Challenges in
Modelling and Computing Planetary Scale Atmospheric Flows
at High Resolution
Rupert Klein
Mathematik & Informatik, Freie Universität Berlin
Challenges in Scienti c Computing
Warwick, July 1st, 2009
Thanks to ...
Ulrich Achatz
(Goethe-Universität, Frankfurt)
Didier Bresch
(Université de Savoie, Chambéry)
Omar Knio
(Johns Hopkins University, Baltimore)
Oswald Knoth
(IFT, Leipzig)
Piotr Smolarkiewicz
(NCAR, Boulder, CO, USA)
MetStröm
Motivation
Asymptotics
Two-Scale Models
Numerics
Conclusions
Motivation
The Challenge: global cloud-resolving models
Motivation
Compressible ow equations
ρt + ∇ · (ρv) = 0
(ρu)t + ∇ · (ρv ◦ u) + P ∇kπ = 0
(ρw)t + ∇ · (ρvw) + P πz = −ρg
Pt + ∇ · (P v) = 0
1
γ
P = p = ρθ ,
π = p/ΓP ,
Γ = cp/R ,
v = u + wk ,
(u · k ≡ 0)
Motivation
Pseudo-incompressible model∗
“sound-proof”
ρt + ∇ · (ρv) = 0
(ρu)t + ∇ · (ρv ◦ u) + P ∇kπ = 0
(ρw)t + ∇ · (ρvw) + P πz = −ρg
× +∇ · (P v) = 0
P ≡ P (z) ,
ρθ = P (z) ,
∆x < 15 km
θ = θ(z) + θ0
∗
Durran (1988)
Motivation
Hydrostatic primitve equations
“vertically sound-proof”
ρt + ∇ · (ρv) = 0
(ρu)t + ∇ · (ρv ◦ u) + P ∇kπ = 0
×
+P πz = −ρg
Pt + ∇ · (P v) = 0
1
P = p γ = ρθ ,
π = p/ΓP ,
Γ = cp/R ,
∆x > 15 km
v = u + wk ,
(u · k ≡ 0)
Motivation
Why not simply solve the full compressible- ow equations?
Simple wave initial data, periodic domain
(integration: implicit midpoint rule, staggered grid, 512 grid pts., CFL = 10)
0.5
0.5
0
0
p
1
p
1
-0.5
-0.5
-1
-1
-0.4
t=0
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
0.5
0.5
0
0
p
1
p
1
-0.5
-0.3
-0.2
-0.1
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
x
-0.5
-1
t=3
-0.4
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
x
Implicit discretization regularizes by slowing down the waves∗
∗
see, e.g., Reich et al. (2007)
Motivation
Why not simply solve the full compressible- ow equations?
Simple wave initial data, periodic domain
(integration: implicit midpoint rule, staggered grid, 512 grid pts., CFL = 10)
0.5
0.5
0
0
p
1
p
1
-0.5
-0.5
-1
-1
-0.4
t=0
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
0.5
0.5
0
0
p
1
p
1
-0.5
-0.3
-0.2
-0.1
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
x
-0.5
-1
t=3
-0.4
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
x
Implicit discretization regularizes by slowing down the waves∗
∗
see, e.g., Reich et al. (2007)
Motivation
10
T. Hundertmark
and S. Reich
Dispersion
relation for linear waves
in a compressible
atmosphere
(a) ! = 10 min
2
10
10
wave fequency [s!1]
0
!2
10
!4
10
!6
external
Lz=80km
Lz=8km
Lz=800m
Lz=80m
0
wave fequency [s!1]
external
Lz=80km
Lz=8km
Lz=800m
Lz=80m
10
10
(b) ! = 10 sec
2
10
!2
10
!4
10
marked: regularized
unmarked: exact dispersion
0
10
2
10
horizontal length scale [km]
!6
4
10
10
marked: regularized
unmarked: exact dispersion
0
10
2
10
horizontal length scale [km]
4
10
Figure 1. Dispersion relations for vertical slice model and its regularized formulation with the equations
being linearized about a stationary isothermal reference state. The regularization parameter, α, is set
equal to α = 10 min in panel (a) and to α = 10 s in panel (b), respectively. Lines corresponding to
different vertical length scales, Lz , are plotted with increasing line width for decreasing Lz . For fixed
Lz , each line represents wave frequency, ω, as a function of horizontal length scale, L x = 2π/kx .
