Towards Geometric Finite-Element Particle-in-Cell Methods for Gyrokinetics
Michael Kraus1,2 (
[email protected]) In collaboration with Eric Sonnendrücker1,2 , Katharina Kormann1,2 , Omar Maj1 , Hiroaki Yoshimura3 , Philip Morrison4 , Joshua Burby5 and Leland Ellison6 1 2 3 4 5 6
Max-Planck-Institut für Plasmaphysik, Garching Technische Universität München, Zentrum Mathematik Waseda University, School of Science and Engineering University of Texas at Austin, Institute for Fusion Studies Courant Institute of Mathematical Sciences, New York University Princeton Plasma Physics Laboratory
Outline 1
Geometric Numerical Integration
2
Discrete Differential Forms
3
Discrete Poisson Brackets
4
Variational Integrators
5
Summary and Outlook
6
Guiding Centre Dynamics
Structure-Preserving Discretisation geometric structure: global property of differential equations, which can be defined independently of particular coordinate representations 1 e.g., topology, conservation laws, symmetries, constraints, identities preservation of geometric properties is advantageous for numerical stability and crucial for long time simulations bounds global error growth and reduces numerical artifacts various families Lie group integrators, discrete Euler-Poincaré methods integral preserving schemes, discrete variational derivative method, discrete gradients discrete differential forms and mimetic methods symplectic and multisymplectic methods variational and Poisson integrators 1
Christiansen, Munthe-Kaas, Owren: Topics in Structure-Preserving Discretization, Acta Numerica 2011
Geometric Structures of the Vlasov-Maxwell System Vlasov equation in Lagrangian coordinates ˙ s = Vs , X
˙ s = es E(t, Xs ) + e Vs × B(t, Xs ) V c
fs (t, Xs (t), Vs (t)) = fs (Xs (0), Vs (0))
Maxwell’s equations in Eulerian coordinates ∂E = ∇ × B − J, ∂t
∇ · E = −ρ,
∂B = −∇ × E, ∂t
∇ · B = 0,
ρ(t, x) =
∑
∫ es
s
J(t, x) =
∑
dv fs (t, x, v), ∫
es
dv fs (t, x, v) v
s
the spaces of electrodynamics have a deRham complex structure Poisson structure (antisymmetric bracket, Jacobi identity) variational structure (Hamilton’s action principle) energy, momentum and charge conservation (Noether theorem)
Differential Forms mathematical language of vector analysis too limited to provide an intuitive description of electrodynamics (only two types of objects) ϕ: scalar field E: field intensity (integrated along a one-dimensional path) B: flux density (integrated over a two-dimensional surface) ρ: charge density (integrated over a three-dimensional volume)
tensor analysis is concise and general, but very abstract subset of tensor analysis: calculus of differential forms, combining much of the generality of tensors with the simplicity of vectors in three dimensional space: four types of forms 0-forms 1-forms 2-forms 3-forms
Λ0 : Λ1 : Λ2 : Λ3 :
scalar quantities (functions) vectorial quantities (line elements) vectorial quantities (surface elements) scalar quantities (volume elements)
Maxwell’s Equations and the deRham Complex electromagnetic fields as differential forms ϕ ∈ Λ0 (Ω),
A, E ∈ Λ1 (Ω),
B, J ∈ Λ2 (Ω),
ρ ∈ Λ3 (Ω)
exterior derivative d : Λi → Λi+1 generalises grad, curl, div the spaces of Maxwell’s equations build a deRham sequence for geometers d
d
d
0 → Λ0 (Ω) − → Λ1 (Ω) − → Λ2 (Ω) − → Λ3 (Ω) → 0 for analysts grad
curl
div
0 → H1 (Ω) −−−→ H(curl, Ω) −−→ H(div, Ω) −−→ L2 (Ω) → 0 exactness: the range of di : Λi → Λi+1 equals the kernel of di+1 dd• = 0,
curl grad • = 0,
div curl • = 0
Discrete deRham Complex discrete deRham complex d
d
d
0 → Λ0 (Ω) − → Λ1 (Ω) − → Λ2 (Ω) − → Λ3 (Ω) → 0 ↓ πh0
↓ πh1 d
↓ πh2 d
↓ πh3 d
0 → Λ0h (Ω) − → Λ1h (Ω) − → Λ2h (Ω) − → Λ3h (Ω) → 0 the discrete spaces Λih ⊂ Λi are finite element spaces of differential forms, building an exact deRham sequence compatibility: projections commuted with the exterior derivative by translating geometrical and topological tools, which are used in the analysis of stability and well-posedness of PDEs, to the discrete level one can show that exactness and compatibility guarantee stability2 2
Arnold, Falk, Winther: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155, 2006. Arnold, Falk, Winther: Finite Element Exterior Calculus: From Hodge Theory to Numerical Stability, Bulletin of the AMS 47, 281-354, 2010.
