Energy conserving time integration methods for the incompressible Navier-Stokes equations · Energy...

Energy conserving time integration methods for the incompressible

Navier-Stokes equations

B. Sanderse

Paper presented at the “Verwer65 - Workshop on time integration of ODEs and PDEs”,

Amsterdam, 17-19 January, 2011

ECN-M-11-005 January 2011

Energy conserving time integration methods for the incompressible Navier-Stokes equations

B. Sanderse

donderdag 20 januari 2011


Introduction

• Accurate and efficient numerical simulation of turbulence with Large Eddy Simulation

• Requirements for discretization:• low numerical diffusion• high order spatial discretization• stable long-term integration

• Energy conserving discretizations1:• Correctly capture the turbulence energy spectrum and cascade• Non-linear stability bound

1: B. Perot (2010), Discrete conservation properties of unstructured mesh schemes, Annual Review Fluid Mechanics.


Discretization of fluid flows

The incompressible Navier-Stokes equations describe fluid flow:

To compute flows of practical interest, we need:• A numerical approximation to the NS equations• A turbulence model

∇ · u = 0,

∂u

∂t+ (u ·∇)u = −∇p+ ν∇2u.


Energy properties of incompressible NS

The incompressible Navier-Stokes equations:

possess a number of mathematical properties, e.g.

• The convective operator is skew-symmetric• The diffusive operator is symmetric• The divergence and gradient operator are related

∇ · u = 0,

∂u

∂t+ (u ·∇)u = −∇p+ ν∇2u.


Energy properties of incompressible NS

• In inviscid flow energy is an invariant:

• The NS equations are then time-reversible

• Time-reversibility has been proposed as test for energy conservation1

1: Duponcheel et al. (2008), Time reversibility of the Euler equations as benchmark for energy conserving schemes, JCP

dk

dt= −ν(∇u,∇u) ≤ 0, k = 1

2�u�2


• Finite volume method on staggered Cartesian grid• Exact conservation of mass• Compatible divergence and gradient operator,

no pressure-velocity decoupling• Skew-symmetric convective operator• Energy is conserved:

a stable discretization on any grid!

• Fourth order accuracy: Verstappen & Veldman, JCP 2003

Spatial discretization

xixi!1

!xi!1

yj

yj!1

!yj!1pi,j ui,j

vi,jvi,j

Figure 3.4: Staggered grid layout with p-centered control volume "pi,j .

xixi!1

yj

yj!1

pi,j pi+1,jui,j ui+1,j

vi,jvi,j

Figure 3.5: u-centered control volume "ui,j .

xixi!1

yj

yj!1

pi,j

pi,j+1

ui,j

vi,j

vi,j+1

Figure 3.6: v-centered control volume "vi,j .

25


• Method of lines gives non-autonomous semi-discrete DAE system of index 2:

• Continuous properties are mimicked in a discrete sense:

Spatial discretization

!du

dt+ C(u, u)! !Du+Gp = g2(t)

Mu = g1(t)

(Gp, u) = −(Mu, p)

(C(u, v), w) = −(v, C(u,w)) → (C(u, u), u) = 0

0 = g(y, t)

y� = f(y, z, t)


• Current practice: multistep or multistage methods with pressure correction and explicit convection• Not energy conserving, not time-reversible

• Research questions:• Energy conservation and/or time reversibility with Runge-

Kutta methods?• Order reduction when applying Runge-Kutta methods to the

incompressible NS equations?• Can implicit methods be more efficient than explicit

methods?

Temporal discretization


• General implicit/explicit Runge-Kutta method:

• supplemented with consistent initial conditions:

where is the Laplacian; should be bounded.

Runge-Kutta methods

Ui = un +∆ts�

j=1

aijF (Uj , pj , tj)

MUi = g1(ti)

un+1 = un +∆ts�

i=1

biF (Ui, pi, ti)

Mun+1 = g1(tn+1)

Mu0 = g1(t0)

L = MG

Lp0 = M(−C(u0) + νDu0 + g2(t0))− g1(t

0)

L−1


• In literature there is confusion on how to handle the pressure• The theory on DAEs gives a guideline: half-explicit methods1

• Equations for the stages:

Explicit methods

Ui = un +∆ti−1�

j=1

aijF (Uj , pj , tj)

MUi = g1(ti)

F (Ui, pi, ti) = −C(Ui) + νDUi −Gpi + g2(ti)

1: Hairer et al. (1989), The numerical solution of differential-algebraic systems by Runge-Kutta methods

F (Ui, ti) = −C(Ui) + νDUi + g2(ti)


• Poisson equation for the pressure:

• Similar to pressure correction• Butcher array and only appear for unsteady BCs

Explicit methods

∆t

ai+1,i∆tLpi = M

un +∆ti−1�

j=1

ai+1,jF (Uj , pj , tj) +∆tai+1,iF (Ui, ti)

−g1(ti+1)

Lpi = MF (Ui, ti)−i�

j=1

f(A)g1(ti+1)− g1(tn)

∆t1 ≤ i ≤ s


• Example: classical RK4

Explicit methods

Lp1 = MF1 −g1(tn+1/2)− g1(tn)

12∆t

Lp2 = MF2 −g1(tn+1/2)− g1(tn)

12∆t

Lp3 = MF3 −g1(tn+1/2)− g1(tn)

∆t

Lp4 = MF4 − 4g1(tn+1)− g1(tn)

∆t+ 3

g1(tn+1/2)− g1(tn)12∆t

approximations to g1


• No equation for ; is not of the expected order• Higher order accurate with additional Poisson equation:

• This does not increase computational costs: solution of is the same as .