courtesy: Sebastian Reich, Potsdam University
60 km
Breaking wave-test for anelastic models (Smolarkiewicz & Margolin (1997))
Absorption layers
d = 0 ... 1/600s
U = 10 m/s
N = 0.01 1/s
0 km
“Witch of Agnesi”
h = 628 m, l = 1000m
-60 km
0 km
60 km
0 km
60 km
Breaking wave-test for anelastic models (Smolarkiewicz & Margolin (1997))
-60 km
0 km
60 km
Motivation
Asymptotics
Two-Scale Models
Numerics
Conclusions
Asymptotics: planetary scale
Hydrostatic Primitive Equations: ... aspect-ratio asymptotics
ρt + ∇ · (ρv) = 0
(ρu)t + ∇ · (ρv ◦ u) + P ∇kπ = 0
P πz = −ρg
Pt + ∇ · (P v) = 0
1
γ
P = p = ρθ ,
π = p/ΓP ,
Γ = cp/R ,
v = u + wk ,
(u · k ≡ 0)
Asymptotics: convective scale
Characteristic (inverse) time scales
dimensional
advection
sound
:
:
dimensionless
uref
hsc
1
p
√
pref /ρref
ghsc
=
hsc
hsc
√
ghsc 1
=
uref
ε
s
internal waves :
N=
√
ghsc
uref
g dθ
θ dz
For single-scale models with advection & internal waves:∗
hsc dθ
= O(M2 )
θ dz
or
∆θ ∼ 0.3 K
s
s
hsc dθ 1
=
dz
ε
θ
hsc dθ
θ dz
Asymptotics: convective scale
Characteristic (inverse) time scales
dimensional
advection
sound
:
:
dimensionless
uref
hsc
1
p
√
pref /ρref
ghsc
=
hsc
hsc
√
ghsc 1
=
uref
ε
s
internal waves :
N=
√
ghsc
uref
g dθ
θ dz
s
s
hsc dθ 1
=
dz
ε
θ
hsc dθ
θ dz
For single-scale models with advection & internal waves:∗
hsc dθ
= O(ε2)
θ dz
or
∆θ ∼ 0.3 K
∗
Ogura & Phillips (1962)
Asymptotics: convective scale
Ogura & Phillips' (1962) Anelastic Model:
× ∇ · (ρv) = 0
(ρv)t + ∇ · (ρv ◦ v) + ρ∇π = ρθ0gk
(ρθ0)t + ∇ · (ρθ0v) = 0
For single-scale models with advection & internal waves:∗
hsc dθ
= O(ε2)
θ dz
or
∆θ ∼ 0.3 K
∗
Ogura & Phillips (1962)
Asymptotics: convective scale
More realistic regimes with three time scales
Mach number and strati cation
uref
= ε 1,
cref
hsc dθ
= O(εµ) ,
θ dz
(0 < µ < 2)
Rescaled dependent variables
π = π ε(z) + επ̃ ,
θ = 1 + εµθ(z) + εν+µθ̃ ,
| {z }
ε
θ
(µ = 2 (1 − ν))
Asymptotics: convective scale
1
dθ
θ̃τ + ν w̃
= −ṽ · ∇θ̃
ε
dz
1 θ̃
1 ε
= −ṽ · ∇ṽ − ε1−ν θ̃∇π̃ .