Spline Differential Forms electrostatic potential ϕh ∈ Λ0h (Ω) ∑ ϕh (t, x) = ϕi,j,k (t) Λ0i,j,k (x) i,j,k
zero-form basis Λ0h (Ω)
{
= span
}
Spi (x1 ) Spj (x2 ) Spk (x3 )
the i-th basic splines (B-spline) of order p is defined by Spi (x) = where S1i (x)
xi+p − x p−1 x − xi Sp−1 (x) + S (x) i xi+p−1 − xi xi+p − xi+1 i+1 {
=
1 x ∈ [xj , xj+1 ) 0 else
Spline Differential Forms electric field intensity Eh ∈ Λ1h (Ω) ∑ ei,j,k (t) Λ1i,j,k (x) Eh (t, x) = i,j,k
one-form basis
{ Dp (x1 ) Sp (x2 ) Sp (x3 ) i j k , Λ1h (Ω) = span 0 0 0 Spi (x1 ) Dpj (x2 ) Spk (x3 ) , 0 } 0 0 Spi (x1 ) Spj (x2 ) Dpk (x3 )
spline differentials d p S (x) = Dpi (x) − Dpi+1 (x), dx i
Dpi (x)
Sp−1 (x) =p i xi+p − xi
Spline Differential Forms magnetic flux density Bh ∈ Λ2h (Ω) ∑ bi,j,k (t) Λ2i,j,k (x) Bh (t, x) = i,j,k
two-form basis
{ Sp (x1 ) Dp (x2 ) Dp (x3 ) i j k , Λ2h (Ω) = span 0 0 0 Dpi (x1 ) Spj (x2 ) Dpk (x3 ) , 0 } 0 0 Dpi (x1 ) Dpj (x2 ) Spk (x3 )
spline differentials d p S (x) = Dpi (x) − Dpi+1 (x), dx i
Dpi (x)
Sp−1 (x) =p i xi+p − xi
Spline Differential Forms charge density ρh ∈ Λ3h (Ω) ρh (t, x) =
∑
ρi,j,k (t) Λ3i,j,k (x)
i,j,k
three-form basis
{ } Λ3h (Ω) = span Dpi (x1 ) Dpj (x2 ) Dpk (x3 )
spline differentials d p S (x) = Dpi (x) − Dpi+1 (x), dx i
Dpi (x) = p
Sp−1 (x) i xi+p − xi
Morrison-Marsden-Weinstein Bracket Vlasov-Maxwell noncanonical Hamiltonian structure ] ∫ ( ) δF δG ∂ δF δG ∂ δG δF {F, G}[f , E, B] = dx dv f , + dx dv f · − · δf δf ∂v δf δE ∂v δf δE ( ( ) ∫ ) ∫ ∂ δF ∂ δG δF δG δG δF + dx dv f B· × + dx ·∇× − ·∇× ∂v δf ∂v δf δE δB δE δB ∫
[
Hamiltonian: sum of the kinetic energy of the particles, the electrostatic field energy and the magnetic field energy ∫ ∫ ) 1 1 ( 2 H= |v| f (x, v) dx dv + |E(x)|2 + |B(x)|2 dx 2 2 time evolution of any functional F[f , E, B] d F[f , E, B] = {F, H} dt
Discretisation of the Vlasov-Maxwell Poisson System finite-dimensional representation of the fields f , E, B discretisation of the brackets [ ] ∫ ( ) ∫ δF δG ∂ δF δG ∂ δG δF {F, G}[f , E, B] = dx dv f , + dx dv f · − · δf δf ∂v δf δE ∂v δf δE ( ) ∫ ( ) ∫ ∂ δF ∂ δG δF δG δG δF + dx + dx dv f B· × ·∇× − ·∇× ∂v δf ∂v δf δE δB δE δB
