Conservative, skew symmetric compact finite
difference schemes for the compessible Navier
Stokes equations
Jan Schulze, Julius Reiss & Jörn Sesterhenn
Fachgebiet für Numerische
Fluidmechanik TU Berlin
INδAM, Catania
June, 2010
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
1 / 92
What’s it about?
Supersonic Jet Noise
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
2 / 92
experiment (Seiner 1984)
Screech - spectrum
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
3 / 92
Jet – noise
vortex-pairing
1. noise source
Schlieren visualization of an overexpanded jet.
Van Dyke
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
4 / 92
Jet – noise
vortex-pairing
1. noise source
Schlieren visualization of an overexpanded jet.
Yyshock – diamondsYy
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
4 / 92
Jet – noise
vortex-pairing
1. noise source
Schlieren visualization of an overexpanded jet.
vortex-pairing ⇒ noise
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
4 / 92
Jet – noise
shock-induced
2. noise source
Schlieren visualization of an overexpanded jet.
shock – mixing layer interaction ⇒ noise
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
4 / 92
Jet – noise
screech
3. noise source
Schlieren visualization of an overexpanded jet.
screech ⇒ noise up to 160 dB!
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
4 / 92
Jet – noise
screech
3. noise source
Schlieren visualization of an overexpanded jet.
screech ⇒ noise up to 160 dB!
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
4 / 92
Project – objective
Shape optimization – Screech minimization
Shape optimization of a jet-nozzle to reduce jet noise.
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
5 / 92
Project – objective
Objective function – Minimization of screech
Objective function in frequency space.
J=
λ1
Z
|
+
λ2
Z
ko
ku
Z
∞
e−ikt p′ (t)dt
−∞
dk
{z
}
J1
ko
ku
∂
∂k
Z
|
Sesterhenn (TU Berlin)
2
Skew symmetric compact FD schemes
∞
e−ikt p′ (t)dt
−∞
{z
J2
2
dk
}
June, 2010
6 / 92
Project – objective
Adjoint equations (Shape optimization)
Adjoint shape optimization.
noise
measure
noise
geometry to control
adjoint NS − backwards
NS − forwards
nozzle
nozzle
vortices in the mixing layer
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
7 / 92
Computing Supersonic Jet Noise
DNS/LES of the Flow Field
High order schemes, preserving the dispersion relation
Truely nonreflecting boundary condition
Proper shock treatment / Shock turbulence interaction
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
8 / 92
Computing Supersonic Jet Noise
DNS/LES of the Flow Field
High order schemes, preserving the dispersion relation
Truely nonreflecting boundary condition
Proper shock treatment / Shock turbulence interaction
Half-Way Solution:
run
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
8 / 92
Computing Supersonic Jet Noise
DNS/LES of the Flow Field
High order schemes, preserving the dispersion relation
Truely nonreflecting boundary condition
Proper shock treatment / Shock turbulence interaction
Understanding/Optimization/Control of Jet Noise:
Modal analysis
Adjoint equations (with proper shock treatment)
Skew-symmetric formulation
Optimization algorithm
Feedback Control
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
8 / 92
Computing Supersonic Jet Noise
DNS/LES of the Flow Field
High order schemes, preserving the dispersion relation
Truely nonreflecting boundary condition
Proper shock treatment / Shock turbulence interaction
Understanding/Optimization/Control of Jet Noise:
Modal analysis
Adjoint equations (with proper shock treatment)
Skew-symmetric formulation
Optimization algorithm
Feedback Control
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
8 / 92
Overview
1
Objectives
2
Some DNS Results
Supersonic Jet Noise
Adjoint Optimization
3
Fully Conservative, Skew Symmetric FD Schemes
Need for skew-symmetric, conservative formulations
What does Skew Symmetry mean?
Motivating Example: Burgers’ Equation
Euler/Navier-Stokes Equations
4
Conclusion
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
9 / 92
Overview
1
Objectives
2
Some DNS Results
Supersonic Jet Noise
Adjoint Optimization
3
Fully Conservative, Skew Symmetric FD Schemes
Need for skew-symmetric, conservative formulations
What does Skew Symmetry mean?
Motivating Example: Burgers’ Equation
Euler/Navier-Stokes Equations
4
Conclusion
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
10 / 92
Direct Numerical Simulation of a supersonic jet
Nozzle: velocity profile
planar jet
6th order compact in x and y ; z (periodic)
4th order Runge Kutta or exponential-Krylov in time.
Nozzle: velocity profile
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
11 / 92
Direct Numerical Simulation of a supersonic jet
Nozzle: velocity profile
2040 × 1020 × 144 ≈ 300 · 106
points @ 1020 CPU’s
MJet = 1.35
pJet /p∞ = 0.9 (over-expanded)
100 000 CPUh
ReD = 30 000
SGI-Altix 4700 @ LRZ
example
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
12 / 92
Direct Numerical Simulation of a supersonic jet
Nozzle: velocity profile
Time history
Spectrum
120
p′ /p′max
1
100
SPL [dB]
80
0
-1
21
60
22
23
24
t/Ts
40
20
0
1e-2
1e0
1e1
f /f0
•: semi-empirical law;
frequency only (Powell 1953)
run
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
13 / 92
Overview
1
Objectives
2
Some DNS Results
Supersonic Jet Noise
Adjoint Optimization
3
Fully Conservative, Skew Symmetric FD Schemes
Need for skew-symmetric, conservative formulations
What does Skew Symmetry mean?
Motivating Example: Burgers’ Equation
Euler/Navier-Stokes Equations
4
Conclusion
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
14 / 92
Why use Adjoint methods?
Solution of J = hg, ui subject to
Lu = f
for p RHS f will cost p solutions + 1 scalar product
Alternatively solve
L∗ v = g
with hu, L∗ v i := hLu, v i
Then
hg, ui = hL∗ v , ui = hv , Lui = hv , f i
for p RHS f will cost 1 solution + p scalar products
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
15 / 92
Objective Function
J = λ1
Z
ko
ku
Z
∞
e
−∞
Sesterhenn (TU Berlin)
−ikt
u(t)dt
2
dk + λ2
Z
ko
ku
∂
∂k
Skew symmetric compact FD schemes
Z
∞
e
−∞
−ikt
u(t)dt
2
June, 2010
dk
16 / 92
Adjoint Methods for Optimization
Problem Formulation
Cost function
J=
Z
q T Qq dt
q: state-vector
need to satisfy constraints like:
governing equations (e.g. Navier–Stokes) and boundary
conditions.
Constraint: Full nonlinear state equations
N (q, Φ) = 0
N : nonlinear operator; e.g. Navier–Stokes
Φ: control
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
17 / 92
Adjoint Methods for Optimization
Optimality Condition
Lagrange function (enforce variables)
L = J − N (q, Φ), q +
q + : adjoint
Z Z variable (Lagrange multiplier)
hp, qi =
pq dΩdt
Optimality condition
∂L
=0
∂Φ
∂
∂J
+
=
N (q, Φ), q
∂Φ
∂Φ
gradient depending on adjoint state q + !
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
18 / 92
Adjoint Methods for Optimization
Adjoint Equation
Adjoint Equation
∂L
=0
∂q
⇒ N + (q, q + ) = 0
Steepest descent
Φn+1 = Φn − αn
Sesterhenn (TU Berlin)
∂J
∂Φ
Skew symmetric compact FD schemes
June, 2010
19 / 92
Adjoint Methods for Optimization
Some Issues
Adjoint equations need to be solved backwards in time.
Adjoint equations depend on forward solution q (N + (q, q + )).
Implementation of adjoint equation can be three times as
expensive as direct equation.
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
20 / 92
Adjoint Methods for Optimization
Example: Burgers Equation
Burgers–equation
∂u
∂u
1 ∂2u
=0
+u
−
∂t
∂x
Re ∂x 2
Adjoint Burgers–equation
−
Sesterhenn (TU Berlin)
∂u +
∂u +
1 ∂2u+
=0
−u
−
∂t
∂x
Re ∂x 2
Skew symmetric compact FD schemes
June, 2010
21 / 92
Optimization method
Conjugate Gradient
increase convergence for optimization areas with narrow and flat
valleys.
e.g.: Polak–Ribiere Algorithm
example
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
22 / 92
Adjoint Methods for Optimization
Optimization Loop
Adjoint shape optimization.
noise
measure
noise
geometry to control
adjoint NS − backwards
NS − forwards
nozzle
nozzle
vortices in the mixing layer
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
23 / 92
Adjoint Shape Optimization
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
24 / 92
Overview
1
Objectives
2
Some DNS Results
Supersonic Jet Noise
Adjoint Optimization
3
Fully Conservative, Skew Symmetric FD Schemes
Need for skew-symmetric, conservative formulations
What does Skew Symmetry mean?
Motivating Example: Burgers’ Equation
Euler/Navier-Stokes Equations
4
Conclusion
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
25 / 92
Aeroacosutic Radiation by Shock Turbulence
Interaction
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
26 / 92
Aeroacosutic Radiation by Shock Turbulence
Interaction
R ATIO
OF
KOLMOGOROV
SCALE TO SHOCK THICKNESS
δu ≈
for weak shocks:
ν 6.89
c M −1
approximation for isotropic turbulence:
ε ≈ 15
2
νurms
λ2
(ν 3 /ε)1/4
1 M1 − 1 p
η
≈
≈
Reλ
δu
(ν/c)(6.89/(M1 − 1))
8 Mt
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
27 / 92
Skew Symmetric discretisation
Aim: Calculation and optimisation of flows with shocks and sound
Needed:
High precision/High order schemes
Numerical efficient code
Precise adjoints
→ Finite difference schemes
But: Correct shock treatment relies on conservation laws.
Therefore a conserving high order FD scheme is needed.
→ Skew symmetric schemes
Talking not about Gibbs phenomenon
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
28 / 92
Skew Symmetric discretisation
The core ideas
Skew symmetry . . .
is defined with respect to a scalar product:
(u, Av ) = (A† u, v ) ≡ −(Au, v )
is a property of many operators, e.g.
Z
Z
u(x)∂x v (x)dx = − (∂x u(x)) v (x)dx + [B.T .]
is defined for linear and non-linear operators, e.g.
A = [v (x)∂x + ∂x v (x)]
is the key to conservation properties
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
29 / 92
Skew Symmetric Operators
"∂x is skew-symmetric"
(∂x u, v ) = − (u, ∂x v ) + B.T .
"∂xx is symmetric"
(∂xx u, v ) = − (∂x u, ∂x v ) = (u, ∂xx v )
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
30 / 92
Implications for ut + λux = 0
1
∂t (u, u) =
2
1
{(∂t u, u) + (u, ∂t u)}
2
1
=
{(−λ∂x u, u) + (u, −λ∂x u)}
2
λ
= − {(∂x u, u) + (u, ∂x u)} = 0
2
−→ Conservation of energy !
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
31 / 92
Discretized equation
u̇ + λDu = 0
(u, v ) := u T v
1d
(u, u) =
2 dt
1
{(u̇, u) + (u, u̇)}
2
1
{(−λDu, u) + (u, −λDu)}
=
2
o
λn
= −
(Du)T u + u T Du
2
o
λn T T
= −
u (D + D)u
2
−→ Conservation of energy if D T = −D!
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
32 / 92
Skew Symmetric discretisation
Skew symmetry in the discrete case . . .
corresponds to a skew symmetric matrix
is often violated, e.g.:




1 −1
0 −1
1  1 0 −1
1  0 1 −1
 c

Du =
,D =



∆x
∆x
..
.. ..
.. ..
.
.
.
.
.
1
is often violated in for non-linear terms, as product rule is violated
implies to the conservation of a discrete energy
can be constructed by a Galerkin ansatz for the linear and
non-linear case, were analytic properties imply discrete structures
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
33 / 92
Skew Symmetric discretisation
Nonlinear Terms
How about ∂t + ∂x u 2 /2 = 0? Is (∂x u) skew-symmetric?
Z
Z
(v , (∂x u)w) =:
v (∂x u)w dx = − ∂x v (uw) dx
=
(u∂x , w)
−→ No !
(∂x u)† = −(u∂x )
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
34 / 92
Skew Symmetric discretisation
Example: The Burgers’ Equation
u2
2
= ∂t u + u∂x u
0 = ∂t u + ∂x
o
→
∂t u +
The (kinetic) energy is E = 21 (u, u) =
∂t E
∂t E
∂t E
1
2
R
1
[u∂x + ∂x u] u = 0
3
u 2 dx
= (u, ∂t u)
1
= − (u, [u∂x + ∂x u] u)
3
1
2
= + ([u∂x + ∂x u] u, u) − [u 3 ]xx1O
3
3
Thus, we get ∂t E = (u, ∂t u) = − 31 [u 3 ]xx1O
Skew symmetry implies conservation
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
35 / 92
Discrete Version 21 (DU)u
define U := diag(u)
(DU)T = U T D T = −UD T
NOT −DU!!!
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
36 / 92
Energy Conservation for u̇ + 31 (UD + DU)u = 0
d T
u u = u̇ T u + u T u̇
dt
o
1n
= −
((UD + DU)u)T u + u T (UD + DU)u)
3
o
1n T
u (−DU − UD + UD + DU) u
= −
3
= 0
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
37 / 92
Momentum Conservation for u̇ + 31 (UD + DU)u = 0
o
d T
1 n T
1 u = −
(1 (UD + DU)u
dt
3




