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Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Compatible Finite Element Discretizations of Geometric Systems

Onno Bokhove

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

School of Mathematics, Leeds 2013

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

1 Introduction 2 Non-Autonomous Systems 3 NCP time flux 4 Spatial NCP? 5 Conclusion

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

• 1. Introduction

Numerical modelling of nonlinear waves and currents is often adequately done using conservative, Hamiltonian fluid dynamics, 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 with wave-makers [used for testing model offshore structures] wave-sloshing validations in a table-top Hele-Shaw cell with linear momentum damping. The question is how we can derive stable time integrators?

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

• 2. Non-Autonomous Hamiltonian Systems

Simulation (2D) of waves in MARIN’s wave tank: Workings of wave-maker

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Hele-Shaw Wave Tank

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

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Maths of MARIN’s Wave Tank

Mathematical formulation via Miles’ variational principle: = δ T L[φ, h, t]dt (1) = δ T L

xw(t)

φs∂th − 1 2g(h + b − H)2 − b+h

b

1 2|∇φ|2dzdx − b+h

b

dxw dt 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.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

FEM of MARIN’s Wave Tank

FEM formulation of Miles’ variational principle. FEM test/basis functions ˜ ϕj(x, z, t), ˆ ϕk(x, t) with i, j in D, 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) ˆ ϕk(x, t) in VP = δ T L[φj, hj, t]dt (3) = δ T φkMkl dhl dt − φkDkl dhl dt − . . . −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.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

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 L

• φs∂th − 1

2g(h − H0)2

dx − L ˜

γh 1 2|∇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 of the free surface“at rest”. Non-autonomous due to damping/integrating factor.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

FEM of Hele Shaw Wave Tank

Substitution potential flow Ansatz (¯ u, ¯ w) = (∂xφ, ∂zφ) into 2D Navier-Stokes eqns gives damped water waves: = δ T L[φj, hj, t]dt (6) = δ T

• φkMkl

dhl dt −1 2g(hk − H)Mkl(hl − H) −1 2φiAij(hk)φj

• e3νt/l2dt.

(7) Explicit time dependence in e3νt/l2 due to damping.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Damped Water Waves: Model vs. Data

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

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Autonomous Variational/Hamiltonian System

Both discretizations are succinctly summarized as: = δ T

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

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Autonomous Variational/Hamiltonian System

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

• pTM dq

dt − 1 2pTSp − 1 2qTMq

• dt

⇐ ⇒ M dq dt = Sp = ∂H ∂p , M dp dt = −Mq = −∂H ∂q (9) for Hamiltonian H = 1 2(pTSp + qTMq). (10)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Damped nonlinear oscillator

Toy example = δ T

• 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) = 1

2p2 + 1 2q2 + 1 4q4.

Note the integrating factor s.t.: d(peγt) dt = −(q + q3)eγt. (12)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Damped nonlinear oscillator

Dynamics becomes linear in long-time limit. The transformation q = Qe−γt/2 p = Pe−γt/2 (13) shows from = δ T P dQ dt − ˜ Hdt that for t → ∞ d ˜ H dt = 0 with (14) ˜ H = 1 2P2 + 1 2γPQ + 1 2Q2 + 1 4Q4e−γt = (1 2p2 + 1 2γpq + 1 2q2 + 1 4q4)eγt.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Conservative Products

Goal: to derive stable variational time integrators with time discontinuous FEM Finite elements in time. ph = pjϕj(t) and qh = qjϕj(t) expanded in piecewise continuous 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!

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Conservative Products

Consider p dq

dt or g(u) du dt = dQ(u) dt

with u = u(p, q). Dal Maso, LeFloch and Murat (1995) define g(u)du dt = lim

ǫ→0 g(uǫ)duǫ

dt (15) Introduce a Lipschitz continuous path φ : [0, 1] → ℜ with φ(0) = uL and φ(1) = uR with limits uL, uR at td: Moreover, jump depends on path φ(τ): g(uǫ)duǫ dt → Cδ(t − td) with C = 1 g(φ)τ dφ dτ (τ)dτ

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

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) | dφ

dτ (τ; uL, uR)| ≤ K|uL − uR|

Theorem by DLM: There is a unique real-valued bounded Borel measure µ on ]a, b[ such that: if u is discontinuous at a position td ∈]a, b[ then µ({td}) = 1 g(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 dq

dt –term in VP.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

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 = δ

N−1

• n=0

tn+1

tn

• ph

dqh dt − H(ph, qh, t)

• eγtdt

+

N

• n=−1

1 φp(τ; pL, pR)dφq dτ (τ; qL, qR)dτ (18)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Conservative Products: VP

Using DLM-theorem, discrete variational principle = δ

N−1

• n=0

tn+1

tn

• ph

dqh dt − H(ph, qh, t)