discretisation of functionals ∫ ∫ ) 1 1 ( 2 H= |v| f (x, v) dx dv + |E(x)|2 + |B(x)|2 dx 2 2 time discretisation d F[f , E, B] = {F, H} dt
Discretisation of the Fields particle-like distribution function for Np particles labeled by a, fh (x, v, t) =
Np ∑
( ) ( ) wa δ x − xa (t) δ v − va (t) ,
a=1
with weights wa , particle positions xa and particle velocities va 1-form and 2-form basis functions (vector-valued) 1,1 2,1 Λα (x) Λα (x) 1,2 2,2 1 2 Λα (x) = Λα (x) = Λα (x) , Λα (x) Λ1,3 Λα2,3 (x) α (x) semi-discrete electric field Eh and magnetic field Bh ∑ ∑ Eh (t, x) = eα (t) Λ1α (x), Bh (t, x) = bα (t) Λ2α (x) α∈Z3
with coefficient vectors e and b
α∈Z3
Discretisation of the Distribution Function functionals of the distribution function, F[f ], restricted to particle-like distribution functions, Np ∑ ( ) ( ) fh (x, v, t) = wa δ x − xa (t) δ v − va (t) , a=1
become functions of the particle phasespace trajectories, ˆ a , va ) F[fh ] = F(x replace functional derivatives with partial derivatives ˆ ˆ ∂F ∂ δF ∂ δF ∂F = wa = wa and ∂xa ∂x δf (xa ,va ) ∂va ∂v δf (xa ,va ) rewrite kinetic bracket as semi-discrete particle bracket [ ] ∑ ( ) ∫ ∂ δF ∂ δG δF δG ∂ δF ∂ δG = wa dx dv f , · − · δf δf ∂x δf ∂v δf ∂v δf ∂x δf (xa ,va ) a ˆ ˆ ∂F ∑ 1 ( ∂F ˆ ∂G ˆ) ∂G = · − · wa ∂xa ∂va ∂xa ∂va a
Discretisation of the Electrodynamic Fields semi-discrete electric field Eh and magnetic field Bh ∑ ∑ eα (t) Λ1α (x), Bh (x) = bα (t) Λ2α (x) Eh (x) = α∈Z3
α∈Z3
functionals F[E] and F[B], restricted to the semi-discrete fields Eh and ˆ ˆ Bh , can be considered as functions F(e) and F(b) of the finite element coefficients ˆ F[Eh ] = F(e),
ˆ F[Bh ] = F(b)
functional derivatives of linear and quadratic functionals F[Eh ] and ˆ ˆ F[Bh ] can be replaced with partial derivatives of F(e) and F(b), ˆ δF[Eh ] ∑ ∂ F(e) = (M1−1 )αβ Λ1β (x), δE ∂eα α,β
with mass matrices ∫ (M1 )αβ = dx Λ1α (x) Λ1β (x),
ˆ δF[Bh ] ∑ ∂ F(b) 2 = (M−1 2 )αβ Λβ (x) δB ∂bα α,β
∫ (M2 )αβ =
dx Λ2α (x) Λ2β (x)