1
T
T
Uu
= −
u
Du
+
1
D
|{z}
| {z }

3


=0 (telescope-sum)
=0 (skew-symmetry)
= 0
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
38 / 92
telescope-sum
Example: Finite Volume Discretzation



0 −1
f1
1  1 0 −1
  f2 



∆x
..
.. ..
.
.
.
1
In general
1T D = 0
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Skew symmetric compact FD schemes
June, 2010
39 / 92
telescope-sum
Reminder: Consistency
Taylor Series
u(x ± h) = u + u ′ h + u ′′ h2 /2 + u ′′′ h3 /6 + . . .
leads via αui−1 + βui + γui+1 to
(α + β + γ) ui + h(−α + γ)ui′ + . . .
|
{z
}
D·1=0
Constency + Skew symmetry implies telescope-property
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
40 / 92
Time Integration
Integrate
1
∂t u + (DU + UD)u = 0
3
Use trapezoidal rule
1
1
1
u n+1 − u n 1
+ (DU n+ 2 + U n+ 2 D)u n+ 2 = 0
∆t
3
With
1
u n+ 2 :=
Sesterhenn (TU Berlin)
u n+1 + u n
2
Skew symmetric compact FD schemes
June, 2010
41 / 92
Time Integration
Energy Conservation
1
u n+ 2
T u n+1 − u n 1 1
1
1
1 T
(DU n+ 2 + U n+ 2 D)u n+ 2 = 0
+
(u n+ 2
∆t
3
With
2
u n+1 − (u n )2
u n+1 + u n u n+1 − u n
=
=0
2
∆t
2∆t
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
42 / 92
Skew Symmetric discretisation
Galerkin ansatz
f (x) ≈ f̃ (x) =
N
X
f̂i τi (x)
i=1
where the amplitudes f̂i are selected by a consistency condition, in our
case the collocation condition
f̃ (xi ) ≡ f (xi )
This defines a matrix transformation for the coefficients and the
function values
f = T f̂j
with transformation matrix
Tij = τj (xi )
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
43 / 92
Skew Symmetric discretisation
Galerkin ansatz
Operators a discretised by mapping on the ansatz-functions
(testfunctions=ansatzfunctions), e.g., f (x) = ∂x F (x)
Z x1
Z x1
vfdx =
v ∂x Fdx
x0
X
v̂i f̂j
ij
x0
Z
ti (x)tj (x)dx =
|
{z
}
X
ij
Ŝij
v̂i F̂j
Z
ti (x)∂x tj (x)
|
{z
}
Âij
S f̂ = ÂF̂
or D = S −1 A. S defines the scalar product.
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
44 / 92
Skew Symmetric discretisation
Galerkin ansatz
Operators a discretised by mapping on the ansatz-functions
(testfunctions=ansatzfunctions), e.g., f (x) = ∂x F (x)
Z x1
Z x1
vfdx =
v ∂x Fdx
x0
X
v̂i f̂j Ŝij
ij
x0
=
X
v̂i F̂j Âij
ij
S f̂ = ÂF̂
or D = S −1 A. S defines the scalar product.
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
44 / 92
Skew Symmetric discretisation
Galerkin ansatz
The derivative S f̂ = ÂF̂
A is antisymmetric
Z
Z
Aij = ti (x)∂x tj (x) = − tj (x)∂x ti (x) + [B.T .]
x
x
D = S −1 A is antisymmetric with respect to S
(u, Dv )S = u t SDv = u t SS −1 Av = u t Av = −v t SS −1 Au = −(v , Du)S
Complete freedom in choosing the ansatzfunctions τi
S f̂ = ÂF̂ is compact derivative for τ (x) with compact support
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
45 / 92
Skew Symmetric discretisation
Galerkin ansatz: B-Splines
b-splines have compact support:
1.0
0
B-Spline
Ableitung
6th Order
10
-1
10
0.5
-2
10
0.0
-3
10
-4
10
-0.5
-5
10
-1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-6
10
1
10
Figure: B-Spline and its first derivative
A and S have band structure: Derivative S f̂ = ÂF̂ is found to be
(fi+1 + fi−1 ) +
1
120 (fi+2
+ fi−2 ) =
5
12 (Fi+1
− Fi−1 ) +
1
24 (Fi+2
− Fi−2 )
→ compact derivative, 6th order
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
46 / 92
Overview
1
Objectives
2
Some DNS Results
Supersonic Jet Noise
Adjoint Optimization
3
Fully Conservative, Skew Symmetric FD Schemes
Need for skew-symmetric, conservative formulations
What does Skew Symmetry mean?
Motivating Example: Burgers’ Equation
Euler/Navier-Stokes Equations
4
Conclusion
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
47 / 92
Skew Symmetric discretisation
Burgers’ Equation
Same procedure for Burgers equation (in skew symmetric notation)
1
∂t u + [∂x u + u∂x ]u − ν(∂x )2 u = 0
3
leading to
1
∂t ûj Sij + ûj ûk Aijk = ν ûj A2ij
3
with
Ŝij
Âijk
Â2ij
Sesterhenn (TU Berlin)
=
=
=
Z
Z
Z
τi (x)τj (x)dx,
τi (x) ∂x τj (x) + τj (x)∂x τk (x)dx
τi (x)(∂x )2 τj (x)dx
Skew symmetric compact FD schemes
June, 2010
48 / 92
Skew Symmetric discretisation
Burgers’ Equation
=
Âijk
Z
τi (x) ∂x τj (x) + τj (x)∂x τk (x)dx
antisymmetric by definition for given j:

..

.






Âj = 





Sesterhenn (TU Berlin)
0
7
− 240
1
− 80
0
0
7
240
0
31
− 80
7
− 120
0
1
80
31
80
0
− 31
80
1
− 80
0
7
120
31
80
0
7
− 240
0
0
1
80
7
240
0
..
Skew symmetric compact FD schemes
.












June, 2010
49 / 92
Skew Symmetric discretisation
Burgers’ Equation
Construct u-dependent matrix:
Âû =
X
ûi Ai
i


 u1 ·


u2 ·
û
 = 

u3 ·



Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes

..
.