ṽ τ + ν ε k + θ ∇π̃
ε θ
ε
ε
1
dπ
π̃τ
+
γΓπ ε∇ · ṽ + w̃
= −ṽ · ∇π̃ − γΓπ̃∇ · ṽ
ε
dz
For the linear system:
Conservation of weighted quadratic energy
Control of time derivatives by initial data (τ = O(1))
... consider internal wave scalings for τ = O(εν ):
ϑ=
τ
ν ,
ε
π ∗ = εν−1π̃ ,
Asymptotics: convective scale
1
dθ
θ̃τ + ν w̃
= −ṽ · ∇θ̃
ε
dz
1 θ̃
1 ε
= −ṽ · ∇ṽ − ε1−ν θ̃∇π̃ .
ṽ τ + ν ε k + θ ∇π̃
ε θ
ε
ε
1
dπ
π̃τ
+
γΓπ ε∇ · ṽ + w̃
= −ṽ · ∇π̃ − γΓπ̃∇ · ṽ
ε
dz
For the linear system:
Conservation of weighted quadratic energy
Control of time derivatives by initial data (τ = O(1))
... consider internal wave scalings for τ = O(εν ):
ϑ=
τ
,
ν
ε
π ∗ = εν−1π̃ ,
Klainerman, Majda (1982) ... Dutrifoy, Majda, Schochet (2006,2008)
Asymptotics: convective scale
Linearized compressible / pseudo-incompressible systems
dθ
dz
θ̃ϑ + w̃
ṽ ϑ +
εµ πϑ∗ +
θ̃
ε
ε
θ
= 0
k + θ ∇π ∗
= 0
ε
dπ
γΓπ ε∇ · ṽ + w̃
dz
= 0
Vertical mode expansion (separation of variables)



∗

Θ 0 0 0
θ̃
 0 U∗ 0 0 
 ũ 
  (ϑ, x, z) = 
(z) exp (i [ωϑ − λ · x])
∗
 0 0 W 0 
 w̃ 
0 0 0 Π∗
π∗
Asymptotics: convective scale
Relation between compressible and pseudo-incompressible vertical modes


d 
1
λ2
1 λ2N 2 ∗
1 dW ∗ 
∗
−
+ ε εW = 2 ε ε W
dz 1 − εµ ω2ε/λ2 2 θεP ε dz
ω θ P
θ P
c
εµ = 0: pseudo-incompressible case
regular Sturm-Liouville problem for internal wave modes
εµ > 0: compressible case
nonlinear Sturm-Liouville problem ...
ω 2/λ2
= O(1) :
ε2
c
perturbations of pseudo-incompressible modes & EVals
Asymptotics: convective scale


2 2
d 
1
1 dW ∗ 
λ2
1
λ
N
∗
∗
−
+
W
=
W
ε
ε
ε
ε
ε
ε
2
2
dz 1 − εµ ω ε/λ2 θ P dz
ω2 θ P
θ P
c
Internal wave modes
ω 2 /λ2
cε 2
= O(1)
• pseudo-inc. modes/EVals = compressible modes/EVals + O(εµ)
• phase errors remain small for
• validity for tadv = O(1)
⇒
†
ϑ = tadv /εν < O(ε−µ)
ν − µ = 1 − 23 µ > 0
The pseudo-incompressible model remains relevant for strati cations
1 dθ
< O(ε2/3)
⇒
∆θ|h0 sc <
50 K
∼
θ dz
not merely up to O(ε2) as in Ogura-Phillips (1962)
† rigorous proof pending → work with D. Bresch
Asymptotics: convective scale
Anelastic
Anelastic
∇ · (ρv) = 0
× +∇ · (ρv) = 0
θ0
(ρv)t + ∇ · (ρv ◦ v) + ρ∇π =
ρgk
θ
θ0
v t + v · ∇v + ∇π =
gk
θ
θt + v · ∇θ = 0
Pt + ∇ · (P v) = 0
ρ(z)θ = P ,
θ0 = θ(z) − θ(z)
θ = θ(z) + θ0
baroclinic torque / modi ed divergence
Pseudo-incompressible
(1/θ)t + v · ∇(1/θ) = 0
ρt + ∇ · (ρv) = 0
