Skew-Symmetric Schemes for compressible and
incompressible flows
Julius Reiss
Fachgebiet für Numerische
Fluiddynamik
TU Berlin
CEMRACS
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
1 / 46
shock
acoustic
Use Finite Difference or use Finite Volume?
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
2 / 46
shock
acoustic
Use Finite Difference or use Finite Volume?
→ skew symmetric, conservative FD
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
2 / 46
Overview
1
Burgers equation
2
Euler/Navier-Stokes Equations
3
Time discretization
4
Arbitrarily Transformed Grids
5
Fluxes & Boundary conditions
6
Incompressible flows
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
3 / 46
The concept
∂t u + ∂x f (u) = 0
Discretised:
∂t u + D u u = 0
∂t
R
u dx
1T D u = 0
∂t
R
telescoping sum
u 2 dx
(D u )T = −D u
J. Reiss (TU Berlin)
skew symmetry
Skew-Symmetric Schemes
August 8th , 2012
4 / 46
The concept
∂t u + ∂x f (u) = 0
Discretised:
∂t u + D u u = 0
∂t
R
u dx
1T D u = 0
∂t
R
telescoping sum −→ Momentum Conservation
u 2 dx
(D u )T = −D u
J. Reiss (TU Berlin)
skew symmetry −→ Kin. Energy Conservation
Skew-Symmetric Schemes
August 8th , 2012
4 / 46
Literature: Skew Symmetry
Feiereisen W.C.Reynolds J.H. Ferziger, Numerical simulation of a
compressible homogeneous, turbulent shear flow,
NASA-CR-164953; SU-TF-13
E. Tadmor, Skew-Selfadjoint Form for Systems of Conservation
Laws, J. Math. Ana. Appl. 103, p428, (1984)
Y. Morinishi and T. S. Lund and O. V. Vasilyev and P. Moin, Fully
conservative higher order finite difference schemes for
incompressible flow JCP 143, p 90, (1998)
R. W. C. P. Verstappen, A. E. P. Veldman. Symmetry-preserving
discretization of turbulent flow. JCP 187, p343 , (2003)
J.C. Kok, A high-order low-dispersion symmetry-preserving
finite-volume method. . . JCP 228, p 6811, (2009)
Y. Morinishi, Skew-symmetric form of convective terms. . . , JCP
229, p276 (2010)
S. Pirozzoli, Generalized conservative approximations of split
convective derivative operators, p7180 JCP 229, (2010)
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
5 / 46
Skew–Symmetric discretisation
Simple example: Burgers’ equation
divergence and convection form
∂t u+ ∂x
∂t u+
u2
2
u∂x u
=0
(D)
=0
(C)
by (2 · [D] + [C])/3:
skew–symmetric form
1
∂t u + [∂x u · +u∂x ·]u = 0
3
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
(S)
August 8th , 2012
6 / 46
Skew–Symmetric discretisation
Burgers’ in skew–symmetric form
1
∂t u + [∂x u · +u∂x ·]u = 0
3
Discrete
∂t u +
1
(DU + UD) u = 0
{z
}
|3
Du
U = diag(u), derivative D with D T = −D
∂t u + D u u = 0
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
7 / 46
Skew Symmetric discretisation
Kinetic Energy Conservation
1 T
2u u
=
1
2
P
i
ui2
∂t uT u = (∂t u)T u + uT ∂t u
= −(D u u)T u − uT D u u
= −uT (D u )T + D u u
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
8 / 46
Skew Symmetric discretisation
Kinetic Energy Conservation
1 T
2u u
=
1
2
P
i
ui2
∂t uT u = (∂t u)T u + uT ∂t u
= −(D u u)T u − uT D u u
= −uT (D u )T + D u u
Symmetry of transport term D u
(D u )T =
1
1
(DU + UD)T = (U T D T + D T U T ) = −D u
3
3
U = diag(u) = U T ,
J. Reiss (TU Berlin)
D T = −D
Skew-Symmetric Schemes
August 8th , 2012
8 / 46
Skew Symmetric discretisation
Kinetic Energy Conservation
1 T
2u u
=
1
2
P
i
ui2
∂t uT u = (∂t u)T u + uT ∂t u
= −(D u u)T u − uT D u u
= −uT (D u )T + D u u = 0
{z
}
|
=0
Symmetry of transport term D u
(D u )T =
1
1
(DU + UD)T = (U T D T + D T U T ) = −D u
3
3
Skew symmetry implies conservation of Ekin
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
8 / 46
Skew Symmetric discretisation
Momentum Conservation
1T u =
P
i
1 · ui
∂t 1T u =
1T ∂t u
= −1T D u u
Telescoping
1T D u u =
1
1 T
(1 D U + 1T UD)u = |uT{z
Du} = 0
3 |{z}
3
=0
=0
with D T = −D
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
9 / 46
Skew Symmetric discretisation
Momentum Conservation
1T u =
P
i
1 · ui
∂t 1T u =
1 T ∂t u
T u
= −1
D u} = 0
| {z
=0
Telescoping
1T D u u =
1 T
1
(1 D U + 1T UD)u = |uT{z
Du} = 0
3 |{z}
3
=0
=0
Telescoping sum property implies conservation of momentum