June, 2010
50 / 92
Skew Symmetric discretisation
Burgers’ Equation
Construct u-dependent matrix:
Âû =
X
ûi Ai
i


Au = 

..
..
.
.
−
−bi −ai−
..
.
0 +ai+ +bi+
..
..
..
.
.
.


 = −(Au )T

with
ai± =
bi± =
Sesterhenn (TU Berlin)
7 i±2 31 i±1 31 i
7 i∓1
u
+
u
+
u +
u
240
80
80
240
1 i
7 i±1
1 i±2
u +
u
+
u
80
120
80
Skew symmetric compact FD schemes
June, 2010
50 / 92
Skew Symmetric discretisation
Burgers’ Equation
Ŝ∂t û + 13 Aû û = νA2 û
Derivative is 6th order for const deriv, 4th for mixed terms
Still needed: Time Integration scheme → later
Energy conserving!
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
51 / 92
Skew Symmetric discretisation
Burgers’ Equation
Ŝ∂t û + 13 Aû û = νA2 û
Derivative is 6th order for const deriv, 4th for mixed terms
Still needed: Time Integration scheme → later
Energy conserving!
. . . but what about momentum conservation?
Remember starting point:
1
∂t u + [∂x u + u∂x ]u − ν(∂x )2 u = 0
3
Not in divergence form!
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
51 / 92
Skew Symmetric discretisation
Burgers’ Equation: Momentum Conservation
Assume
1̃(x) = 1̂i τ (x) ≡ 1∀x
then
X
i
1̂i Âijk =
Z
Z
1 ∂x τj + τj ∂x τk dx = τj ∂x τk + [B.T .] = Âjk + [B.T .]
and therefore 1t Au u = u t Au + [uu]x = 2/3[uu]x
1
0 = (1̂, Ŝ∂t û) + (1̂, Âû û) − ν(1̂, A2 û)
3
1
= ∂t (1̂, Ŝ û) + ûj ûk [τj (x)τk (x)] − ν ûj [∂x τj (x)]
2
This is a generalised telescoping sum property.
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
52 / 92
Skew Symmetric discretisation
Numerical results: Burgers’ Equation
3.5
3.0
2.5
2.0
1.5
1.0
0.50
30
25
20
15
10
5
00.0
1
1.0
3
1.5
4
2.0
2.5
Time integration by midpoint
rule1
1
0.5
2
5
6
E_kin
momentum
3.0
3.5
4.0
⊗
Verstappen, Veldman, J. Com. Phys 187, p. 343 (2003)
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
53 / 92
Skew Symmetric discretisation
Numerical results: Burgers’ Equation
Time= 0.000000 (nu=0.000000)
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.00
1
2
3
4
5
Time= 0.640000 (nu=0.000000)
3.0
6
7
1.00
1
2
Time= 1.990000 (nu=0.000000)
3.0
4
3
5
6
7
Ekin
30
25
2.5
20
2.0
15
10
1.5
5
1.00
1
2
3
4
5
6
7
0
0.0
0.5
1.0
1.5
2.0
Time integration by midpoint rule2
2
Verstappen, Veldman, J. Com. Phys 187, p. 343 (2003)
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
54 / 92
Skew Symmetric discretisation
Side Remark: Finte Volume
How does conservative FD compare with FV?
Example: Burgers’ Equation:
conservation
momentum
energy
conservative FD
by telescoping sum
by skew symmetry
Finite Volume
enforced by use of fluxes
usually not fulfilled
Both schemes have correct shock treatment, but conservative FD has
inherent stability!
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
55 / 92
Overview
1
Objectives
2
Some DNS Results
Supersonic Jet Noise
Adjoint Optimization
3
Fully Conservative, Skew Symmetric FD Schemes
Need for skew-symmetric, conservative formulations
What does Skew Symmetry mean?
Motivating Example: Burgers’ Equation
Euler/Navier-Stokes Equations
4
Conclusion
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
56 / 92
Skew Symmetric discretisation
Euler Equations
∂t ̺ + ∂x (̺u) = 0
∂t (̺u) + ∂x (̺u 2 ) + ∂x p = 0
= 0
∂t (̺e + ̺u 2 /2) + ∂x ̺u(e + p/̺ + u 2 /2)
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
57 / 92
Skew Symmetric discretisation
Euler Equations
∂t ̺ + ∂x (̺u) = 0
∂t (̺u) + ∂x (̺u 2 ) + ∂x p = 0
= 0
∂t (̺e + ̺u 2 /2) + ∂x ̺u(e + p/̺ + u 2 /2)
(Anti)-Symmetries:
∂t ̺ + ∂x (̺u) = 0
1
1
(∂t ̺ + ̺∂t ) u + (∂x u̺ + u̺∂x ) u + ∂x p = 0
2
2
1
γ
∂t p +
∂x (up) − u∂x p = 0
γ−1
γ−1
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
57 / 92
Skew Symmetric discretisation
Euler Equations
∂t ̺ + ∂x (̺u) = 0
1
1
(∂t ̺ + ̺∂t ) u + (∂x u̺ + u̺∂x ) u + ∂x p = 0
2
2
1
γ
∂t p +
∂x (up) − u∂x p = 0
γ−1
γ−1
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
58 / 92
Skew Symmetric discretisation
Euler Equations
∂t ̺ + ∂x (̺u) = 0
1
1
(∂t ̺ + ̺∂t ) u + (∂x u̺ + u̺∂x ) u + ∂x p = 0
2
2
1
γ
∂t p +
∂x (up) − u∂x p = 0
γ−1
γ−1
with the Galerkin ansatz
Ŝ∂t ̺ˆ + B̂ û ̺ˆ = 0
[∂t Ŝ ̺ˆ + Ŝ ̺ˆ∂t ]û + Â̺ˆû û + 2Âp̂ = 0
γ
1
Ŝ∂t p̂ +
B û p̂ − Ĉ û p̂ = 0
γ−1
γ−1
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
58 / 92
Skew Symmetric discretisation
Navier-Stokes Equations
Navier-Stokes Equation is similar
∂t ̺ + ∂x (̺u) = 0
1
1
(∂t ̺ + ̺∂t ) u + (∂x u̺ + u̺∂x ) u + ∂x p = µ∂x2 u
2
2
1
γ
∂t p +
∂x (up) − u∂x p = µ∂x u∂x u + λ∂x T
γ−1
γ−1
with the Galerkin ansatz
Ŝ∂t ̺ˆ + B̂ û ̺ˆ = 0
[∂t Ŝ ̺ˆ + Ŝ ̺ˆ∂t ]û + Â̺ˆû û + 2Âp̂ = µÂ2 û
γ
1
Ŝ∂t p̂ +
B û p̂ − Ĉ û p̂ = µC̃ û û + λÂT (ˆ
̺, p̂)
γ−1
γ−1
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
59 / 92
Skew Symmetric discretisation
Euler Equations
Ŝ∂t ̺ˆ + B̂ û ̺ˆ = 0
[∂t Ŝ ̺ˆ + Ŝ ̺ˆ∂t ]û + Â̺ˆû û + 2Âp̂ = 0
1
γ
B û p̂ − Ĉ û p̂ = 0
Ŝ∂t p̂ +
γ−1
γ−1
What is new
set of equations: conservation of energy depends on momentum
depends on mass equation
non-constant time derivative
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
60 / 92
Skew Symmetric discretisation
Euler Equations: Time integration
P
Galerkin ansatz in time ui (t) = n uin ω n (t)
n−1/2
n−1/2
+ ui
)
with linear splines in time, ui (tn ) = 21 (ui
Ŝ∂t ̺ˆ + B̂ û ̺ˆ = 0
becomes
q
Apq Ŝ ̺ˆq + S pqr B̂ û ̺ˆr = 0
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
61 / 92
Skew Symmetric discretisation
Euler Equations: Time integration
P
Galerkin ansatz in time ui (t) = n uin ω n (t)
n−1/2
n−1/2
+ ui
)
with linear splines in time, ui (tn ) = 21 (ui
Ŝ∂t ̺ˆ + B̂ û ̺ˆ = 0
becomes
1
Ŝ ̺ˆn−1 − ̺ˆn+1 =
2∆t
1 û n−1
1 û n−1
n+1
n
n
+ B̂ û ̺ˆn−1 +
+ 6B̂ û + B̂ û
B̂
B̂
̺ˆn
12
12
1 û n
n+1
B̂ + B̂ û
+
̺ˆn+1
12
Procedure is 2 Step, symmetric, conserving and implicit.
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
61 / 92
Skew Symmetric discretisation
Euler Equations: Conserved quantities
Mass
1
1Ŝ (ˆ
̺N+1 + ̺ˆN ) − (ˆ
̺0 + ̺ˆ1 ) = 0
2
Momentum
̺ˆN−2 S + ̺ˆN−1 S û N−1 + ̺ˆN−1 S + ̺ˆN S û N
− LB + dt · R(N, N − 1)
4
and Energy
1
1̂S
γ−1
(p̂N
+
2
p̂N )
+
Sesterhenn (TU Berlin)
N−1
N−1
û N
û N Ŝ ̺ˆ +̂S ̺ˆ
4
− LB + dt · R(N, N − 1)
Skew symmetric compact FD schemes
June, 2010
62 / 92
Skew Symmetric discretisation
Euler Equations: Numerical results
rho
1.10
1.05
1.00
0.95
0.90
0.85
1.0
0.5
0.0
0.5
1.0
u
0.10
0.05
0.00
0.05
0.10
1.0
0.5
0.0
0.5
1.0
0.5
0.0
0.5
1.0
p
1.10
1.05
1.00
0.95
0.90
0.85
1.0
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
⊗
June, 2010
63 / 92
Skew Symmetric discretisation
Euler Equations: Numerical results
0.00035
+5.1278
0.00030
0.00025
pot. Energy
kin. Energy (shifted)
0.00020
total Energy
0.00015
0.00010
0.00005
0.0
0.2
0.4
0.6
0.8
1.0
Energy conserved to 10−15
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
64 / 92
Skew Symmetric discretisation
Euler Equations: Time integration
1
Ŝ ̺ˆn−1 − ̺ˆn+1 =
2∆t
1 û n−1
1 û n−1
n
n
n+1
B̂
B̂
+ B̂ û ̺ˆn−1 +
+ 6B̂ û + B̂ û
̺ˆn
12
12
1 û n
n+1
+
B̂ + B̂ û
̺ˆn+1
12
The energy equation is similar (D u = γB û + (1 − γ)Ĉ û )
1
Ŝ p̂n−1 − p̂n+1 =
2∆t
1 û n−1
1 û n−1
n
n
n+1
D̂
D̂
p̂n
+ D̂ û p̂n−1 +
+ 6D̂ û + D̂ û
12
12
1 û n
n+1
D̂ + D̂ û
p̂n+1
+
12
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
65 / 92
Skew Symmetric discretisation
Euler Equations: Time integration
Momentum equation similar, just some more terms
1 ̺ˆn−1
n
n
n+1
(Ŝ
+ Ŝ ̺ˆ )û n−1 − (Ŝ ̺ˆ + Ŝ ̺ˆ )û n+1
4∆t
1 ̺n−1 u n
n n−1
n−1 n−1
n n
2Â
+ 2Â̺ u + 3Â̺ u + 3Â̺ u û n−1
=
120
1 ̺n+1 u n+1
n−1 n−1
n+1 n
n−1 n
+
2Â
+ 2Â̺ u + 3Â̺ u + 3Â̺ u
120
+3Â̺
n u n+1
+ 3Â̺
n u n−1
+ 24Â̺
n un
1 ̺n+1 u n
n n+1
n+1 n+1
n n
2Â
+ 2Â̺ u + 3Â̺ u + 3Â̺ u û n+1
120
1 n−1 2 n 1 n+1
+Â
p̂
+ p̂ + p̂
6
3
6
û n
+
(still everything sparse!)
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
66 / 92
with all matrices with the structure