• does not influence and can be computed as post-processing step

Explicit methods

pspn+1

p1pn+1

pn+1un+1

Lpn+1 = MFn+1 − g1(tn+1)


• Recall the general form:

Implicit Runge-Kutta methods

Ui = un +∆ts�

j=1

aijF (Uj , pj , tj)

MUi = g1(ti)

un+1 = un +∆ts�

i=1

biF (Ui, pi, ti)

Mun+1 = g1(tn+1)


• Energy conservation (periodic BC, ):

• : , , • :


(Fi, Ui) = 0mij = bibj − biaij − bjaji mij = −mji

(un+1, un+1) = (un, un) + 2∆ts�

i=1

bi (Fi, Ui) + ...

∆t2s�

i,j=1

mij (Fi, Fj) .

MUi = 0

ν = 0

(C(Ui, Ui), Ui) = 0 M = −G∗


• Time reversibility

• Conditions for coefficients:

• Time reversibility and energy conservation conditions are satisfied by Gauss methods


un+ci → −un+1−ci

aij + as+1−i,s+1−j = bj ,

bi = bs+1−i,

ci = 1− cs+1−i


• Butcher tableaus:

• : algebraic stability

• Convergence order1:

Implicit Runge-Kutta methods: Gauss

12

121

(a) s = 1

12 −

√36

14

14 −

√36

12 +

√36

14 +

√36

14

12

12

(b) s = 2

ODE DAE u DAE ps odd

2ss+ 1 s− 1

s even s s− 2

1: Hairer et al. (1989), The numerical solution of differential-algebraic systems by Runge-Kutta methods

mij = 0


• Solution of non-linear system with Newton linearization


�I + a11∆tJ(U1) G

a11∆tM 0

� �∆F1

∆p1

�= R

I + a11∆tJ(U1) a12∆tJ(U1) G 0a21∆tJ(U2) I + a22∆tJ(U2) 0 Ga11∆tM a12∆tM 0 0a21∆tM a22∆tM 0 0

∆F1

∆F2

∆p1∆p2

= R


• For general there is no guarantee that • Additional projection1:

• Alternative:Stiffly accurate RK methods: , e.g. Radau IIA or Lobatto IIIA/C methods, not energy conserving


g1(t) Mun+1 = g1(tn+1)

asi = bi, cs = 1

1: Ascher & Petzold (1991): Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM J. Num. Anal.

un+1 = un +∆ts�

i=1

biFi

un+1 = un+1 −Gφ

Lφ = Mun+1 − g1(tn+1)


• Pressure can be solved as for explicit methods

• Alternatively:

• Order reduction expected


pi = pn +∆ts�

j=1

aijZj

Zi =1

∆t

s�

j=1

ωij(pj − pn)

pn+1 = pn +∆ts�

i=1

biZi


• Sacrificing energy conservation: pressure correction• Provisional velocity:

• Pressure solve:

• Velocity update:

Implicit Runge-Kutta methods - PC

un+1 − u∗

∆t= −1

2G∆p

u∗ − un

∆t= F (

un + u∗

2)−Gpn

1

2L∆p =

1

∆t(Mu∗ − g1(t

n+1))


• An energy error is introduced since • The scheme remains unconditionally stable, but only in a linear

sense1

• Time reversibility is unaffected

Implicit Runge-Kutta methods - PC

Mu∗ �= g1(tn+1)

1: van Kan (1986): A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J.Sci.Stat.Comput.

�un+1�2 + ∆t2

8�Gpn+1�2 ≤ �un�2 + ∆t2

8�Gpn�2.


• Sacrificing time-reversibility: extrapolated convecting velocity

• : independent of the time level of the convecting velocity

• First order:

• Second order:

• Still unconditionally stable

• Linear system

• Not time reversible

Implicit Runge-Kutta methods - linear

(C(u, u), u) = 0

C(un, un+1/2)

C�32u

n − 12u

n−1, un+1/2�


• Energy conservation and time reversibility

• Computational cost

Summary

time reversibley n

energy conservingy IRK IRK - linear (ERK-cons)n IRK - PC ERK

Table 4: Overview energy conservation and time reversibility.

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) ERK2

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) ERK4

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) IRK2

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) IRK4

Figure 1: Stability regions.

