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OnnoBokhove

Introduction

Non-AutonomousSystems

NCP time flux

Spatial NCP?

Conclusion

Compatible Finite Element Discretizations ofGeometric Systems

Onno Bokhove

School of Mathematics, University of Leedswith Elena Gagarina, Vijaya Ambati & Shavarsh Nurijanyan (Twente)

School of Mathematics, Leeds 2013

CompatibleSchemes

OnnoBokhove

Introduction

Non-AutonomousSystems

NCP time flux

Spatial NCP?

Conclusion

1 Introduction

2 Non-Autonomous Systems

3 NCP time flux

4 Spatial NCP?

5 Conclusion

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Introduction

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Spatial NCP?

Conclusion

1. Introduction

Numerical modelling of nonlinear waves and currents is oftenadequately done using conservative, Hamiltonian fluiddynamics, even in the presence of some forcing and damping.In the modelling of two laboratory experiments,non-autonomous Hamiltonian/variational systems emerge:

investigation of freak waves in wave tanks withwave-makers [used for testing model offshore structures]

wave-sloshing validations in a table-top Hele-Shaw cellwith linear momentum damping.

The question is how we can derive stable time integrators?

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Conclusion

2. Non-Autonomous Hamiltonian Systems

Simulation (2D) of waves in MARIN’s

wave tank:

Workings of wave-maker

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Introduction

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NCP time flux

Spatial NCP?

Conclusion

Hele-Shaw Wave Tank

Simulation of damped, sloshing waves: initial conditions inmodel & experiment.

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NCP time flux

Spatial NCP?

Conclusion

Maths of MARIN’s Wave Tank

Mathematical formulation via Miles’ variational principle:

0 = δ

∫ T

0L[φ, h, t]dt (1)

= δ

∫ T

0

∫ L

xw (t)φs∂th −

1

2g(h + b − H)2

−∫ b+h

b

1

2|∇φ|2dzdx −

∫ b+h

b

dxwdt

φwdzdt (2)

with potential φ = φ(x , z , t) such that velocity(u,w)T = ∇φ = (∂xφ, ∂zφ)T

free-surface φs(x , t) ≡ φ(x , z = h + b, t) at ∂Ds , depth h

specified wave-maker piston xw (t) with φw ≡ φ(xw , z , t).

non-autonomous due to piston wave-maker.

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NCP time flux

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Conclusion

FEM of MARIN’s Wave Tank

FEM formulation of Miles’ variational principle.

FEM test/basis functions ϕj(x , z , t), ϕk(x , t) with i , j inD, k , l at free surface ∂Ds & m at wave maker.

Substitute φh(x , z , t) = φj(t)ϕj(x , z , t),hh(x , t) = hk(t)ϕk(x , t) in VP

0 = δ

∫ T

0L[φj , hj , t]dt (3)

= δ

∫ T

0φkMkl

dhldt− φkDkl

dhldt− . . .

−1

2g(hk + bk − H)Mkl(hl + bl − H)

−1

2φiAijφj − wm(t)φmdt. (4)

Mkl ,Dkl ,Aij wm depend on {hk(t), t}: mesh movement.

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Maths of Hele Shaw Wave Tank

Substitution potential flow Ansatz (u, w) = (∂xφ, ∂zφ)into 2D Navier-Stokes eqns gives damped water waves:

0 = δ∫ T

0

(∫ L0

(φs∂th − 1

2g(h − H0)2)dx

−∫ L

0

∫ γh0

12 |∇φ|

2dzdx

)e3νt/l2dt (5)

Use experiment to validate linear momentum damping.

Tilt tank till at rest: then drop it to create a linear tilt ofthe free surface“at rest”.

Non-autonomous due to damping/integrating factor.

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Conclusion

FEM of Hele Shaw Wave Tank

Substitution potential flow Ansatz (u, w) = (∂xφ, ∂zφ)into 2D Navier-Stokes eqns gives damped water waves:

0 = δ

∫ T

0L[φj , hj , t]dt (6)

= δ

∫ T

0

(φkMkl

dhldt

−1

2g(hk − H)Mkl(hl − H)

−1

2φiAij(hk)φj

)e3νt/l2dt. (7)

Explicit time dependence in e3νt/l2 due to damping.