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
9 / 46
Time Integration
Implicit midpoint rule1
Fully discrete
u n+1 − u n 1 u n+1/2 n+1/2
+ D
u
=0
∆t
3
with u n+1/2 = 12 (u n + u n+1 )
1
Verstappen, Veldman, J. Com. Phys 187, p. 343 (2003)
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
10 / 46
Time Integration
Implicit midpoint rule1
Fully discrete
u n+1 − u n 1 u n+1/2 n+1/2
+ D
u
=0
∆t
3
with u n+1/2 = 12 (u n + u n+1 )
Multiplying by (u n+1/2 )T
(u n+1/2 )T (u n+1 − u n ) +
1 n+1/2 T u n+1/2 n+1/2
(u
) D
u
{z
}
3|
=0
=
=
1
1 n
(u + u n+1 )T (u n+1 − u n )
2
(u n+1 )2 (u n )2
−
=0
2
2
Verstappen, Veldman, J. Com. Phys 187, p. 343 (2003)
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
10 / 46
Numerical example: Burgers’ Equation
⊗⊗
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
11 / 46
Overview
1
Burgers equation
2
Euler/Navier-Stokes Equations
3
Time discretization
4
Arbitrarily Transformed Grids
5
Fluxes & Boundary conditions
6
Incompressible flows
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
12 / 46
Skew Symmetric discretisation
Euler Momentum Equation
divergence and convection form
∂t (%u) + ∂x (%u 2 ) + ∂x p = 0
(D)
%∂t (u) + %u∂x (u) + ∂x p = 0
(C)
by ([D]+[C])/2
Skew symmetric form
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = 0
2
2
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
(S)
August 8th , 2012
13 / 46
Skew Symmetric discretisation
Euler Equations
∂t % + ∂x (u%) = 0
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = 0
2
2
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
14 / 46
Skew Symmetric discretisation
Euler Equations
∂t % + ∂x (u%) = 0
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = 0
2 2
p
p
+ ∂x u
+p
∂t
+
γ−1
γ−1
%u 2 /2
∂t
%u 2 /2 + ∂x u
= 0
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
14 / 46
Skew Symmetric discretisation
Skew symmetric Euler Equations
∂t % + ∂x (u%) = 0
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = 0
2
2
1
γ
∂t p +
∂x (up) − u∂x p = 0
γ−1
γ−1
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
15 / 46
Skew Symmetric discretisation
Skew symmetric Euler Equations
∂t % + ∂x (u%) = 0
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = 0
2
2
1
γ
∂t p +
∂x (up) − u∂x p = 0
γ−1
γ−1
Discretised
∂t % + (DU)% = 0
1
1
(∂t % + %∂t )u + (DUR + RUD)u + Dp = 0
2
2
1
γ
∂t p +
(DU)p − (UD)p = 0
γ−1
γ−1
with U = diag(u), R = diag(%), Derivative D = −D T
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
15 / 46
Skew Symmetric discretisation
Skew symmetric Navier–Stokes Equations
∂t % + ∂x (u%) = 0
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = ∂x τ
2
2
1
γ
∂t p +
∂x (up) − u∂x p = −u∂x τ + ∂x τ
γ−1
γ−1
Discretised
∂t % + (DU)% = 0
1
1
(∂t % + %∂t )u + (DUR + RUD)u + Dp = Dτ
2
2
1
γ
∂t p +
(DU)p − (UD)p = −UDτ + Dτ u
γ−1
γ−1
with U = diag(u), R = diag(%), Derivative D = −D T , τ = µ∂x u
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
16 / 46
Conservation, time continuous
∂t % + DU% = 0
1
1
(∂t % + %∂t )u + (DUR + RUD) u + Dp = 0
{z
}
2
2|
D u%
γ
1
∂t p +
DUp − UDp = 0
γ−1
γ−1
1T (mass) = 0
+
=0
T
T
1 (innerE) + u (mom) = 0
1T (mom)
u T (mass)/2
J. Reiss (TU Berlin)
→
→
→
mass Conservation
Momentum Conservation
Energy Conservation
Skew-Symmetric Schemes
August 8th , 2012
17 / 46
Conservation, time continuous
∂t % + DU% = 0
1
1
(∂t % + %∂t )u + (DUR + RUD) u + Dp = 0
{z
}
2
2|
D u%
γ
1
∂t p +
DUp − UDp = 0
γ−1
γ−1
1T (mass) = 0
+
=0
T
T
1 (innerE) + u (mom) = 0
1T (mom)
u T (mass)/2
→
→
→
mass Conservation
Momentum Conservation
Energy Conservation
1
1
∂t 1T p + u T (∂t % + %∂t )u = ∂t
γ−1
2
γ
1
+
1T DUp + u T D u% u = 0
γ−1
2
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
1
T
T
1 p + u (%u)/2
γ−1
August 8th , 2012
17 / 46
Numerical example 1D
Time Integration: Leapfrog–like
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
18 / 46
Time discretization
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = 0
2
2
One Step methods:
Morinishi’s rewriting
1
2
(∂t ρ · +ρ∂t ·) u
=
√ √
ρ∂t ρu
use adopted midpoint rule
−→ similar to Morinishi , Subbareddy et. al., JCP 2009