.. .. ..
.
.
.
1
 b − a − di a + b +
i
i
i
i
f 
.. .. ..
.
.
.




(-1)
were for B̂ û we have f = 240 and
di
ai±
bi±
= +31(ui+1 − ui−1 ) + ui+2 − ui−2 ,
= ±(31ui + 62ui±1 + 7ui±2 ),
= ±(ui + 7ui±1 + 2ui±2 ),
(0)
were for Ŝ ̺ˆ we have f = 840 and
di
ai±
bi±
= 288̺i + 86(̺i+1 + ̺i−1 ) + ̺i+2 + ̺i−2 ,
= 5̺i∓1 + 86̺i + 86̺i±1 + 5̺i±2 ,
= ̺i + 5̺i±1 + ̺i±2 ,
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
(1)
June, 2010
67 / 92
with all matrices with the structure

.. .. ..
.
.
.
1
 b − a − di a + b +
i
i
i
i
f 
.. .. ..
.
.
.




(2)
were for Â̺ˆû we have f = 3360 and
di
ai±
= 0
(3)
= ± 11(̺i±1 ui∓1 + ̺i±2 ui + ̺i∓1 ui±1 + ̺i ui±2 )
+18(̺i±2 ui±2 + ̺i∓1 ui∓1 )
+69(̺i±1 ui±2 + ̺i±2 ui±1 + ̺i∓1 ui + ̺i ui∓1 )
bi±
+474(̺i±1 ui + ̺i ui±1 ) + 748(̺i±1 ui±1 + ̺i ui )
= ± 3(̺i±2 ui + ̺i ui±2 ) + 10(̺i±2 ui±2 + ̺i ui )
+29(̺i±1 ui±2 + ̺i±1 ui + ̺i±2 ui±1 + ̺i ui±1 )
+138̺i±1 ui±1 ,
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
(4)
(5)
68 / 92

1

f 
..
.
..
.
bi− ai−
..
.
di ai+ bi+
.. .. ..
.
.
.