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Damped Water Waves: Model vs. Data

Measure free surface & calculate potential energy P(t):

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Non-Autonomous Variational/Hamiltonian System

Both discretizations are succinctly summarized as:

0 = δ

∫ T

0

(pTM

dq

dt− 1

2pTAp

−1

2qTMq − pTDq − w(t)Tp

)f (t)dt (8)

MARIN’s tank: f (t) = 1,A = A(q, t),M = M(q, t),D = D(q, t),w(t) 6= 0.

Hele-Shaw tank: w(t) = D = 0, f (t) = exp (3νt/l2).

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Non-Autonomous Variational/Hamiltonian System

Note: Newton’s equations for coupled linear oscillators inlimit A(q) = S (constant) & f (t) = 1,w(t) = 0:

0 =

∫ T

0

(pTM

dq

dt− 1

2pTSp − 1

2qTMq

)dt

⇐⇒ Mdq

dt= Sp =

∂H

∂p,

Mdp

dt= −Mq = −∂H

∂q(9)

for Hamiltonian

H =1

2(pTSp + qTMq). (10)

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Damped nonlinear oscillator

Toy example

0 = δ

∫ T

0

(pdq

dt− H(p, q)

)eγtdt

δ(peγt) :dq

dt= p =

∂H

∂p

δq :dp

dt+ γp = −(q + q3) = −∂H

∂q(11)

with energy/Hamiltonian H = H(p, q) = 12p

2 + 12q

2 + 14q

4.

Note the integrating factor s.t.:

d(peγt)

dt= −(q + q3)eγt . (12)

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Damped nonlinear oscillator

Dynamics becomes linear in long-time limit. Thetransformation

q = Qe−γt/2 p = Pe−γt/2 (13)

shows from

0 = δ

∫ T

0PdQ

dt− Hdt

that for t →∞dH

dt= 0 with (14)

H =1

2P2 +

1

2γPQ +

1

2Q2 +

1

4Q4e−γt

= (1

2p2 +

1

2γpq +

1

2q2 +

1

4q4)eγt .

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Non-Conservative Products

Goal: to derive stable variational time integrators with timediscontinuous FEM

Finite elements in time.

ph = pjϕj(t) and qh = qjϕj(t) expanded in piecewisecontinuous fashion, e.g.:

What to do with derivatives pTMdq/dt at the jumps?

No staggered C-grid in t: crux lies in choice numerical flux!

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Non-Conservative Products

Consider p dqdt or g(u)dudt 6=

dQ(u)dt with u = u(p, q).

Dal Maso, LeFloch and Murat (1995) define

g(u)du

dt= lim

ε→0g(uε)

duε

dt(15)

Introduce a Lipschitz continuous path φ : [0, 1]→ < withφ(0) = uL and φ(1) = uR with limits uL, uR at td :

Moreover, jump depends on path φ(τ):

g(uε)duε

dt→ Cδ(t − td) with C =

∫ 1

0g(φ)τ

dφ

dτ(τ)dτ

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Non-Conservative Products

DLM assume a fixed family of paths with:(i) φ(0; uL, uR) = uL, φ(1; uL, uR) = uR ,(ii) φ(0; uL, uL) = uL,(iii) |dφdτ (τ ; uL, uR)| ≤ K |uL − uR |Theorem by DLM: There is a unique real-valued boundedBorel measure µ on ]a, b[ such that: if u is discontinuousat a position td ∈]a, b[ then

µ({td}) =

∫ 1

0g(φ)(τ ; ul , uR)

dφ

dτ(τ ; uL, uR)dτ. (16)

µ is the nonconservative product of g(u) by du/dt.

DGFEM in 3D & 4D: Rhebergen et al (2008ab, 2009)

Idea is to explore NCP for p dqdt –term in VP.

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Non-Conservative Products: VP

Choice of path: open question.

We generally chose a linear path:

φ(τ ; uL, uR) = uL + τ(uR − uL). (17)

Partition time in time slabs [tn, tn+1]

Using DLM-theorem, variational principle becomes

0 = δ

N−1∑n=0

∫ tn+1

tn

(ph

dqhdt− H(ph, qh, t)

)eγtdt

+N∑

n=−1

∫ 1

0φp(τ ; pL, pR)

dφqdτ

(τ ; qL, qR)dτ (18)

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Non-Conservative Products: VP

Using DLM-theorem, discrete variational principle

0 = δ

N−1∑n=0

∫ tn+1

tn

(ph

dqhdt− H(ph, qh, t)

)eγtdt

+N∑

n=−1

∫ 1

0φp(τ ; pL, pR)

dφqdτ

(τ ; qL, qR)dτ

For quadratic & linear paths (γ = 0):φp = pL + 2a1τ + 3a2τ

2 & φq = qL + τ(qR − qL) s.t.:∫ 1

0φp(τ ; pL, pR)

dφqdτ

(τ ; qL, qR)dτ = (αpL+βpR)(qR−qL)

with 0 ≤ α, β ≤ 1, i.e., jump in q× weighted mean in p.