generalises to higher order
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
19 / 46
Time discretization
1
1
(∂t % · +%∂t ·) u + (∂x u% · +%u∂x ·) u + ∂x p = 0
2
2
One Step methods:
Morinishi’s rewriting
1
2
(∂t ρ · +ρ∂t ·) u
=
√ √
ρ∂t ρu
use adopted midpoint rule
−→ similar to Morinishi , Subbareddy et. al., JCP 2009
generalises to higher order
√
J. Reiss (TU Berlin)
1
√
ρ∂t ( ρu) + D uρ u + Dx p = 0
2
Skew-Symmetric Schemes
August 8th , 2012
19 / 46
Time discretization
√
1
√
ρ∂t ρ + B u ρ = 0
2
1 uρ
√
√
ρ∂t ( ρu) + D u + Dx p = 0
2
1
γ
∂t p +
Bup − C up = 0
γ−1
γ−1
√
√
√
√
ρn+1/2
ρn+1 −
√ n
ρ
1 n+a
+ B u ρn+b = 0
ρn+1/2
∆t
2
√ n+1
√ n
ρu
−
ρu
1 n+a n+b
+ D u ρ u n+1/2 + Dx pn+c = 0
∆t
2
√
√
ρn+1/2 = 21 ( ρn + ρn+1 )
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
20 / 46
Time discretization
√
1
√
ρ∂t ρ + B u ρ = 0
2
1 uρ
√
√
ρ∂t ( ρu) + D u + Dx p = 0
2
1
γ
∂t p +
Bup − C up = 0
γ−1
γ−1
√
√
√
√
ρn+1/2
ρn+1 −
√ n
ρ
1 n+a
+ B u ρn+b = 0
ρn+1/2
∆t
2
√ n+1
√ n
ρu
−
ρu
1 n+a n+b
+ D u ρ u n+1/2 + Dx pn+c = 0
∆t
2
√
√
ρn+1/2 = 21 ( ρn + ρn+1 )
J. Reiss (TU Berlin)
u n+1/2 =
Skew-Symmetric Schemes
√
n+1 +(√ρu)n
1 ( ρu)
√
2
ρn+1/2
August 8th , 2012
20 / 46
Time discretization
Higher Order
√
1
√
ρ∂t ρ + B u ρ = 0
2
1 uρ
√
√
ρ∂t ( ρu) + D u + Dx p = 0
2
1
γ
∂t p +
Bup − C up = 0
γ−1
γ−1
Implicit Midpoint rule is Gauss-collocation method
All Gauss-collocation methods preserve skew-symme. & cons.†
Midpoint (2nd order)
1
2
1
2
1
Midpoint
(4th order)
√
1
3
1
2 − √6
4√
3
1
3
1
+
+
2
6
4
6
1
2
†Brouwer, Reiss, in prep.
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
1
4
√
−
3
6
1
4
1
2
August 8th , 2012
21 / 46
Overview
1
Burgers equation
2
Euler/Navier-Stokes Equations
3
Time discretization
4
Arbitrarily Transformed Grids
5
Fluxes & Boundary conditions
6
Incompressible flows
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
22 / 46
Euler Equations, more dimensions
Skew–symmetric momentum equation
√
1
√
ρ∂t ( ρuα ) +
∂xβ ρuβ · +ρuβ ∂xβ · uα + ∂xα p = 0.
2
Main problem: Keep skew symmetric structure on distorted grids.a
Local base eα = ∂ξα r, r = r(ξ, η, ζ) = (x, y , z)T
Use divergence as
∂uβ
= ∇u =
∂xβ
1X
∂ξα (eβ × eγ )u
J α,cy
=
1X
(eβ × eγ )∂ξα u.
J α,cy
a
Veldman, Rinzema, Playing with nonuniform grids, J. Engin. and Math.26,
p 119, (1992), VV2003
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
23 / 46
Euler Equations, more dimensions
Skew–symmetric momentum equation
√
1
√
ρ∂t ( ρuα ) +
∂xβ ρuβ · +ρuβ ∂xβ · uα + ∂xα p = 0.
2
Main problem: Keep skew symmetric structure on distorted grids.
Local base eα = ∂ξα r, r = r(ξ, η, ζ) = (x, y , z)T
Use divergence as
∂uβ
∂xβ
=
1X
∂ξα (eβ × eγ )u
J α,cy
=
1X
(eβ × eγ )∂ξα u.
J α,cy
[∂xβ ρuβ · + ρuβ ∂xβ ·]uα
∼
X
η
eη
y
eξ
ξ
x
[∂ξα (eβ × eγ )u%· + u%(eβ × eγ )∂ξα ·]uα
α,cy
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
23 / 46
Euler Equations, more dimensions
Skew–symmetric momentum equation
√
1
√
ρ∂t ( ρuα ) +
∂xβ ρuβ · +ρuβ ∂xβ · uα + ∂xα p = 0.
2
Main problem: Keep skew symmetric structure on distorted grids.
Local base eα = ∂ξα r, r = r(ξ, η, ζ) = (x, y , z)T
Use divergence as
∂uβ
∂xβ
=
1X
∂ξα (eβ × eγ )u
J α,cy
=
1X
(eβ × eγ )∂ξα u.
J α,cy
η
eη
y
eξ
ξ
x
[∂xβ ρuβ · + ρuβ ∂xβ ·]uα
∼
X
[∂ξα (eβ × eγ )u%· + u%(eβ × eγ )∂ξα ·]uα
α,cy
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
23 / 46
Euler Equations in 2D
Semidiscrete
J∂t ρ + B u ρ = 0
1
√
√
J ρ∂t ( ρu) + D uρ u + Dx p = 0
2
1 uρ
√
√
J ρ∂t ( ρv ) + D v + Dy p = 0
2
1
γ
J
∂t p +
Bu p − C u p = 0
γ−1
γ−1
with
B u = Dξ Ũ + Dη Ṽ
D uρ = (Dξ ŨR + R ŨDξ ) + (Dη Ṽ R + R Ṽ Dη )
C u = UDx + VDy
where Ũ = (UYη − VXη ) and Ṽ = (VXξ − UYξ )
and Dx = Dξ Yη − Dη Yξ , Dy = . . .
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
24 / 46
Overview
1
Burgers equation
2
Euler/Navier-Stokes Equations
3
Time discretization
4
Arbitrarily Transformed Grids
5
Fluxes & Boundary conditions
6
Incompressible flows
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
25 / 46
Boundary conditions
Flux vs. Value, 1D mass equation
−→
∂t ρ + Dx uρ = 0
∂t 1T ρ + 1T Dx uρ = 0
| {z }
bT