Semi-Discrete Poisson Bracket semi-discrete Poisson bracket ( ) ˆ ˆ ∂F ∑ 1 ˆ ∂G ˆ ∂ F ∂ G ˆ d [xa , va , eα , bα ] = ˆ G} {F, · − · wa ∂xa ∂va ∂xa ∂va a ( ) ˆ ˆ ∂F ∑ ∑ ∂F ˆ ∂G ˆ ∂G −1 −1 1 1 + · (M1 )αβ Λβ (xa ) − · (M1 )αβ Λβ (xa ) ∂va ∂eα ∂va ∂eα a α,β ( ) ˆ ∑∑ ˆ 1 ∂F ∂G 2 + bα (t) Λα (xa ) · × wa ∂va ∂va a α ( ) ˆ ˆ ∑ ˆ ˆ ∂F ∂G ∂G ∂F −1 −1 −1 −1 T T + (M1 )αβ Rβη (M2 )ηγ − (M1 )αβ Rβη (M2 )ηγ ∂eα ∂bγ ∂eα ∂bγ α,β,γ,η
rotation matrix (decomposable into mass matrix M2 and incidence matrix I) ∫ Rαβ = dx Λ2α (x) · ∇ × Λ1β (x), R = M2 I
Semi-Discrete Poisson System semi-discrete equations of motion ˆ d, ˆ d, x˙ p = {xp , H} v˙ p = {vp , H}
ˆ d b˙ = {b, H}
ˆ d, e˙ = {e, H}
with discrete Hamiltonian ∑ 1 ∑ 1 ∑ 2 ˆ=1 H |va | wa + eα (t) (M1 )αβ eβ (t) + bα (t) (M2 )αβ bβ (t) 2 a 2 2 α,β
α,β
ˆ Poisson system: y˙ = P(y) ∇H(y) with y = (xp , vp , e, b) ˆ 0 M−1 0 0 ∂ H/∂xp xp p −1 ˆ b T (t, xp ) M−1 d B (Λ1 (xp ))T M−1 0 p 1 h vp = −Mp ∂ H/∂vp −1 −1 1 T ˆ 0 −M1 (Λ (xp )) 0 M1 I dt e ∂ H/∂e −1 ˆ b 0 0 −IM1 0 ∂ H/∂b P is anti-symmetric and satisfies the Jacobi identity if div Bh = 0 and ) ∂Λ1ki (xa ) ∂Λ1kj (xa ) ∑ ( b 2 Λα (xa ) Iαk for all a, i, j, k, = − j i ∂xa ij ∂xa α
recursion relation for splines, evaluated at all particle positions
Splitting Methods Hamiltonian splitting3 ˆ=H ˆp + H ˆp + H ˆp + H ˆE + H ˆB H 1 2 3 with ˆp = H i
1 2
(vip )T Mp vip ,
ˆE = H
1 2
eT M1 e,
ˆB = H
1 2
bT M2 b
split semi-discrete Vlasov-Maxwell equations into five subsystems ˆ p }d , y˙ = {y, H i
ˆ E }d , y˙ = {y, H
ˆ B }d y˙ = {y, H
the exact solution of each subsystem constitutes a Poisson map ∫t
∫t ˆ p }d dt, φt,E (y0 ) = y0 + {y, H i
φt,pi (y0 ) = y0 + 0 3
ˆ E }d dt, ... {y, H 0
Crouseilles, Einkemmer, Faou. Hamiltonian splitting for the Vlasov– Maxwell equations. Journal of Computational Physics 283, 224–240, 2015. Qin, He, Zhang, Liu, Xiao, Wang. Comment on “Hamiltonian splitting for the Vlasov–Maxwell equations”. arXiv:1504.07785, 2015. He, Qin, Sun, Xiao, Zhang, Liu. Hamiltonian integration methods for Vlasov–Maxwell equations. arXiv:1505.06076, 2015.