• eγtdt

+

N

• n=−1

1 φp(τ; pL, pR)dφq dτ (τ; qL, qR)dτ For quadratic & linear paths (γ = 0): φp = pL + 2a1τ + 3a2τ 2 & φq = qL + τ(qR − qL) s.t.: 1 φp(τ; pL, pR)dφq dτ (τ; qL, qR)dτ = (αpL+βpR)(qR −qL) with 0 ≤ α, β ≤ 1, i.e., jump in q× weighted mean in p.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Conservative Products: Symplectic Euler

Recap: = δ

N−1

• n=0

tn+1

tn

• ph

dqh dt − H(ph, qh, t)

• eγtdt

+

N

• n=−1

(αpL + βpR)(qR − qL)eγt∗ (19) Symplectic Euler (SE) for piecewise constant basis functions: pheγt = 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)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

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) − ∆tn 2 γ(pn + pn+1) MP

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Conservative Products: Symplectic Euler

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

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP: Hele Shaw Wave Tank revisited

Measure free surfaces:

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP: Hele Shaw Wave Tank revisited

Simulations vs. measurements (damped potential flow):

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP: Hele Shaw Wave Tank revisited

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

5 10 15 20 2.6 2.7 t Emod(t) 5 10 15 20 2.6 2.7 2.8 t Emod(t)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP: Stormer-Verlet

Stormer-Verlet (SV) for piecewise linear basis functions with α = 1, β = 0 and ζ ∈ [−1, 1]: ph = pn+1

L

(1 − ζ) 2 + pn+1

R

(1 + ζ) 2 qh = qn+1/2(1 + ζ) − qnζ. (21)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP: Stormer-Verlet

Discrete Lagrangian (γ = 0): Lh(ph, qh) =

N

• n=−1

(qn+1 + qn − 2qn+1/2)pn+1

R N−1

• n=0

(pn+1

L

+ pn+1

R

)(qn+1/2 − qn) −∆tn 2

• H(pn+1

L

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

R

, qn+1/2)

• Stormer-Verlet with pn+1

L

= pn

R continuous & q is DG:

δpn+1

L

: qn+1/2 = qn + ∆tpn+1

L

/2 δqn+1/2 : pn+1

R

= pn+1

L

− ∆t 2

• qn+1/2 + (qn+1/2)3

δpn+1

R

: qn+1 = qn+1/2 + ∆tpn+1

R

/2

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP: Stormer-Verlet Simulations

Driven wave focusing in MARIN’s wave tank. Entire wave tank MARIN with false vertical wall instead of beach. Comparison with measured data MARIN:

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP: midpoint rule

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

• pn+1/2(1 − ζ)/2 + pnζ
• eγtn+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).

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

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

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Spatial NCP & Extended Clebsch variables?

Example “1D/symmetric” shallow water equations: ∂tu − hvq = −∂x 1 2(u2 + v2) + gh

• ∂tv + u∂xv

= ∂th + ∂x(hu) = with PV q = ∂xv/h (23) Variational principle using Clebsch potentials/LCs: = δ T

• −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)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Extended Clebsch Variables: Gauge Symmetry

Symmetry in Hamiltonian suggests reduction phase space. Consider variations of H with constant density δh = 0 & leaving velocity invariant = 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

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Extended Clebsch Variables: Gauge Symmetry/PV

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

• hδ(∂tφ + π1∂ta1 + π2∂ta2)dxdydt

= T

• h∂G

∂t dxdydt = − T

• G [h∂(x, y)/∂(π2, a2)]

∂t π2da2dt. (25) Hence, 1D potential vorticity conserved: Dq Dt = 0 with q = 1 h ∂(π2, a2) ∂(x, y) = ∂xv h . (26)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP & Gauge Symmetry

Consider DGFEM basis function in space —x: means. In contrast to Cotter’s approach variables are defined on the element (rod). Crux lies in choice of numerical flux. Discrete variational principle follows, α = 1, 2: 0 = δ

N−1

• k=0

∆xk T −1 2h(u2

k + v2 k ) + hkvkπ2k

+hk( ˙ φk + π1k ˙ a1k + π2k ˙ a2k) + 1 2g(hk − H)2 +

N−1

• k=−1

1 hu(τ; φk+1, φk)dφφ dτ (τ; φk+1, φk) +huπα(τ; φk+1, φk)dφaα dτ (τ; φk+1, φk)dτ (27)

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

NCP & Challenge

Challenge: find an allowable path for the NCP numerical flux with δhk = 0 such that δuk = 0, with: hkuk ≡ ∂ ∂uk 1 hu(τ; φk+1, φk)dφφ dτ (τ; φk+1, φk) +huπα(τ; φk+1, φk)dφaα dτ (τ; φk+1, φk)dτ + 1 hu(τ; φk, φk−1)dφφ dτ (τ; φk, φk−1) +huπα(τ; φk, φk−1)dφaα dτ (τ; φk, φk)dτ

• Why? It means there is a symmetry such that phase space

can be reduced from h, φ, a1, a2, π1, π2 to h, u, v.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Conclusion

Established NCP-FEM-derivation of classic symplectic time 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 under investigation for wave problems: Challenge: NCP-DGFEM in space for variational principles with Clebsch variables.

Compatible Schemes Onno Bokhove Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

References

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