θ0
v t + v · ∇v + θ∇π =
gk
θ
θ0
(ρv)t + ∇ · (ρv ◦ v) + P ∇π =
ρgk
θ
∇ · (P v) = 0
× +∇ · (P v) = 0
ρ(z)θ = P ,
θ = θ(z) + θ0
relevant for deep atmospheres / large scales∗
∗
see, e.g., Smolarkiewicz & Dörnbrack (2007)
Asymptotics: convective scale
Anelastic
Boussinesq approximation
× +∇ · (ρv) = 0
θ0
(ρv)t + ∇ · (ρv ◦ v) + ρ∇π =
ρgk
θ
Pt + ∇ · (P v) = 0
ρ(z)θ = P ,
θ = θ(z) + θ0
∇·v = 0
v t + v · ∇v + ∇π = θ gk
θt + v · ∇θ = 0
θ0 = θ(z) − θ(z)
zero-Mach, variable density ow
Pseudo-incompressible
ρt + ∇ · (ρv) = 0
θ0
(ρv)t + ∇ · (ρv ◦ v) + P ∇π =
ρgk
θ
ρt + v · ∇ρ = 0
1
v t + v · ∇v + ∇π = (ρ − ρ) gk
ρ
× +∇ · (P v) = 0
∇·v = 0
θ = θ(z) + θ0
Small scale limits
ρ(z)θ = P ,
Asymptotics: convective scale
Cold air blobs at small scales
-8
-6
-4
-2
0
x [m]
2
4
6
8
10
9
8
7
6
5
4
3
2
1
0
-10
-8
-6
-4
-2
0
x [m]
2
4
6
8
-6
-4
-2
0
x [m]
2
4
6
8
10
-8
-6
-4
-2
0
x [m]
2
4
6
8
10
10
9
8
7
6
5
4
3
2
1
0
θ1/θ2 = 0.5
-10
z [m]
-8
θ1/θ2 = 0.9
-10
10
10
9
8
7
6
5
4
3
2
1
0
-10
10
9
8
7
6
5
4
3
2
1
0
10
z [m]
z [m]
-10
z [m]
Pseudo-incompressible
z [m]
z [m]
Anelastic
10
9
8
7
6
5
4
3
2
1
0
-8
-6
-4
-2
0
x [m]
2
4
6
8
10
10
9
8
7
6
5
4
3
2
1
0
θ1/θ2 = 0.1
-10
-8
-6
-4
-2
0
x [m]
2
4
6
8
10
Motivation
Asymptotics
Two-Scale Models
Numerics
Conclusions
Two-Scale Models
Dispersion relation for linear waves in a compressible atmosphere
2
10
external
Lz=80km
Lz=8km
Lz=800m
Lz=80m
vertical acoustics
0
wave fequency [s 1]
10
10
2
10
4
10
6
Lamb wave
internal waves
exact dispersion
0
10
2
10
horizontal length scale [km]
4
10
Goal:
Keep the external and internal waves, eliminate vertical acoustics
courtesy: Sebastian Reich, Potsdam University
Two-Scale Models
Durran, JFM, (08); Arakawa & Konor, MWR, (09)∗
ρ∗t + ∇ · (ρ∗v) = 0
ut + u · ∇v + wv z + θ∇k (πh + π 0) = 0
wt + u · ∇w + wwz + θ (πh + π 0)z = −g
θt + u · ∇θ + wθz = 0
Z
∗
ρ = ρ(θ, πh) ,
z
πh = πS (t, x) −
zS
g
dz ,
θ
compressible barotropic dynamics for πS
∗
(the gist only!)
Motivation
Asymptotics
Two-Scale Models
Numerics
Conclusions
Numerics
Why not simply solve the full compressible equations?
Competing approaches:
model codes
• Split-explicit / multi-rate methods, e.g.,
– Runge-Kutta (slow) + forward-backward (fast), e.g.,
Wicker & Skamarock, MWR, (98), ... ;
MM5, LM, WRF ...
– Multirate in nitesimal schemes, peer methods
Wensch et al., BIT, (09);
ASAM, ...