b
T
−1 1
 −1 0
 2

− 21
= 1T 



1
2


3 1
1 3

0
 = (− , , 0, · · · , 0, − , )

2
2
2 2
.. .. .. 
.
.
.
−1 1
1
2
mass flux over boundaries:
3
1
3
1
+ (ρu)N − (ρu)N−1
= −f0 + fN
bT uρ = − (ρu)1 − (ρu)2
2
2
2
2
Flux no-zero even for u1 = 0
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
26 / 46
Boundary conditions
Would like u1 = 0 ⇔ f0 = 0
− 12
 −1
 2

Wu 0 = 



1
2
0
− 12

1
2
0
..
.
1
2
..
.
..
− 12
1
2
.



 u.


−→ bT = (−1, 0, . . . , 0, 1).
weigh- matrix W (norm)
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
27 / 46
Boundary conditions
Would like u1 = 0 ⇔ f0 = 0
− 12
 −1
 2

Wu 0 = 



1
2
0
− 12

1
2
0
..
.
1
2
..
.
..
− 12
1
2
.



 u.


−→ bT = (−1, 0, . . . , 0, 1).
weigh- matrix W (norm)
1
1
Wij = diag( , 1, 1, . . . , 1, )
2
2
Summation By Parts (SBP)-Property
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
27 / 46
Boundary conditions: SBP
For SBP derivatives there is a norm, such that2
Telescoping sum is broken at boundary
Skew Symmetry only broken at D1,1 and DN,N
Boundary flux of our scheme depends entirely
on boundary values,
addition flux to enforce BC
Enforcing BC e.g. in Carpenter3 give global energy estimates.
(Work in progress. . . )
2
3
Strand, JCP 110, p47, 1994
Carpenter et al, JCP 108, p272, 1993
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
28 / 46
Skew Symmetric discretisation
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
29 / 46
Skew Symmetric discretisation
Without SBP
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
30 / 46
Skew Symmetric discretisation
With SBP
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
31 / 46
Skew Symmetric discretisation
Conservation with boundary
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
32 / 46
Skew Symmetric discretisation
First Conclusions
Skew symmetry leads to kinetic energy conservation
. . . with telescoping sum property full conservation
strict skew symmetry & perfect conservation on transformed grids
Procedure easy to implement & numerically efficient
Derivatives of any order in space
Time stepping of any order
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
33 / 46
Skew Symmetric discretisation
First Conclusions
Skew symmetry leads to kinetic energy conservation
. . . with telescoping sum property full conservation
strict skew symmetry & perfect conservation on transformed grids
Procedure easy to implement & numerically efficient
Derivatives of any order in space
Time stepping of any order
Outlook
Implementation for shock acoustic simulations in progress
Big DNS of turbulent flows
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
33 / 46
Bonus Part
Schemes for the incompressible Navier-Stokes equation
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
34 / 46
Bonus Part
Schemes for the incompressible Navier-Stokes equation
How to avoid odd-even decoupling?
Idea:
energy and momentum conserving, and collocated, by
I
I
use of skew symmetry
&
the combination of non-symmetric derivatives
Grid transformations preserving these properties
I
general in 2D
I
restricted in 3D
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
34 / 46
Incompressible Navier-Stokes