17

nonlinear SP linear SP nonlinear LaplaceERK s 0 0 0 s

IRK - steady BC 1 0 0 1IRK - unsteady BC 1 0 0 2

IRK - linear 0 1 0 2IRK - PC 0 0 1 2


Summary

time reversibley n

energy conservingy GLRKn GLRKn - linear, ERKcn GLRKn - pressure correction ERK

Table 4: Overview energy conservation and time reversibility.

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) ERK2

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) ERK4

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) IRK2

−5 −4 −3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) IRK4

Figure 1: Stability regions.

17


• Taylor vortex: Order of accuracy

• Shear-layer roll-up: Energy conservation and time reversibility

• Corner flow: Order of accuracy and efficiency

Results


• Analytical solution on a domain of

• Boundary conditions: periodic or unsteady Dirichlet

• Temporal error evaluated at

Taylor vortex

u(x, y, t) = − sin(πx) cos(πy) e−2π2νt,

v(x, y, t) = cos(πx) sin(πy) e−2π2νt,

p(x, y, t) =1

4(cos(2πx) + cos(2πy))e−4π2νt

t = 1

[ 14 , 214 ]× [ 14 , 2

14 ]


Taylor vortex; velocity field

x

y

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Energy error

Taylor vortex; periodic BC

ν = 1/100

20x20 volumes

10 2 10 1

10 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! k

ERK2IRK2IRK2 linearIRK2 PCERK4IRK4

4

2


Taylor vortex; periodic BC

ν = 1/100

Pressure error

20x20 volumes

10 2 10 1

10 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! p

ERK2IRK2IRK2 linearIRK2 PCERK4IRK4IRK 4 no p solve

4

2


Energy error

Taylor vortex; Dirichlet BC

ν = 1/100

20x20 volumes

10 2 10 1

10 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! k

ERK2IRK2IRK2 linearIRK2 PCERK4IRK4IRK4 no projection

2

4


Taylor vortex; Dirichlet BC

ν = 1/100

Pressure error

20x20 volumes

10 2 10 1

10 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! p

ERK2IRK2IRK2 linearIRK2 PCERK4IRK4IRK4 no projection

2

4


• Analytical solution on a domain of

• Boundary conditions: periodic• Inviscid: • Time reversal at

Shear-layer roll-up

[0, 2π]× [0, 2π]

u =

tanh

�y−π/2

δ

�y ≤ π,

tanh�

3π/2−yδ

�y > π,

v = ε sin(x),

p = 0.

ν = 0t = 8


• Implicit midpoint• 100x100 volumes•

Shear-layer roll-up; time reversibility

∆t = 0.05


Shear-layer roll-up; energy conservation

0 2 4 6 8 10 12 14 1610

8

6

4

2

0

2x 10 14

!t

! k



IRK2 - PC

0 2 4 6 8 10 12 14 168

7

6

5

4

3

2

1

0

1x 10 3

!t

! k


• Implicit midpoint• 10x10 volumes•


∆t = 1


• Explicit RK 4• 100x100 volumes•


∆t = 0.05



ERK4

0 2 4 6 8 10 12 14 163.5

3

2.5

2

1.5

1

0.5

0

0.5x 10 7

!t

! k

ERK4


• 2nd order Adams-Bashforth• • 100x100 volumes


∆t = 0.05



32x32 volumes

t = 4

10 2 10 110 16

10 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! k

ERK2IRK2IRK2 linearIRK2 PCERK4IRK4

2

4

3


• ‘stiff’ problem; boundary layers,• domain • 20x20 volumes• simulation until

Corner flow

[0, 1]× [0, 1]ν = 1/10

t = 1

u = v = 0

u = v = 0

∂v

∂x= 0

u = 0, v = − sin(π(x3 − 3x2 + 3x))e1−1/t

−p+ ν∂u

∂x= 0


Corner flow

10 4 10 3 10 2 10 1 10010 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! k

ERK2ERK4


Corner flow

10 4 10 3 10 2 10 1 10010 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! k

IRK2IRK2 linearIRK2 PCAB CN


Corner flow

10 4 10 3 10 2 10 1 10010 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! k

IRK4IRK4 no projection


Corner flow

10 4 10 3 10 2 10 1 10010 14

10 12

10 10

10 8

10 6

10 4

10 2

!t

! k

ERK2IRK2IRK2 linearIRK2 PCAB CNERK4IRK4IRK4 no projection

4

2


Corner flow: computational time

10 1 100 101 102 10310 14

10 12

10 10

10 8

10 6

10 4

10 2

CPU

! k

ERK2IRK2IRK2 linearIRK2 PCAB CNERK4IRK4IRK4 no projection


• Time-reversible and energy conserving integration of the NS equations can be achieved with Gauss methods

• Order reduction is observed for unsteady boundary conditions only, and can be cured with an extra projection step. Both the differential and algebraic variable obtain the classical order.

• Cheaper methods can be constructed by linearizing or splitting, leading to loss of time-reversibility or energy conservation properties.

• Optimization of matrix solvers and investigation of turbulent flow will shed more light on the usefulness of these properties.

Conclusions


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