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Non-Conservative Products: Symplectic Euler

Recap:

0 = δ

N−1∑n=0

∫ tn+1

tn

(ph

dqhdt− H(ph, qh, t)

)eγtdt

+N∑

n=−1

(αpL + βpR)(qR − qL)eγt∗

(19)

Symplectic Euler (SE) for piecewise constant basisfunctions: phe

γt = pn+1eγtn+1, qh = qn & α = 1, β = 0:

Lh(ph, qh) =N−1∑n=0

−∆tnH(pn+1, qn)eγtn+1

+N∑

n=−1

(qn+1 − qn)pn+1eγtn+1

(20)

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Non-Conservative Products: Symplectic Euler

Symplectic Euler (SE) for toy model:

δqn : pn+1 = e−γ∆tnpn −∆tn(qn + (qn)3)

δ(pn+1eγtn+1

) : qn+1 = qn + ∆tnpn+1.

Compare SE w. forward/backward Euler & midpoint:

pn+1 = pn −∆tn(qn + (qn)3)−∆tnγpn FE

pn+1 = pn −∆tn(qn + (qn)3)−∆tnγpn+1 BE

pn+1 = pn −∆tn(qn + (qn)3)− ∆tn2γ(pn + pn+1) MP

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Non-Conservative Products: Symplectic Euler

Comparison (blue: SE, red: SE-FE, black: SE-BE):

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NCP: Hele Shaw Wave Tank revisited

Measure free surfaces:

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NCP: Hele Shaw Wave Tank revisited

Simulations vs. measurements (damped potential flow):

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NCP: Hele Shaw Wave Tank revisited

SE vs. SE-BE(implications correct numerical damping for inertial ranges?):

0 5 10 15 202.6

2.7

t

Em

od(t

)

0 5 10 15 202.6

2.7

2.8

t

Em

od(t

)

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NCP: Stormer-Verlet

Stormer-Verlet (SV) for piecewise linear basis functionswith α = 1, β = 0 and ζ ∈ [−1, 1]:

ph = pn+1L

(1− ζ)

2+ pn+1

R

(1 + ζ)

2

qh = qn+1/2(1 + ζ)− qnζ. (21)

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NCP: Stormer-Verlet

Discrete Lagrangian (γ = 0):

Lh(ph, qh) =N∑

n=−1

(qn+1 + qn − 2qn+1/2)pn+1R

N−1∑n=0

(pn+1L + pn+1

R )(qn+1/2 − qn)

−∆tn2

(H(pn+1

L , qn+1/2) + H(pn+1R , qn+1/2)

)Stormer-Verlet with pn+1

L = pnR continuous & q is DG:

δpn+1L : qn+1/2 = qn + ∆tpn+1

L /2

δqn+1/2 : pn+1R = pn+1

L − ∆t

2

(qn+1/2 + (qn+1/2)3

)δpn+1

R : qn+1 = qn+1/2 + ∆tpn+1R /2

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NCP: Stormer-Verlet Simulations

Driven wave focusing in MARIN’s wave tank.

Entire wave tank MARIN with false vertical wall instead ofbeach.

Comparison with measured data MARIN:

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NCP: midpoint rule

Midpoint (MP): for piecewise linear basis functions withα = 1, β = 0 and ζ ∈ [−1, 1]:

pheγt =

(pn+1/2(1− ζ)/2 + pnζ

)eγt

n+1/2,

qh = qn+1/2(1 + ζ)− qnζ. (22)

pn+1/2 =1

2(pn+1 + pn) qn+1/2 =

1

2(qn+1 + qn).