∂t ρ+∂xα ρuα = 0
1
√
√
ρ∂t ( ρuα ) + ∂xβ ρuβ · +ρuβ ∂xβ · uα + ∂xα p = ∂β τα,β
2
1
γ
∂t p +
∂x (uβ p) − uβ ∂xβ p = 0
γ−1
γ−1 β
∂t uα +
1
∂xβ uβ · +uβ ∂xβ · uα + ∂xα p = ν∆uα
2
∂xα uα = 0
α, β = 1, 2, 3
Pressure Poission Equation:
∂xα ∂xα p = ∂xα −D u uα + ν∆uα
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
35 / 46
Incompressible Navier-Stokes
∂t uα +
1
∂xβ uβ · +uβ ∂xβ · uα + ∂xα p = ν∆uα
2
∂xα uα = 0
α, β = 1, 2, 3
Pressure Poission Equation:
∂xα ∂xα p = ∂xα −D u uα + ν∆uα
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
36 / 46
Incompressible Navier-Stokes
∂t uα +
1
∂xβ uβ · +uβ ∂xβ · uα + ∂xα p = ν∆uα
2
∂xα uα = 0
α, β = 1, 2, 3
Pressure Poission Equation:
∂xα ∂xα p = ∂xα −D u uα + ν∆uα
Discrete
∂xα ∂xα p ∼ Dα Gα p
(Gp)i ∼ (pi+1 − pi−1 ) & (Du)i ∼ (ui+1 − ui−1 )
|
{z
}
DGp ∼ (pi+2 − 2pi + pi−2 )
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
36 / 46
Incompressible Navier-Stokes
∂t uα +
1
∂xβ uβ · +uβ ∂xβ · uα + ∂xα p = ν∆uα
2
∂xα uα = 0
α, β = 1, 2, 3
Pressure Poission Equation:
∂xα ∂xα p = ∂xα −D u uα + ν∆uα
Discrete
∂xα ∂xα p ∼ Dα Gα p
(Gp)i ∼ (pi+1 − pi−1 ) & (Du)i ∼ (ui+1 − ui−1 )
|
{z
}
DGp ∼ (pi+2 − 2pi + pi−2 )
Odd-even decoupling, nasty to solve!
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
36 / 46
Avoiding decoupling of ∆p
Collocated Grid
Staggered Grid
vi,j+1/2
vi,j
ui-1/2,j pi,j ui+1/2,j
pi,j ui,j
vi,j-1/2
+ Simple
+ No extra damping
+ Nice Boundary
+ Same geometry factors
- Boundary non-simple
- Different geometry factors for
- Upwind: Large damping
- Rhie-Chow: Smaller Damping
p, uα
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
37 / 46
Avoiding decoupling of ∆p
Collocated Grid
Staggered Grid
vi,j+1/2
vi,j
ui-1/2,j pi,j ui+1/2,j
pi,j ui,j
vi,j-1/2
+ Simple
+ No extra damping
+ Nice Boundary
+ Same geometry factors
- Boundary non-simple
- Different geometry factors for
- Upwind: Large damping
- Rhie-Chow: Smaller Damping
p, uα
Skew Symmetric Schemes
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
37 / 46
Avoiding decoupling of ∆p
Collocated Grid
Staggered Grid
vi,j+1/2
vi,j
ui-1/2,j pi,j ui+1/2,j
pi,j ui,j
vi,j-1/2
+ Simple
+ No extra damping
+ Nice Boundary
+ Same geometry factors
- Boundary non-simple
- Different geometry factors for
- Upwind: Large damping
- Rhie-Chow: Smaller Damping
p, uα
Skew Symmetric Schemes
J. Reiss (TU Berlin)
— ??? —
Skew-Symmetric Schemes
August 8th , 2012
37 / 46
Discretisation
∂xα uα = 0
∂t uα +
1
∂xβ uβ · +uβ ∂xβ · uα + ∂xα p = ν∆uα
2
Dα uα = 0
1
∂t uα + (Dβ Uβ + Uβ Gβ ) uα + Gα p = νLuα
|2
{z
}
Du
Transport term is skew symmetric
D u T = 21 (Uβ DβT + GβT Uβ ) ≡ −D u
provided
J. Reiss (TU Berlin)
Dβ = −GβT
not necessary skew sym.!
Skew-Symmetric Schemes
August 8th , 2012
38 / 46
Discrete Poission Equation
Dα uα = 0
u
∂t uα + D uα + Gα p = νLuα
Dα Gα p = Dα −D u uα + νLuα
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
39 / 46
Discrete Poission Equation
Dα uα = 0
u
∂t uα + D uα + Gα p = νLuα
Dα Gα p = Dα −D u uα + νLuα
Dβ = −GβT
| {z }
→
Energy cons.!
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
39 / 46
Discrete Poission Equation
Dα uα = 0
u
∂t uα + D uα + Gα p = νLuα
Dα Gα p = Dα −D u uα + νLuα