(6)
and were for Ĉ û we have f = 240 and
di
ai±
bi±
Sesterhenn (TU Berlin)
= ui−2 − ui+2 − 31ui+1 + 31ui−1
= ±(7ui∓1 + 62ui + 31ui±1 )
= ±(2ui + 7ui±1 + ui±2 )
Skew symmetric compact FD schemes
(7)
June, 2010
69 / 92
Overview
1
Objectives
2
Some DNS Results
Supersonic Jet Noise
Adjoint Optimization
3
Fully Conservative, Skew Symmetric FD Schemes
Need for skew-symmetric, conservative formulations
What does Skew Symmetry mean?
Motivating Example: Burgers’ Equation
Euler/Navier-Stokes Equations
4
Conclusion
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
70 / 92
Conclusion & Outlook
Our Goal in Computation of Supersonic Jet Noise includes
◮
Simulation of supersonic Jet-Noise
◮
Adjoint Simulation of the Navier-Stokes Equations
◮
skew-symmetic formulation of the Navier-Stokes Equations (with
shock treament)
enables
◮
Optimisation of Supersonic Flow
needed
◮
skew symmetric filter for the computation of non-resolved shocks
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
71 / 92
End
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
72 / 92
Wave decomposition for isentropic flow
1 ∂p
c 2 ∂t
1 ∂p
c 2 ∂x
Continuity eq.
∂̺
∂t
+̺
∂u
+u
∂x
∂̺
∂x
=0
Momentum eq.
∂u
1 ∂p
∂u
+u
+
=0
∂t
∂x
̺ ∂x
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
73 / 92
Wave decomposition for isentropic flow
1 ∂p
c 2 ∂t
1 ∂p
c 2 ∂x
Continuity eq.
∂̺
∂t
∂u
+̺
+u
∂x
∂̺
∂x
=0
·
c
̺
Momentum eq.
∂u
1 ∂p
∂u
+u
+
=0
∂t
∂x
̺ ∂x
±
∂u
∂t
±
1 ∂p
̺c ∂t
Sesterhenn (TU Berlin)
+ (u ± c)
∂u
∂x
±
1 ∂p
̺c ∂x
=0
Skew symmetric compact FD schemes
June, 2010
73 / 92
Acoustic Waves
∂u
1 ∂p
±
∂t
̺c ∂t
Sesterhenn (TU Berlin)
+ (u ± c)
1 ∂p
∂u
±
∂x
̺c ∂x
Skew symmetric compact FD schemes
=0
June, 2010
74 / 92
Acoustic Waves
∂u
1 ∂p
±
∂t
̺c ∂t
+ (u ± c)
1 ∂p
∂u
±
∂x
̺c ∂x
=0
homentropic flow
∂R ±
∂R ±
+λ
=0
∂t
∂x
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
74 / 92
Application to the Euler Equations
1 ∂p
∂u
+
∂t
̺c ∂t
+ (u + c)
∂u
1 ∂p
+
∂x
̺c ∂x
=0
1 ∂p
∂u
−
∂t
̺c ∂t
+ (u − c)
∂u
1 ∂p
−
∂x
̺c ∂x
=0
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
75 / 92
Application to the Euler Equations
X + := (u + c)
1 ∂p
∂u
+
∂x
̺c ∂x
1 ∂p
∂u
+
∂t
̺c ∂t
+ (u+c)
∂u
1 ∂p
+
∂x
̺c ∂x
=0
1 ∂p
∂u
−
∂t
̺c ∂t
+ (u−c)
1 ∂p
∂u
−
∂x
̺c ∂x
=0
X
Sesterhenn (TU Berlin)
−
:= (u − c)
∂u
1 ∂p
−
∂x
̺c ∂x
Skew symmetric compact FD schemes
June, 2010
75 / 92
Euler Equations in Wave Form
∂p
̺c
=−
X+ + X−
∂t
2
1
∂u
= − X+ − X−
∂t
2
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
76 / 92
Wave-Expression of the Navier–Stokes Eqn.
∂p
∂t
∂u
∂t
∂v
∂t
∂w
∂t
∂s
∂t
̺c +
=
−
=
−
=
−
=
−
=
R
s
s
s
− X +Y +Z
+
p
2
1
2
(X
(X
+
X +
X
Sesterhenn (TU Berlin)
v
w
+X
−X
1
2
+Y
−
−
(Y
+
w
+
) + (Y
)+Y
u
−Y
−
1
+
2
(Z
+
+Y
+Z
u
)+Z
v
−
) + (Z
+
+
+
+Z
1 ∂τ1j
−
∂s
p
s
s
s
+X +Y +Z
) +
Cv
∂t
̺ ∂xj
1 ∂τ2j
̺ ∂xj
1 ∂τ3j
) +
̺ ∂xj
!
∂qi
+Φ
−
∂xi
−Z
−
Skew symmetric compact FD schemes
June, 2010
77 / 92
Adjoint Equations in Characteristic Wave Formulation
DNS-Code in characteristic wave
formulation
Euler equations (u)
∂u
1
= − X+ + X−
∂t
2
direct wave
X
±
= (u ± c)
1 ∂p ∂u
±
̺c ∂x
∂x
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
78 / 92
Adjoint Equations in Characteristic Wave Formulation
DNS-Code in characteristic wave
formulation
adjoint DNS-Code in adjoint
characteristic wave formulation
Euler equations (u)
adjoint Euler equations (u ∗ )
∂u
1
= − X+ + X−
∂t
2
−
direct wave
X ± = (u ± c)
∂u ∗
1
= − X +∗ − X −∗
∂t
2
adjoint wave
1 ∂p ∂u
±
̺c ∂x
∂x
Sesterhenn (TU Berlin)
∂p∗
∂u ∗
X ±∗ = (−u ± c) −̺c
±
∂x
∂x
Skew symmetric compact FD schemes
June, 2010
78 / 92
Adjoint Equations in Characteristic Wave Formulation
DNS-Code in characteristic wave
formulation
adjoint DNS-Code in adjoint
characteristic wave formulation
Euler equations (u)
adjoint Euler equations (u ∗ )
∂u
1
= − X+ + X−
∂t
2
−
direct wave
∂u ∗
1
= − X +∗ − X −∗
∂t
2
adjoint wave
X ± = (u ± c)
1 ∂p ∂u
±
̺c ∂x
∂x
∂p∗
∂u ∗
X ±∗ = (−u ± c) −̺c
±
∂x
∂x
analogous for adjoint compressible Navier–Stokes equations
adds tons of source–terms (continuous approach)
