00
-0
0.5
5e
-0. -05
000
15 0
0
1.0
0
05
5e-
0.5
1.0
Spiegelman: Columbia University, February 18, 2002
54
2-D Diffusion
Errors for ADI scheme
1.0
0 0
0
0
0.5
0
0
0
0
0
0.0
0.0
1.0
0.5
1.0
0
0
0
0
0.5
0
0.5
1.0
0
0
0
0
0
0
0
0.0
0.0
0.0
1.0 0.0
1.0
0.5
1.0
0.5
1.0
-0.001
0
0
0
0
0.5
0
0.0
0.0
01
01
0
0
0
0
0.0
0.0
0.5
0
0
0
0.5
0.0
1.0 0.0
0.5
1.0
Spiegelman: Columbia University, February 18, 2002
55
Structure of the 5-point Laplacian Operator as a sparse matrix
0
50
100
150
200
0
50
100
nz = 1065
150
200
Spiegelman: Columbia University, February 18, 2002
56
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
nz = 217
40
50
Spiegelman: Columbia University, February 18, 2002
57
Structure of the inverse of the 5-point Laplacian Operator as a
sparse matrix
20
40
60
80
100
120
140
160
180
200
220
50
100
150
200
Spiegelman: Columbia University, February 18, 2002
58
Testing Laplace Solvers: the sin-cell test
Solution
1.0
0.5
0
-0.5
0
0.5
-0.5
0
0.5
0
0.0
0.0
0.5
Errors
1.0
Spiegelman: Columbia University, February 18, 2002
59
1.0
0
15
00
0.0
0.0001
-0.0001
0
0.5
0
-0
15
00
0
-0.
0
-0
5
01
00
-0.
-0.0001
0.0001
0
0
15
00
0.0
0.0
0.0
0.5
1.0
Spiegelman: Columbia University, February 18, 2002
60
Convergence Behaviour of optimal SOR
4
10
2
L2 Norm of residual
10
0
10
10
10
10
10
10
−2
−4
−6
−8
−10
0
50
100
iterations
150
200
Spiegelman: Columbia University, February 18, 2002
61
A nested Multi-level Grid
(A 3-level grid)
Spiegelman: Columbia University, February 18, 2002
62
V-cycles and Full Multi-Grid (FMG) cycles
4
V-cycle
coarsest grid
S
3
R
R
2
R
R
1
S
S
R
R
R
R
finest grid
R
R
S
S 4
R
R
R
R
coarsest grid
R 3
R
2
FMG-Vcycle
1
R
R
finest grid
Spiegelman: Columbia University, February 18, 2002
63
Multi-Grid storage scheme ala Briggs
grid 1
2
34
1
A
5
u
rhs
res
ip(1)
ip(2)
ip(3) ip(4)
Spiegelman: Columbia University, February 18, 2002
64
A V-cycle in the Briggs Scheme
Going up!
Spiegelman: Columbia University, February 18, 2002
65
grid 1
u
initial Guess
relax Npre times
2
34
0
0
0
fine
rhs
then calculate residual
then restrict
res
ip(1)
ip(2)
u
ip(3) ip(4)
0
relax Npre times
rhs
then calculate residual
then restrict
res
u
rhs
res
solve
u
Spiegelman: Columbia University, February 18, 2002
66
A V-cycle in the Briggs Scheme
Coming Down!
Spiegelman: Columbia University, February 18, 2002
67
grid 1
2
34
u
add back
relax Npost times
rhs
interpolate
correction
interp
coarse
ip(1)
ip(2)
ip(3) ip(4)
u
relax
add
rhs
interpolate
interp
u
much improved guess
relax
add
rhs
interpolate
interp
fine
Spiegelman: Columbia University, February 18, 2002
68
The Big test!