.


1 
−1 1 0 
Dβ = −GβT = ∆h


| {z }
..
.
→ Energy cons.!

..
Not upwind!
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
39 / 46
Discrete Poission Equation
Dα uα = 0
u
∂t uα + D uα + Gα p = νLuα
Dα Gα p = Dα −D u uα + νLuα

.


1 
−1 1 0 
Dβ = −GβT = ∆h


| {z }
..
.
→ Energy cons.!
−→ Dβ Gβ =

..

..
 .
1 
−2
∆h  1


1 

..
.
Not upwind!
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
39 / 46
Discrete Poission Equation
Dα uα = 0
u
∂t uα + D uα + Gα p = νLuα
Dα Gα p = Dα −D u uα + νLuα

.


1 
−1 1 0 
Dβ = −GβT = ∆h


| {z }
..
.
→ Energy cons.!
−→ Dβ Gβ =

..

..
 .
1 
−2
∆h  1


1 

..
.
Not upwind!
Dβ =
J. Reiss (TU Berlin)
..
.
1/6 −1 1/2 1/3 0
Skew-Symmetric Schemes
!
August 8th , 2012
39 / 46
Numerical example
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
40 / 46
Numerical example
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
40 / 46
Numerical example
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
40 / 46
Transformed Grids
Dα uα = 0
1
∂t uα + (Dβ Uβ + Uβ Gβ ) uα + Gα p = νLuα
2
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
41 / 46
Transformed Grids
Dα uα = 0
1
∂t uα + (Dβ Uβ + Uβ Gβ ) uα + Gα p = νLuα
2
D̄β Mα,β uα = 0
J∂t uα +
1
D̄γ Mβ,γ Uβ + Uβ Mβ,γ Ḡγ uα + Mα,γ Ḡγ p = νLuα
2
D̄β = −ḠβT
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
41 / 46
Transformed Grids
Dα uα = 0
1
∂t uα + (Dβ Uβ + Uβ Gβ ) uα + Gα p = νLuα
2
D̄β Mα,β uα = 0
J∂t uα +
1
D̄γ Mβ,γ Uβ + Uβ Mβ,γ Ḡγ uα + Mα,γ Ḡγ p = νLuα
2
D̄β = −ḠβT
∆ p ∼ D̄β
J. Reiss (TU Berlin)
Mα,β Mα,γ
Ḡγ p = . . .
J
Skew-Symmetric Schemes
August 8th , 2012
41 / 46
Transformed Grids: Conservation?
D̄β Mα,β uα = 0
1
J∂t uα +
D̄γ Mβ,γ Uβ + Uβ Mβ,γ Ḡγ uα + Mα,γ Ḡγ p = νLuα
2
D̄β = −ḠβT
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
42 / 46
Transformed Grids: Conservation?
D̄β Mα,β uα = 0
1
J∂t uα +
D̄γ Mβ,γ Uβ + Uβ Mβ,γ Ḡγ uα + Mα,γ Ḡγ p = νLuα
2
D̄β = −ḠβT
Energy conservation
uαT Mα,γ Ḡγ p
= pT ḠγT Mα,γ uα
= −pT D̄γ Mα,γ uα
= 0
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
42 / 46
Transformed Grids: Conservation?
D̄β Mα,β uα = 0
1
J∂t uα +
D̄γ Mβ,γ Uβ + Uβ Mβ,γ Ḡγ uα + Mα,γ Ḡγ p = νLuα
2
D̄β = −ḠβT
Energy conservation
Momentum conservation
1T Mα,γ Ḡγ p
uαT Mα,γ Ḡγ p
= pT ḠγT Mα,γ uα
= pT ḠγT mα,γ
= −pT D̄γ Mα,γ uα
= −pT D̄γ mα,γ
= 0
= ?
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
42 / 46
Transformed Grids: Momentum Conservation
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
43 / 46
Transformed Grids: Momentum Conservation
2D
D̄γ mα,γ =
Provided
J. Reiss (TU Berlin)
yξ = D̄ξ y
D̄ξ yη − D̄η yξ
...
=0
yη = D̄η y
Skew-Symmetric Schemes
August 8th , 2012
43 / 46
Transformed Grids: Momentum Conservation
2D
D̄γ mα,γ =
Provided
yξ = D̄ξ y
D̄ξ yη − D̄η yξ
...
=0
yη = D̄η y
3D
D̄γT mα,γ =
D̄ξ (yη zζ − yζ zη ) + D̄η (yζ zξ − yξ zζ ) + D̄ζ (yξ zη − yη zξ )
...
Only zero for restricted transformations:
x(ξ, η), y (ξ, η), z(ζ)
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
43 / 46
Second Conclusions
We found
Schemes for the incompressible Navier-Stokes equation, wich are
collocated, energy and momentum conserving, by
I
I
use of skew symmetry
&
the combination of non-symmetric derivatives
Grid transformations preserving these properties
I
general in 2D
I
restricted in 3D
Very simple to implement
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