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
78 / 92
Adjoint Methods for Optimization
Adjoint equation for u +
∂
u
∂t
+
= −γ p
∂
p
+
+ 2/3
∂x
∂ µ ∂ v+
∂y ∂x
̺
∂ µ ∂ v+
∂x ∂y
−
̺
2
2
µ ∂ u+
∂ µ ∂ w+
∂ µ ∂ w+
µ ∂ u+
∂y 2
∂z 2
− ∂x ∂z
+ 2/3 ∂z ∂x
−
−
−
̺
̺
̺
̺
2
2
2
∂ µ
∂ µ ∂
µ ∂ u+
v+
µ ∂
µ ∂ ̺ ∂
µ ∂
w+
∂x∂y
∂y
∂y
+
+
∂x 2
∂x∂z
v − u + 4/3 ∂x
u −
− 2/3
− 2/3
− 2 1/3
− 1/3
̺
̺
̺
̺
̺2
∂x
̺ ∂x
+
∂
∂
∂
∂
∂
∂
∂
∂
µ
s
u+
v
µ
µ
µ ∂ ̺ ∂ +
µ
̺ ∂ +
µ
̺ ∂ +
µ
∂y
∂x ∂y
∂z
∂x
∂z
2
w − 2 1/3 ∂x
w − 2 ∂z
u −
− 2 1/3 ∂z
− 1/3
− 1/3
−
̺T
̺
̺2
∂x
̺
̺2
∂z
̺
̺2
∂z
∂
∂
+
∂ µ
∂
∂
∂
∂
∂
u − 2/3
v − 2/3 ∂ w ∂ s+
µ4/3
s
µ
µ
̺ ∂ +
µ ∂ +
w +
u
∂x
∂y
∂z ∂x
∂x
∂x
∂z
∂x
∂z
∂z
2 1/3
v −w +
u −2
− 1/3
−2
−
̺
̺2
∂y
̺ ∂z
̺T
̺T
∂ µ
∂ µ
∂ ̺ ∂
∂ s
∂ µ
∂2 µ
µ ∂ ̺ ∂
µ
p
∂
∂
∂
∂y
∂y
∂y
+
+
+
+
∂x
−
− 4/3
p−
p + ∂x p − ∂x∂z
−
v +
u −2
u − 2 4/3 ∂x
u −γ
̺ ∂y
̺
̺2
∂y
̺
̺2
∂x
∂x
∂x
Cv
̺
2
∂2 µ
2
∂
∂ µ ∂ ̺
∂ µ ∂ ̺
∂ µ ∂ ̺
µ
µ ∂ ̺ ∂ ̺
µ ∂
̺
∂
∂
2
∂y 2
∂y ∂y
+
∂x ∂z
∂x∂z
5/3 ∂x ∂z
+ 4/3
− 2/3
−
ww − 8/3 ∂x
− 4 ∂x ∂x
+
v +2
−3
+
̺2
̺3
̺2
∂x
̺
̺2
∂y
̺
̺2
2
2
2
∂
∂2 µ
∂
∂
2
∂
µ
̺
∂ µ ∂ ̺
̺
̺
µ
µ
µ
̺
µ ∂ ̺2
µ ∂ ̺2
∂
2
∂y
∂y 2
∂x
∂x 2
∂z
∂z 2 u + −
w + 2 ∂z
− 3 ∂z ∂z
+ 8/3
− 4/3
+2
−
+2
−
∂z
̺
̺2
̺3
̺2
̺3
̺2
̺3
̺2
∂ µ4/3 ∂ u − 2/3 ∂ v − 2/3 ∂ w
µ4/3 ∂ u − 2/3 ∂ v − 2/3 ∂ w ∂ ̺
∂x
∂x
∂y
∂z
∂x
∂y
∂z ∂x
2
−2
−
̺T
̺2 T
2
2
2
∂
∂
∂
∂ µ ∂ u+ ∂ v
∂
∂
∂
∂
u − 2/3
µ4/3
v − 2/3
w
µ4/3
u − 2/3
v − 2/3
w
T
∂x∂y
∂x∂z
∂x
∂y
∂z ∂x
∂y ∂y
∂x
∂x 2
+2
+2
−
2
̺T 2
̺T
̺T
2
2
∂ u+ ∂
∂
∂
∂
∂
∂
∂
v
µ
∂
∂
∂
∂
∂
µ
µ
u+
v
̺
u+
v
T
µ
w +
u
w +
u ∂ ̺
µ
∂x∂y
∂y
∂x ∂y
∂y
∂x ∂y
∂y 2
∂z − 2
∂x
∂z ∂z
2
−2
+2
+ 2 ∂z ∂x
−
̺2 T
̺T 2
̺T
̺T
̺2 T
2
2
∂ ̺ ∂ ̺
∂2 ̺
∂ µ ∂ ̺
∂2 µ
µ ∂
w + ∂ u
µ
µ
µ ∂ w + ∂ u ∂ T
∂
∂
∂x∂z
∂x∂y
∂x ∂y
∂x ∂y
∂x∂y
+
+
∂x
∂z ∂z + 2
∂z 2 −
2
ss −
− 5/3
+ 4/3
− 2/3
−
vv
̺T 2
̺T
∂x
̺
̺2
̺3
̺2
∂x
4/3
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
79 / 92
Sponge
Boundary condition
Sponge: efficient boundary condition for aeroacoustic applications
nonreflecting acoustic
reduction of spurious backscattering for leaving hydordynamic
waves
Sponge function (1D)
∂u
= RHS + σ(x) (uref − u)
∂t
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
80 / 92
Sponge
Boundary condition
Sponge: efficient boundary condition for aeroacoustic applications
nonreflecting acoustic
reduction of spurious backscattering for leaving hydordynamic
waves
Sponge function (1D)
∂u
= RHS + σ(x) (uref − u)
∂t
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
80 / 92
Krylov exponential time integration
Motivation
Problem:
linear ODE:
Solution:
dw(t)
dt
= Aw(t);
w(0) = v
w(t) = etA v
A is a stiff matrix i.e. λ|u+c| >> λu
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
81 / 92
Krylov exponential time integration
Motivation
Problem:
linear ODE:
dw(t)
dt
= Aw(t);
w(0) = v
w(t) = etA v
Solution:
A is a stiff matrix i.e. λ|u+c| >> λu
Solution:
increase time-step (CFL =
u·dt
dx )
t
t=0
T
t=0
T
t
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
81 / 92
Krylov exponential time integration
Motivation
Problem:
linear ODE:
dw(t)
dt
= Aw(t);
w(0) = v
w(t) = etA v
Solution:
A is a stiff matrix i.e. λ|u+c| >> λu
Solution:
increase time-step (CFL =
u·dt
dx )
t
T
t=0
numerically unstable
t
T
t=0
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
81 / 92
Krylov exponential time integration
Exponential time integration
method (explicit)
Euler
Runge-Kutta
Exponential integration
Sesterhenn (TU Berlin)
max. CFL-number (time-step)
∼ 0.5
∼ 1.0
→ ∞ (lin. eq.)
Skew symmetric compact FD schemes
June, 2010
82 / 92
Krylov exponential time integration
Exponential time integration
method (explicit)
Euler
Runge-Kutta
Exponential integration
max. CFL-number (time-step)
∼ 0.5
∼ 1.0
→ ∞ (lin. eq.)
Example
linear ODE:
Solution:
Sesterhenn (TU Berlin)
dw(t)
dt
= Aw(t);
w(0) = v
w(t) = etA v
Skew symmetric compact FD schemes
June, 2010
82 / 92
Krylov exponential time integration
Krylov subspace
Problem:
computing the exponential:
w(t) = etA v = v +
(tA)
1! v
+
(tA)2
2! v
+ ...
A is a huge and sparse matrix
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
