[PPT] - An energy-conserving explicit time-integration scheme for nonlinear PowerPoint Presentation

SLIDE 1

An energy-conserving explicit time-integration scheme for nonlinear Hamiltonian dynamics

F. Marazzato, A. Ern, C. Mariotti and L. Monasse

CEA, ENPC and Inria

05/29/2017

Marazzato et al. Energy conservation 05/29/2017 1 / 19

SLIDE 2

Summary

1

Hamiltonian systems

2

Nonlinear Hamiltonian wave equations

3

Local time-stepping

Marazzato et al. Energy conservation 05/29/2017 2 / 19

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Hamiltonian systems

System of N particles Positions q ∈ R3N, mass-matrix M ∈ R3N×3N and velocity momenta p = Mv ∈ R3N . q1, p1 F13 F32 q2, p2 F23 F31 q3, p3 F32 F31 Fext

Marazzato et al. Energy conservation 05/29/2017 3 / 19

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Setting

Hamiltonian H ≡ H(q, p) ∈ R total energy of the system Equations of motion :

        

˙ q = ∂H ∂p ˙ p = −∂H ∂q Separated Hamiltonian dynamics: H(q, p) = 1 2pTM−1p + V (q) Equations of motion :

˙

q = M−1p ˙ p = −∇V (q)

Marazzato et al. Energy conservation 05/29/2017 4 / 19

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Bibliography

Implicit energy-conserving schemes

Averaged vector field methods (second order accuracy) [Quispel and McLaren, 2008]. Higher order generalisation [Hairer, 2010]. Wave equations [Chabassier and Joly, 2010]. Elastodynamics [Hauret and Le Tallec, 2006].

Explicit

Symplectic variable step-size integrator [Hardy, Okunbor and Skeel, 1999]. Time-step choice is not very flexible. Not exactly energy-conserving. Störmer/Verlet. Not energy-conserving for nonlinear Hamiltonians and variable time-step.

        

pn+1/2 = pn − ∆t 2 ∇V (qn) qn+1 = qn + ∆tM−1pn+1/2 pn+1 = pn+1/2 − ∆t 2 ∇V (qn+1)

Marazzato et al. Energy conservation 05/29/2017 5 / 19

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Explicit time-integration scheme

Free-flight idea developed in [Mariotti, 2015]. Mid-point integration rule for the momentum jumps.

      

q(t) = qn + 1 mpn+1/2 (t − tn) 1 2

[p]n+1 + [p]n

= −

tn+1

tn

∇V (q(t))dt qn−1, tn−1 qn, tn pn−1/2 qn+1, tn+1 pn+1/2 [p]n = pn+1/2 − pn−1/2

Marazzato et al. Energy conservation 05/29/2017 6 / 19

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Properties

Symmetric and reversible 2nd order consistent Conditionally stable and thus convergent

Theorem (Discrete Energy Conservation)

With an exact integration of forces, the following discrete modified energy is conserved for any time-step: ˜ Hn = V (qn) + 1 2

pn−1/2T M−1pn+1/2,

∀n ∈ N Moreover, with a proper initialisation: ˜ Hn = H0, ∀n ∈ N

Marazzato et al. Energy conservation 05/29/2017 7 / 19

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Proposition (Energetic CFL condition)

Hn Hamiltonian at tn. |M−1/2[p]n| ≤

8β|H0|,

∀n ∈ N = ⇒ |Hn − H0| ≤ β|H0|, ∀n ∈ N for β ∈ [0, 1] (energetic stability).

Proposition (CFL condition (absolute stability))

The scheme is stable as long as, for every particle : ∆tn ≤ 2

m

|∇2V (qn)|, ∀n ∈ N

Marazzato et al. Energy conservation 05/29/2017 8 / 19

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Fermi-Pasta-Ulam

H(p, q) = 1

2 m

i=1(p2 2i−1+p2 2i)+ ω2 4

m

i=1(q2i −q2i−1)2+m i=0(q2i+1−q2i)4

Ij(xm+j, ym+j) = 1

2

y 2

m+j + ω2x2 m+j

energy of the jth stiff spring with:

xi = (q2i + q2i−1)/ √ 2, yi = (p2i + p2i−1)/ √ 2, xm+i = (q2i − q2i−1)/ √ 2, ym+i = (p2i − p2i−1)/ √ 2 Total oscillatory energy I = I1 + I2 + ... + Im close to constant value since I((x(t), y(t))) = I((x(0), y(0))) + O(ω−1) Figure: Test Fermi-Pasta-Ulam (source : [Hairer, 2006]) Figure: Result for ∆t = 0.001