• Semi-implicit / linearly implicit schemes
– explicit advection, damped 2nd or 1st-order schemes for fast modes, e.g.,
Robert, Japan Met. J., (69), ... ;
UKMO, ...
– linearly implicit Rosenbrock-type methods, e.g.,
Reisner et al., MWR, (05), ...;
ASAM, LANL Hurricane model, ...
• Fully implicit integration
∗
see, e.g., Reich et al. (2007)
Numerics
Why not simply solve the full compressible equations?
1
1
0.5
0.5
0
0
p
p
Simple wave initial data, periodic domain
(integration: implicit midpoint rule, staggered grid, 512 grid pts., CFL = 10)
-0.5
-0.5
-1
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
-0.4
1
1
0.5
0.5
0
0
p
p
t=0
-0.5
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
t=3
-0.5
-1
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
Ideas:
Slave short waves (c∆t/` > 1) to long waves (c∆t/` ≤ 1)
with pseudo-incompressible limit behavior
“super-implicit” scheme
non-standard multi grid
projection method
∗
see, e.g., Reich et al. (2007)
Numerics
Why not simply solve the full compressible equations?
1
1
0.5
0.5
0
0
p
p
Simple wave initial data, periodic domain
(integration: implicit midpoint rule, staggered grid, 512 grid pts., CFL = 10)
-0.5
-0.5
-1
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
-0.4
1
1
0.5
0.5
0
0
p
p
t=0
-0.5
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
t=3
-0.5
-1
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
Ideas:
• Slave short waves (c∆t/` > 1) to long waves (c∆t/` ≤ 1)
• with pseudo-incompressible limit behavior
“super-implicit” scheme
non-standard multi grid
projection method
∗
see, e.g., Reich et al. (2007)
Numerics
Why not simply solve the full compressible equations?
1
1
0.5
0.5
0
0
p
p
Simple wave initial data, periodic domain
(integration: implicit midpoint rule, staggered grid, 512 grid pts., CFL = 10)
-0.5
-0.5
-1
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
-0.4
1
1
0.5
0.5
0
0
p
p
t=0
-0.5
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
t=3
-0.5
-1
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
Ideas:
• Slave short waves (c∆t/` > 1) to long waves (c∆t/` ≤ 1)
• with pseudo-incompressible limit behavior
“super-implicit” scheme
non-standard multi grid
projection method
∗
see, e.g., Reich et al. (2007)
Numerics
(
εÿ + εκẏ + y = cos(t) ,
y(0) = 1 + a
,
ẏ(0) = 0
2
(ε = 0.01)
1
0.8
1.5
0.6
1
0.4
0.2
y-ybal
y
0.5
0
0
-0.2
-0.5
-0.4
-1
-0.6
-1.5
-2
-0.8
0
5
10
15
20
25
30
-1
0
5