83 / 92
Krylov exponential time integration
Krylov subspace
Problem:
computing the exponential:
w(t) = etA v = v +
(tA)
1! v
+
(tA)2
2! v
+ ...
A is a huge and sparse matrix
w(t): elements of a Krylov subspace
Km = span{v, Av, ..., Am−1 v}.
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
83 / 92
Krylov exponential time integration
Krylov subspace
Reformulating the Problem:
find an element of the Krylov subspace!
Km = span{v , Av , ..., Am−1 v }
or more precisely
project w(t) = etA v on Km .
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
84 / 92
Krylov exponential time integration
Krylov subspace
Reformulating the Problem:
find an element of the Krylov subspace!
Km = span{v , Av , ..., Am−1 v }
or more precisely
project w(t) = etA v on Km .
⇒ Arnoldi
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Skew symmetric compact FD schemes
June, 2010
84 / 92
Krylov exponential time integration
Arnoldi
Arnoldi
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
85 / 92
Krylov exponential time integration
Arnoldi
Arnoldi
orthonormal basis Vm
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Hessenberg matrix Hm
Skew symmetric compact FD schemes
June, 2010
85 / 92
Krylov exponential time integration
Arnoldi
Arnoldi
orthonormal basis Vm
Hessenberg matrix Hm
Hm = VmT AVm
Hm is a projection of A on Km with respect to Vm
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
85 / 92
Krylov exponential time integration
Arnoldi – matrix
+
=
A
Sesterhenn (TU Berlin)
Vm
=
Vm
Hm
Skew symmetric compact FD schemes
+hm+1 vm+1 e1T
June, 2010
86 / 92
Krylov exponential time integration
Approximation
Approximation of the matrix exponential:
VmT etA Vm ≈ etHm
w(t) = et A v
exact
Sesterhenn (TU Berlin)
=⇒
w(t) ≈ Vm et Hm e1
approximation
Skew symmetric compact FD schemes
June, 2010
87 / 92
Krylov exponential time integration
Approximation
Approximation of the matrix exponential:
VmT etA Vm ≈ etHm
w(t) = et A v
exact
=⇒
w(t) ≈ Vm et Hm e1
approximation
The trick:
Hm ∈ ℜm×m ↔ A ∈ ℜn×n
m≪n
e.g.
3 ≪ 106
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
87 / 92
Krylov exponential time integration
Application to nonlinear equations
What happens to nonlinear equations?
dw(t)
dt
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= f (w(t))
Skew symmetric compact FD schemes
June, 2010
88 / 92
Krylov exponential time integration
Application to nonlinear equations
Linearized equations:
dw(t)
dt
≈ f (w(tn )) + A(w(t) − w(tn ))
solution of the linearized equations:
df
dw
w(t) ≈ w(tn ) + tϕ(tA)f (w(tn ))
with ϕ(z) =
ez −1
z
Sesterhenn (TU Berlin)
Skew symmetric compact FD schemes
June, 2010
89 / 92
Krylov exponential time integration
Application to nonlinear equations
Linearized equations:
dw(t)
dt
≈ f (w(tn )) + A(w(t) − w(tn ))
solution of the linearized equations:
df
dw
w(t) ≈ w(tn ) + tϕ(tA)f (w(tn ))
with ϕ(z) =
ez −1
z
Krylov approximation:
w(t) ≈ w(tn ) + tVm ϕ(tHm )e1 f (w(tn ))
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Skew symmetric compact FD schemes
June, 2010
89 / 92
Krylov performance
Error
Performance
1e-4
2
1e-5
1.5
err
proc.-time [h] (t/TRK3 )
2.5
1e-6
1
1e-7
0.5
Krylov
Runge Kutta
1e-8
0
0
5
10 15 20
0.5 1
5 10 20
50
CFL
CFL
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2
Skew symmetric compact FD schemes
June, 2010
90 / 92
Krylov Ritz-Values
2 corresp. Ritz-Vectors
Ritz-Values of Hm
1st RV
6.0e5
2.0e5
0.0e0
-2.0e5
-4.0e5
-6.0e5
-1.2e4
-8.0e3
-4.0e3
ℜ(eig(Hm ))
Sesterhenn (TU Berlin)
0.0e0
4.0e3
22nd RV
ℑ(eig(Hm ))
4.0e5
Skew symmetric compact FD schemes
June, 2010
91 / 92
Conclusion
Krylov exponential time integration up to twice as fast as
standard explicit methods.
error in the order of standard explicit methods.
advantageous for fluid flow with a
wide range of temporal scales.
◮
◮
Low Mach number flows
Reactive flows
⇒ Good for aeroacoustic applications.
Schulze et al. 2008. Int. J. Num. Meth. in Fluids
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Skew symmetric compact FD schemes
June, 2010
92 / 92