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

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

Summary Hamiltonian systems 1 Nonlinear Hamiltonian wave equations 2 Local time-stepping 3 Marazzato et al. Energy conservation 05/29/2017 2 / 19

Hamiltonian systems System of N particles Positions q ∈ R 3 N , mass-matrix M ∈ R 3 N × 3 N and velocity momenta p = Mv ∈ R 3 N . q 3 , p 3 F 32 F ext F 23 F 31 q 2 , p 2 F 31 F 13 q 1 , p 1 F 32 Marazzato et al. Energy conservation 05/29/2017 3 / 19

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 2 p T M − 1 p + V ( q ) Equations of motion : q = M − 1 p � ˙ p = −∇ V ( q ) ˙ Marazzato et al. Energy conservation 05/29/2017 4 / 19

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. p n +1 / 2 = p n − ∆ t  2 ∇ V ( q n )     q n +1 = q n + ∆ t M − 1 p n +1 / 2  p n +1 = p n +1 / 2 − ∆ t  2 ∇ V ( q n +1 )   Marazzato et al. Energy conservation 05/29/2017 5 / 19

Explicit time-integration scheme Free-flight idea developed in [Mariotti, 2015]. Mid-point integration rule for the momentum jumps. q ( t ) = q n + 1  m p n +1 / 2 ( t − t n )    � t n +1 1 [ p ] n +1 + [ p ] n � �  = − ∇ V ( q ( t )) dt   2 t n q n − 1 , t n − 1 q n +1 , t n +1 q n , t n p n +1 / 2 [ p ] n = p n +1 / 2 − p n − 1 / 2 p n − 1 / 2 Marazzato et al. Energy conservation 05/29/2017 6 / 19

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: H n = V ( q n ) + 1 p n − 1 / 2 � T M − 1 p n +1 / 2 , � ˜ ∀ n ∈ N 2 Moreover, with a proper initialisation: H n = H 0 , ˜ ∀ n ∈ N Marazzato et al. Energy conservation 05/29/2017 7 / 19

Proposition (Energetic CFL condition) H n Hamiltonian at t n . � | M − 1 / 2 [ p ] n | ≤ 8 β | H 0 | , ∀ n ∈ N | H n − H 0 | ≤ β | H 0 | , = ⇒ ∀ n ∈ N for β ∈ [0 , 1] (energetic stability). Proposition (CFL condition (absolute stability)) The scheme is stable as long as, for every particle : � m ∆ t n ≤ 2 |∇ 2 V ( q n ) | , ∀ n ∈ N Marazzato et al. Energy conservation 05/29/2017 8 / 19

Fermi-Pasta-Ulam 2 i )+ ω 2 H ( p , q ) = 1 � m i =1 ( p 2 2 i − 1 + p 2 � m i =1 ( q 2 i − q 2 i − 1 ) 2 + � m i =0 ( q 2 i +1 − q 2 i ) 4 2 4 I j ( x m + j , y m + j ) = 1 y 2 m + j + ω 2 x 2 � � energy of the j th stiff spring with: m + j 2 √ √ x i = ( q 2 i + q 2 i − 1 ) / 2 , y i = ( p 2 i + p 2 i − 1 ) / 2 , x m + i = √ √ ( q 2 i − q 2 i − 1 ) / 2 , y m + i = ( p 2 i − p 2 i − 1 ) / 2 Total oscillatory energy I = I 1 + I 2 + ... + I m 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

Summary Hamiltonian systems 1 Nonlinear Hamiltonian wave equations 2 Local time-stepping 3 Marazzato et al. Energy conservation 05/29/2017 10 / 19

Motivating example The setting comes from [Chabassier and Joly, 2010] Ω = [0 , 1] and E ( C 2 , strictly convex and growth conditions on ∇E ). Find u : Ω × R + → R 2 such that : ∂ 2 tt u − ∂ x ( ∇E ( ∂ x u )) = 0 BC : u (0 , t ) = 0, u (1 , t ) = 0, IC : u ( x , 0) = u 0 ( x ) , ∂ t u ( x , 0) = u 1 ( x ) � 2 : � H 1 Variational formulation in V = 0 (Ω) d 2 �� � � u · v + ∇E ( ∂ x u ) · ∂ x v = 0 , ∀ v ∈ V , ∀ t > 0 dt 2 Ω Ω {V h , h > 0 } with conforming Lagrange P k finite elements Marazzato et al. Energy conservation 05/29/2017 11 / 19

Time-integration setting Semi-discretization in space by conforming FEM Basis ( ϕ ) i =1 ,.., 2 N for displacements of degrees of freedom The Hamiltonian is written : H ( q , p ) = 1 2 p T M − 1 p + V ( q ) with: � 2 N � � � V ( q ) = E ( q ) i ∂ x ϕ i Ω i =1 Marazzato et al. Energy conservation 05/29/2017 12 / 19

Results for linear wave equations E ( u , v ) = u 2 + v 2 with ( u , v ) displacement vector of the string. 2 Marazzato et al. Energy conservation 05/29/2017 13 / 19

Results for nonlinear wave equations E ( u , v ) = u 2 + v 2 �� � (1 + u ) 2 + v 2 − (1 + u ) − α with 2 0 ≤ α = 0 . 99 < 1 characteristic of the nonlinearity. � t n +1 mid-point quadrature on ∇ V ( q ( t )) dt t n Marazzato et al. Energy conservation 05/29/2017 14 / 19

Results for nonlinear wave equations 2 Relative energy evolution (in %) ~H^n 0.8 H^n 0.6 0.4 E 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 t Marazzato et al. Energy conservation 05/29/2017 15 / 19

Summary Hamiltonian systems 1 Nonlinear Hamiltonian wave equations 2 Local time-stepping 3 Marazzato et al. Energy conservation 05/29/2017 16 / 19

Local time-stepping Local time-stepping H n = H 0 , ∀ n ∈ N without any condition on Theorem 1 ensures that ˜ ∆ 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 δ t = N ∈ N Divide the particles into a "fast" group whose dynamics will be integrated at every δ t and a "slow" group whose dynamics will be integrated only every ∆ t . When the dynamics of all the particles has been integrated over H n is conserved as stated in Theorem 1. N δ t = ∆ t , the energy ˜ Marazzato et al. Energy conservation 05/29/2017 17 / 19

Slow-fast Solar system Setting from [Hairer et al., 2006] ∆ t = 8 δ t = 10 − 3 , m s = 1, m 1 = m 2 = 10 − 2 . � � m s + p T m 1 + p T p T s p s 1 p 1 2 p 2 H ( p , q ) = 1 m s m 1 m s m 2 m 1 m 2 − � q s − q 1 � − � q s − q 2 � − 2 � q 1 − q 2 � m 2 4 Relative errror in energy (in %) "Planete_2.txt" u 2:3 "Planete_1.txt" u 2:3 1x10 -14 "Soleil.txt" u 2:3 "System.txt" u 1:(($2+0.006083333333)/0.006083333333*100) 3 2 5x10 -15 1 Relative error % 0 0 -1 -2 -5x10 -15 -3 -4 -1x10 -14 10 20 30 40 50 60 70 80 90 -5 -5 -4 -3 -2 -1 0 1 2 3 4 Time (s) Marazzato et al. Energy conservation 05/29/2017 18 / 19

Conclusions Nonlinear wave equations Explicit integration Modified energy conservation Hamiltonian systems Easy variable time-stepping Rigorous slow-fast integration Marazzato et al. Energy conservation 05/29/2017 19 / 19