10
15
20
time
y(t)
25
30
35
40
time
y(t) − cos(t)
∗
see, e.g., Reich et al. (2007)
Numerics
εÿ + εκẏ + y = cos(t)
Slow-time asymptotics for ε 1:
y(t) = y (0)(t) + εy (1)(t) + . . . ,
y (0)(t) = cos(t)
y (1)(t) = −(ÿ (0) + κẏ (0))(t)
Associated “super-implicit” discretization (extreme BDF):
y n+1 = cos(tn+1) − ε [(δt + κ) ẏ]∗,n+1
1
1
ẏ n+1 =
y n+1 − y n + y n+1 − 2y n + y n−1
∆t
2
where
u∗,n+1 = 2un − un−1
3
1
(δtu)∗,n+1 =
un − un−1 + (un − 2un−1 + un−2)
∆t
2
∗
see, e.g., Reich et al. (2007)
Numerics
2.5
√
∆t = 7 ε
Implicit midpoint rule
2
1.5
1
y
0.5
0
-0.5
-1
2.5
-1.5
2
-2
1.5
-2.5
0
5
1
10
15
time
20
25
30
y
0.5
0
-0.5
-1
√
∆t = 7 ε
-1.5
-2
-2.5
0
5
10
15
time
20
25
Super-implicit scheme
30
∗
see, e.g., Reich et al. (2007)
Numerics
2
√
∆t = 5.55 ε
Implicit midpoint rule
1.5
1
y
0.5
0
-0.5
-1
2
-1.5
1.5
-2
1
0
10
20
30
40
50
60
time
y
0.5
0
-0.5
-1
√
∆t = 5.55 ε
-1.5
-2
0
10
20
30
40
50
Super-implicit scheme
60
time
∗
see, e.g., Reich et al. (2007)
Numerics
2
√
∆t = 5.55 ε
Blended scheme
1.5
∆y BL= η ∆y IMP +(1 − η) ∆ySU
1
y
0.5
0
-0.5
-1
2
-1.5
1.5
-2
1
0
10
20
30
40
50
60
time
y
0.5
0
-0.5
-1
√
∆t = 5.55 ε
-1.5
-2
0
10
20
30
40
50
BDF2 – for comparison
60
time
∗
see, e.g., Reich et al. (2007)
Numerics
Compressible ow equations:
ρt + ∇ · (ρv) = 0
(ρv)t + ∇ · (ρv ◦ v) + P ∇π = −ρgk
Pt + ∇ · (P v) = 0
1
γ
P = p = ρθ ,
π = p/ΓP ,
Γ = cp/R
∗
see, e.g., Reich et al. (2007)
Numerics
1
1D Linear acoustics:
p
0.5
0
t=0
-0.5
ut + p x = 0
-1
-0.4
pt + c2ux = 0
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
1
0.5
p
Desired:
• remove underresolved modes
• minimize dispersion for marginally
resolved modes
0
t=3
-0.5
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
Strategy:
scale-dependent IMP-SU-Blended scheme via multi grid
∗
see, e.g., Reich et al. (2007)
Numerics
1
1D Linear acoustics:
p
0.5
0
t=0
-0.5
ut + p x = 0
-1
-0.4
pt + c2ux = 0
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
1
0.5
p
Desired:
• remove underresolved modes
• minimize dispersion for marginally
resolved modes
0
t=3
-0.5
-1
-0.4
-0.3
-0.2
-0.1
0
x
0.1
0.2
0.3
0.4
Strategy:
scale-dependent IMP-SU-Blended scheme via multi grid
∗
see, e.g., Reich et al. (2007)
Numerics
Implicit mid-point rule for linear acoustics
1
pn+1 − pn
∂
+ c2 un+ 2 = 0
∆t
∂x
1
un+1 − un
∂
+ pn+ 2 = 0 ,
∆t
∂x
with
X
n+ 12
1
n+1
n
= X
+X
2
1
X n+1 = 2X n+ 2 − X n
or
Implicit problem for half-time uxes
n+ 21
u
∆t ∂ n+ 1
=u −