44 / 46
Conclusions
Skew symmetric Schemes are
Skew symmetric schemes respect kinetic energy
Avoid numerical damping
Stable
Simple to implement
. . . worth a try!
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
45 / 46
Thank you for your attention!
Question?
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
46 / 46
Euler Equations, more dimensions
√
1
√
ρ∂t ( ρuα ) + ∂xβ ρuβ · +ρuβ ∂xβ · uα + ∂xα p = 0.
2
Momentum equation in skew symmetric form, 2D, u1 ≡ u component :
J
√
√ ∂t ( ρu)
ρ
1h
+ (∂ξ %(yη u − xη v ) · +%(yη u − xη v )∂ξ ·)
2
i
+(∂η %(−yξ u + xξ v ) · +%(−yξ u + xξ v )∂η ·) u
+(∂ξ yη − ∂η yξ )p = 0
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
47 / 46
Shock & Artifical Damping
1
1
(∂t R + R∂t )u + (DUR + RUD)u + Dp = Dτ
2
2
Friction Term τ = µDu
DµDu
→
−F σF T u
with adaptive sigama:
1
rth rth σ=
1−
+ 1 − 2
ri
ri
with with shock detector building on
dilatation1
ri = ri (∇u)
4
Bogey et al, JCP 228, 1447, 2009
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
48 / 46
Where are we now?
Shock & Acousic Pulse
%
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
49 / 46
Skew Symmetric discretisation
Euler Equations, time
∂t % + B u % = 0
1
1
[∂t R + R∂t ]u + D u% u + Dx p = 0
2
2
1
γ
∂t p +
Bu p − C u p = 0
γ−1
γ−1
Leap–Frog like scheme
1
4∆t
n
+B u %n = 0
1 n n
(R n+1 + R n )u n+1 − (R n−1 + R n )u n−1 + D u % u n + Dx pn = 0
2
n
B u pn
n
1
1
n+1
n−1
−p
+γ
− C u pn = 0
γ−1 2∆t p
γ−1
1
2∆t
J. Reiss (TU Berlin)
%n+1 − %n−1
Skew-Symmetric Schemes
August 8th , 2012
50 / 46
Skew Symmetric discretisation
Euler Equations, implicit time
mass
1 1 n+1
%
− %n−1 + D (u%)n−1 + 6(u%)n + (u%)n+1 = 0
2∆t
8
velocity
1 n+1
(R
+ R n )u n+1 − (R n−1 + R n )u n−1 +
4∆t
1 1 (u%)n−1 n−1
n
n+1
D
u
+ D (u%) (u n−1 + 4u n + u n+1 ) + D (u%) u n+1
28
1
+ Dx (pn−1 + 2pn + pn+1 ) = 0
4
pressure
1
1 n+1
0=
p
− pn−1 +
γ − 1 2∆t
1
γ
D (up)n−1 + 2(up)n + (up)n+1 − u n D(pn−1 + 2pn + pn+1 )
4γ − 1
4
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
51 / 46
Time Integration
Energy conservation
Time integration by midpoint rule produces energy conservation
u n+1 − u n 1 u n+1/2 n+1/2
+ D
u
=0
∆t
3
Multiplying by (u n+1/2 )T = 21 (u n + u n+1 )T
u n+1 − u n 1 n+1/2 T u n+1/2 n+1/2
1 n
(u + u n+1 )T
+ (u
) D
u
=0
{z
}
2
∆t
3|
=0
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
52 / 46
Time Integration
Energy conservation
Time integration by midpoint rule produces energy conservation
u n+1 − u n 1 u n+1/2 n+1/2
+ D
u
=0
∆t
3
Multiplying by (u n+1/2 )T = 21 (u n + u n+1 )T
u n+1 − u n
1 n
(u + u n+1 )T
2
∆t
=0
gives
1 n+1 T n+1
(u
) u
− (u n )T u n = 0
2
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
52 / 46
Time Integration
Momentum conservation
Time integration by midpoint rule produces momentum conservation
Multiplying by (~1)T = (1, 1, 1, 1, 1, . . . )
u
(~1)T
n+1
− u n 1 ~ T u n+1/2 n+1/2
+ (1) D
u
=0
∆t
3
with
n+1/2 n+1/2
(~1)T D u
u
T
n+1/2
~
= (1) DU
+ (u n+1/2 )T Du n+1/2 = 0
gives
1
2
J. Reiss (TU Berlin)
!
X
i
uin+1
−
X
uin
=0
i
Skew-Symmetric Schemes
August 8th , 2012
53 / 46
Euler Equations, more dimensions
1
1
(∂t % + %∂t ) ui +
∇%~u + %~u · ∇ ui + ∇p = 0.
2
2
Use divergence as
∇u =
1X
(ej × ek )∂ξi u.
J
1X
∂ξi (ej × ek )u =
J
i,cy
i,cy
Momentum equation in skew symmetric form
1
11X
(∂t % + %∂t ) uα +
∂ξi (ej × ek )%u + (ej × ek )%u∂ξi uα
2
2J
i,cy