Timing and errors for solving Poisson problem with Dirichlet Boundaries on
a 2-2 sin cell test
ce s, O3 q o
3
qa c
qau odb db
10
Fishpak
Y12M
MG−Vcycle
FMG
SOR
2
time (cpu seconds)
10
1
0
129
10
10
513
257
10
10
1025
−1
Ni=65
−2
3
10
4
10
5
10
Total Grid−points
6
10
7
10
Spiegelman: Columbia University, February 18, 2002
10
10
average error
10
10
10
10
10
10
69
ce s, O3 q o
−2
qa c
qau odb db
−3
Fishpak
Y12M
MG−Vcycle
FMG
SOR
−4
−5
−6
−7
−8
−9
−10
10
3
10
4
10
5
6
10
10
Total Grid−points
7
10
8
10
Spiegelman: Columbia University, February 18, 2002
70
General Conservation of Mass for chemistry
∂
ρs (1 − φ)cs + ∇· [ρs (1 − φ)Vcs ] = −I + Ds
∂t
(1)
∂
f
f
ρf φc + ∇· ρf φvc = I + Df
∂t
(2)
For Fractional melting I
= (cs /D)Γ, therefore
cs
∂
s
s
ρs (1 − φ)c + ∇· [ρs (1 − φ)Vc ] = − Γ
∂t
D
(3)
cs
∂
f
f
ρf φc + ∇· ρf φvc = Γ
∂t
D
(4)
Spiegelman: Columbia University, February 18, 2002
71
Expanding By chain rule gives
cs
∂
ρs (1 − φ) + ∇· [ρs (1 − φ)V] +
∂t
s
cs
∂c
s
+ V · ∇c = − Γ
ρs (1 − φ)
∂t
D
cf
∂
ρf φ + ∇· [ρf φv] +
∂t
f
cs
∂c
f
+ v · ∇c = Γ
ρf φ
∂t
D
But Conservation of total mass is
∂
ρs (1 − φ) + ∇· [ρs (1 − φ)V] = −Γ
∂t
(5)
∂
ρf φ + ∇· [ρf φv] = Γ
∂t
(6)
Spiegelman: Columbia University, February 18, 2002
72
Therefore we can write the problem as
cs
Ds cs
=− Γ
−c Γ + ρs (1 − φ)
Dt
D
(7)
cs
Df cf
= Γ
c Γ + ρf φ
Dt
D
(8)
s
f
Spiegelman: Columbia University, February 18, 2002
Rearranging
73
Ds c
1
Γ
= −cs
−1
Dt
D
ρs (1 − φ)
s
f
Df c
Γ
c
f
=
−c
Dt
D
ρf φ
s
and Scaling with
φ = φ 0 φ0
w0 0
t
t =
δ
ρs φ0 w0 0
Γ
Γ =
δ
cs = cs0 cs 0
cs0 f 0
f
c
c
=
D
Yields
(9)
(10)
Spiegelman: Columbia University, February 18, 2002
s
φ0
Ds c
=−
cs
Dt
(1 − φ0 φ)
74
1
−1 Γ
D
Df cf
s
f ρs Γ
= c −c
Dt
ρf φ
(11)
(12)
Spiegelman: Columbia University, February 18, 2002
75
Numerics
General Semi-Lagrangian solution of
Dc
= g(c, x, t)
Dt
is
or
1 +
c+ − c−
−
=
g +g
∆t
2
∆t +
−
g +g
c =c +
2
+
−
Therefore for solid concentration can discretize
s
φ0
Ds c
=−
cs
Dt
(1 − φ0 φ)
1
−1 Γ
D
as
cs + − cs (s−) = −(Acs )+ − (Acs )(s−)
Spiegelman: Columbia University, February 18, 2002
where
therefore
1
∆tφ0
( − 1)Γ
A=
2(1 − φ0 φ) D
76
Spiegelman: Columbia University, February 18, 2002
77
Same for melt concentration;
Df cf
s
f ρs Γ
= c −c
Dt
ρf φ
Becomes
c
f+
−c
f (f −)
= [B(cs − cf )]+ + [B(cs − cf )](f −)
where
B=
∆tρs Γ
2ρf φ
(f −)
f (f −)
Therefore
+
(1 + B )c
or
c
f+
=
f+
= (1 − B
)c
+ (Bcs )+ + (Bcs )(f −)
h
i
(f
−)
+ (Bcs )+ + (Bcs )(f −)
(1 − B (f −) )cf
Final Update Scheme:
(1 + B + )
(13)
Spiegelman: Columbia University, February 18, 2002
c
c
f+
=
s+
78
(1 − A(s−) ) s (s−)
c
=
+
(1 + A )
h
i
(f
−)
+ (Bcs )+ + (Bcs )(f −)
(1 − B (f −) )cf
(1 + B + )