p 2,
2 ∂x
n+ 12
n
p
c2∆t ∂ n+ 1
=p −
u 2
2 ∂x
n
1
Eliminate un+ 2
2
2
2
c ∆t ∂
1−
4 ∂x2
p
n+ 12
c2∆t ∂ n
=p −
u
2 ∂x
n
∗
see, e.g., Reich et al. (2007)
Numerics
Implicit mid-point rule for linear acoustics
1
pn+1 − pn
∂
+ c2 un+ 2 = 0
∆t
∂x
1
un+1 − un
∂
+ pn+ 2 = 0 ,
∆t
∂x
with
X
n+ 12
1
n+1
n
= X
+X
2
1
X n+1 = 2X n+ 2 − X n
or
Implicit problem for half-time uxes
n+ 21
u
∆t ∂ n+ 1
=u −
p 2,
2 ∂x
n+ 12
n
p
c2∆t ∂ n+ 1
=p −
u 2
2 ∂x
n
1
Eliminate un+ 2
2
2
2
c ∆t ∂
1−
4 ∂x2
p
n+ 12
c2∆t ∂ n
=p −
u
2 ∂x
n
∗
see, e.g., Reich et al. (2007)
Numerics
Implicit mid-point rule ⇒ super-implicit
1
un+ 2 = un −
p
∆t ∂ n+ 1
p 2
2 ∂x
2
c ∆t ∂ n+ 1 ∆t
= p −
u 2−
2 ∂x
2
n+ 12
n
∂p
∂t
BD,n+ 21
step 1:
1
un+ 2 = un −
1
pn+ 2 = pn −
∆t ∂ n+ 1
p 2
2 ∂x
2
c ∆t ∂ n+ 1 ∆t
u 2−
2 ∂x
2
∂p
∂t
BD,n+ 12
Pressure “projection” equation
2
2
∂ n
c ∆t ∂ n+ 1
2 = c2
p
u +
2
2 ∂x
∂x
∂p
∂t
BD,n+ 21
∗
see, e.g., Reich et al. (2007)
Numerics
Implicit mid-point rule ⇒ super-implicit
1
un+ 2 = un −
p
n+ 12
∆t ∂ n+ 1
p 2
2 ∂x
2
c ∆t ∂ n+ 1 ∆t
= p −
u 2−
2 ∂x
2
n
∂p
∂t
BD,n+ 21
step 1:
1
un+ 2 = un −
1
pn+ 2 = pn −
∆t ∂ n+ 1
p 2
2 ∂x
2
c ∆t ∂ n+ 1 ∆t
u 2−
2 ∂x
2
step 2:
p
n+1
= 2p
n+ 12
−p
n
⇒
p
n+1
= 2p
n+ 21
∂p
∂t
BD,n+ 12
1
1 n+ 1
n−
p 2 +p 2
−
2
∗
see, e.g., Reich et al. (2007)
Numerics
Implicit mid-point rule ⇒ super-implicit
1
un+ 2 = un −
p
n+ 12
∆t ∂ n+ 1
p 2
2 ∂x
2
c ∆t ∂ n+ 1 ∆t
= p −
u 2−
2 ∂x
2
n
∂p
∂t
BD,n+ 21
step 1:
1
un+ 2 = un −
1
pn+ 2 = pn −
∆t ∂ n+ 1
p 2
2 ∂x
2
c ∆t ∂ n+ 1 ∆t
u 2−
2 ∂x
2
step 2:
p
n+1
= 2p
n+ 12
−p
n
⇒
p
n+1
= 2p
n+ 12
∂p
∂t
BD,n+ 12
1
1 n+ 1
n−
p 2 +p 2
−
2
∗
see, e.g., Reich et al. (2007)
Numerics
Scale-dependence via multi-grid
p=
J
X
p(j)
j=1
where
p
(j)
= (1 − P ◦ R) R
j−1
p
with
R : MG restriction
P : MG prolongation
scale-dependent blending
1
X
j
1
(j) (j) n+ 2
η p
=
−
un
un+ 2 =
X
j
n
η (j)p(j) −
∆t ∂ n+ 1
p 2
2 ∂x
2
c ∆t ∂ n+ 1
u 2−
2 ∂x
X
(1 − η (j))
j
∗
∆t
2
(j)
∂p
∂t
BD,n+ 21
see, e.g., Reich et al. (2007)
Numerics
1
implicit midpoint
p
0.5
0
-0.5
-1
-0.4
-0.3
-0.2
-0.1
-0.4
-0.3
-0.2
-0.1
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
x
1
new scheme
p
0.5
0
-0.5
-1
x
1
BDF2
p
0.5
0
-0.5
-1
x
t=0
Numerics
1
implicit midpoint
p
0.5
0
-0.5
-1
-0.4
-0.3
-0.2
-0.1
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
x
1
new scheme
p
0.5
0
-0.5
-1
x
1
BDF2
p
0.5
0
-0.5
-1
-0.4
-0.3
-0.2
-0.1
0
x
t=3
Motivation
Asymptotics
Two-Scale Models
Numerics
Conclusions