X
1
+
∂ξi (ej × ek )p = 0
J
i,cy
α
. . . is indeed skew symmetric!
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
54 / 46
Time discretization
RHS% = 0
∂t % +
1
[∂t R + R∂t ]u +
2
1
∂t p +
γ−1
RHSu = 0
RHSp = 0
Central scheme 3–step
1
4∆t
1
n+1
2∆t %
(R n+1 + R n )u n+1
1
1
γ−1 2∆t
J. Reiss (TU Berlin)
− %n−1
− (R n−1 + R n )u
pn+1 − pn−1
Skew-Symmetric Schemes
+RHS% = 0
n−1
+RHSu = 0
+RHSp = 0
August 8th , 2012
55 / 46
Skew Symmetric discretisation
Euler Equations, time
1
massn+ 2 = 1T
1
momn+ 2 =
n+1/2
etot
=
(%n + %n+1 )
2
(%n + %n+1 )T (u n + u n+1 )
2
2
(u n )T (%n + %n+1 )u n+1
1T
(pn + pn+1 )/2 +
γ−1
4
E.g. momentum conservation
mom
1/2
− mom
N−1/2
=
N−1
X
f1/2 −
n=1
N−1
X
fN−1/2
n=1
→ equivalent to FV for ∆t → 0
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
56 / 46
End
J. Reiss (TU Berlin)
Skew-Symmetric Schemes
August 8th , 2012
46 / 46