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NCP: midpoint rule

Discrete Lagrangian

Lh(ph, qh) =N−1∑n=0

2pn+1/2(qn+1/2 − qn)eγtn+1/2

−∆tnH(pn+1/2, qn+1/2)eγtn+1/2

+N∑

n=−1

(qn+1 + qn − 2qn+1/2)pn+1eγtn+1

Modified midpoint scheme:

pn+1eγtn+1

= pneγtn −∆tn

∂H(pn+1/2, qn+1/2)eγtn+1/2

∂qn+1/2

qn+1 = qn + ∆t∂H(pn+1/2, qn+1/2)

∂pn+1/2

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Spatial NCP & Extended Clebsch variables?

Example “1D/symmetric” shallow water equations:

∂tu − hvq = −∂x(1

2(u2 + v2) + gh

)∂tv + u∂xv = 0

∂th + ∂x(hu) = 0 with PV q = ∂xv/h (23)

Variational principle using Clebsch potentials/LCs:

0 = δ

∫ T

0

∫∫−1

2h(u2 + v2) + h(∂tφ+ π1∂ta1 + π2∂ta2)

+hu(∂xφ+ π1∂xa1 + π2∂xa2) + hvπ2 +1

2g(h − H)2dxdydt

Clebsch potentials (φ, a1, π1π2)(x , t) & a2 = A2(x , t) + y :

δu : u = ∂xφ+ π1∂xa1 + π2∂xa2 & δv : v = π2 (24)

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Extended Clebsch Variables: Gauge Symmetry

Symmetry in Hamiltonian suggests reduction phase space.

Consider variations of H with constant density δh = 0 &leaving velocity invariant

0 = dG = δ(dφ+ π1da1 + π2da2)

⇐⇒ δφ = −G + π1∂G

∂π1+ π2

∂G

∂π2,

δa1 = − ∂G∂π1

, δa2 = − ∂G∂π2

,

δπ1 =∂G

∂a1, δπ2 =

∂G

∂a2

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Extended Clebsch Variables: Gauge Symmetry/PV

Hamilton’s principle under these restricted variations, withG = G (a1, a2, π1, π2, t) arbitrary, becomes (Salmon 1998)

0 =

∫ T

0

∫∫hδ(∂tφ+ π1∂ta1 + π2∂ta2)dxdydt

=

∫ T

0

∫∫h∂G

∂tdxdydt

= −∫ T

0

∫∫G

[h∂(x , y)/∂(π2, a2)]

∂tπ2da2dt.

(25)

Hence, 1D potential vorticity conserved:

Dq

Dt= 0 with q =

1

h

∂(π2, a2)

∂(x , y)=∂xv

h. (26)

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NCP & Gauge Symmetry

Consider DGFEM basis function in space —x : means.

In contrast to Cotter’s approach variables are defined onthe element (rod). Crux lies in choice of numerical flux.

Discrete variational principle follows, α = 1, 2:

0 = δ

N−1∑k=0

∆xk

∫ T

0−1

2h(u2

k + v2k ) + hkvkπ2k

+hk(φk + π1k a1k + π2k a2k) +1

2g(hk − H)2

+N−1∑k=−1

∫ 1

0hu(τ ;φk+1, φk)

dφφdτ

(τ ;φk+1, φk)

+huπα(τ ;φk+1, φk)dφaαdτ

(τ ;φk+1, φk)dτ (27)

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NCP & Challenge

Challenge: find an allowable path for the NCP numericalflux with δhk = 0 such that δuk = 0, with:

hkuk ≡ ∂

∂uk

(∫ 1

0hu(τ ;φk+1, φk)

dφφdτ

(τ ;φk+1, φk)

+huπα(τ ;φk+1, φk)dφaαdτ

(τ ;φk+1, φk)dτ

+

∫ 1

0hu(τ ;φk , φk−1)

dφφdτ

(τ ;φk , φk−1)

+huπα(τ ;φk , φk−1)dφaαdτ

(τ ;φk , φk)dτ

)Why? It means there is a symmetry such that phase spacecan be reduced from h, φ, a1, a2, π1, π2 to h, u, v .

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Conclusion

Established NCP-FEM-derivation of classic symplectictime integrators: SE [DG], SV [C/DG,2nd], MP [-].

But extended derivations to non-autonomous integrators& applied these to driven & damped wave problems

Higher-order & other NCP-DGFEM time integrators underinvestigation for wave problems:

Challenge: NCP-DGFEM in space for variational principleswith Clebsch variables.

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References

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