Marazzato et al. Energy conservation 05/29/2017 9 / 19

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Summary

1

Hamiltonian systems

2

Nonlinear Hamiltonian wave equations

3

Local time-stepping

Marazzato et al. Energy conservation 05/29/2017 10 / 19

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Motivating example

The setting comes from [Chabassier and Joly, 2010] Ω = [0, 1] and E (C2, strictly convex and growth conditions on ∇E). Find u : Ω × R+ → R2 such that : ∂2

ttu − ∂x(∇E(∂xu)) = 0

BC : u(0, t) = 0, u(1, t) = 0, IC : u(x, 0) = u0(x), ∂tu(x, 0) = u1(x) Variational formulation in V =

H1

0(Ω)

2 :

d2 dt2

Ω

u · v

+
Ω

∇E(∂xu) · ∂xv = 0, ∀v ∈ V, ∀t > 0 {Vh, h > 0} with conforming Lagrange Pk finite elements

Marazzato et al. Energy conservation 05/29/2017 11 / 19

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Time-integration setting

Semi-discretization in space by conforming FEM Basis (ϕ)i=1,..,2N for displacements of degrees of freedom The Hamiltonian is written : H(q, p) = 1 2pTM−1p + V (q) with: V (q) =

Ω

E

2N

i=1

(q)i∂xϕi

Marazzato et al.

Energy conservation 05/29/2017 12 / 19

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Results for linear wave equations

E(u, v) = u2+v2

2

with (u, v) displacement vector of the string.

Marazzato et al. Energy conservation 05/29/2017 13 / 19

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Results for nonlinear wave equations

E(u, v) = u2+v2

2

− α

(1 + u)2 + v2 − (1 + u)
with

0 ≤ α = 0.99 < 1 characteristic of the nonlinearity. mid-point quadrature on

tn+1

tn

∇V (q(t))dt

Marazzato et al. Energy conservation 05/29/2017 14 / 19

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Results for nonlinear wave equations 2

0.0 0.2 0.4 0.6 0.8 1.0 t 0.0 0.2 0.4 0.6 0.8 E

Relative energy evolution (in %) ~H^n H^n

Marazzato et al. Energy conservation 05/29/2017 15 / 19

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Summary

1

Hamiltonian systems

2

Nonlinear Hamiltonian wave equations

3

Local time-stepping

Marazzato et al. Energy conservation 05/29/2017 16 / 19

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Local time-stepping

Theorem 1 ensures that ˜ Hn = H0, ∀n ∈ N without any condition on ∆t. It allows ∆t to evolve over a simulation. Ex: Adaptation to stiff forces The only conditions are to respect the previous CFL conditions.

Slow-Fast particle splitting

If system has two time scales ∆t

δt = N ∈ N

Divide the particles into a "fast" group whose dynamics will be integrated at every δt and a "slow" group whose dynamics will be integrated only every ∆t. When the dynamics of all the particles has been integrated over Nδt = ∆t, the energy ˜ Hn is conserved as stated in Theorem 1.

Marazzato et al. Energy conservation 05/29/2017 17 / 19

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Slow-fast Solar system

Setting from [Hairer et al., 2006] ∆t = 8δt = 10−3, ms = 1, m1 = m2 = 10−2. H(p, q) = 1

2

pT

s ps

ms + pT

1 p1

m1 + pT

2 p2

m2

−

msm1 qs−q1 − msm2 qs−q2 − m1m2 q1−q2

5
4
3
2
1

1 2 3 4

5
4
3
2
1

1 2 3 4 "Planete_2.txt" u 2:3 "Planete_1.txt" u 2:3 "Soleil.txt" u 2:3

1x10-14
5x10-15

5x10-15 1x10-14 10 20 30 40 50 60 70 80 90 Relative error % Time (s) Relative errror in energy (in %) "System.txt" u 1:(($2+0.006083333333)/0.006083333333*100)

Marazzato et al. Energy conservation 05/29/2017 18 / 19