Splitting Methods Hamiltonian splitting ˆ=H ˆp + H ˆp + H ˆp + H ˆE + H ˆB H 1 2 3 compositions of Poisson maps are themselves Poisson maps construct Poisson structure preserving integration methods by composition of exact solutions of the subsystems first order time integrator: Lie-Trotter composition Ψh = φh,E ◦ φh,B ◦ φh,p1 ◦ φh,p2 ◦ φh,p3 second order time integrator: symmetric composition Ψh = φh/2,E ◦ φh/2,B ◦ φh/2,p1 ◦ φh/2,p2 ◦ φh,p3 ◦ φh/2,p2 ◦ φh/2,p1 ◦ φh/2,B ◦ φh/2,E
Variational Integrators systematic way to derive structure-preserving discretisation schemes for Lagrangian and Hamiltonian dynamical systems preserved structures discrete symplectic structure good long-time energy behaviour (bounded error) discrete momenta related to symmetries of the discrete Lagrangian discrete Noether theorem provides discrete symmetry condition and discrete form of conservation laws idea discretisation of the Lagrangian and Hamilton’s principle of stationary action application of the discrete action principle to the discrete Lagrangian to obtain discrete Euler-Lagrange equations directly allow for straight-forward derivation of integrators for coupled systems (e.g., coupling of particles and fields for particle-in-cell schemes)
Continuous and Discrete Action Principle q(t)
action: functional of a curve q(t) ∫T A[q(t)] =
( ) L q(t), q˙ (t) dt
q0
varied curves
qT
0
Hamilton’s principle of stationary action: among all possible trajectories the system follows the one that makes the action integral A stationary ( ) ∂L d ∂L δA = 0 → (q, q˙ ) − (q, q˙ ) = 0 ∂q dt ∂ q˙ approximate Lagrangian with finite differences and quadrature formula ( ) qn + qn+1 qn+1 − qn Ld (qn , qn+1 ) = h L , 2 h stationarity of the discrete action: discrete Euler-Lagrange equations δAd = δ
N−1 ∑ n=0
Ld (qn , qn+1 ) = 0
→ D2 Ld (qn−1 , qn ) + D1 Ld (qn , qn+1 ) = 0
Continuous and Discrete Action Principle {qk }
action: functional of a curve q(t) ∫T A[q(t)] =
( ) L q(t), q˙ (t) dt
q0
varied discrete curves
qN
0
Hamilton’s principle of stationary action: among all possible trajectories the system follows the one that makes the action integral A stationary ( ) ∂L d ∂L δA = 0 → (q, q˙ ) − (q, q˙ ) = 0 ∂q dt ∂ q˙ approximate Lagrangian with finite differences and quadrature formula ( ) qn + qn+1 qn+1 − qn Ld (qn , qn+1 ) = h L , 2 h stationarity of the discrete action: discrete Euler-Lagrange equations δAd = δ
N−1 ∑ n=0
Ld (qn , qn+1 ) = 0
→ D2 Ld (qn−1 , qn ) + D1 Ld (qn , qn+1 ) = 0
Continuous Action Principle for Vlasov-Maxwell variations of the action A=
∑∫
∫
dt
∫
dX
[ ] [ ] ˙ − 1 m |V|2 + es ϕ(t, X) dV fs (t, X, V) mV + es A(t, X) · X 2
s
+
1 2
∫
∫ dt
2 [ ] ∂A 2 dx ∇ϕ(t, x) − (t, x) − |∇ × A(t, x)| ∂t
lead to the same equations of motion as the Poisson bracket upon E = −∇ϕ −
∂A , ∂t
B=∇×A
the Vlasov-Maxwell action is (weakly) gauge invariant [ ] [ ] A x, v, A + ∇ψ, ϕ = A x, v, A, ϕ + boundary terms corresponding conservation law: charge conservation ∂ρ +∇·j=0 ∂t
Semi-Discrete Action Principle for Vlasov-Maxwell semi-discrete action (particles, splines, time-continuous) 1 ∑ Ah = N a
∫T
[ ] [ ] 2 dt ma va (t) + ea Ah (t, xa (t)) · x˙ a (t) − 21 ma |va (t)| + ea ϕh (t, xa (t))
0
1 + 2
∫T
∫ dt
2 [ ] ∂Ah 2 dx ∇ϕh (t, x) − (t, x) − |∇ × Ah (t, x)| ∂t
0
same equations of motion as the semi-discrete Poisson bracket, upon ∂Ah , Bh = ∇ × Ah ∂t the semi-discrete action is still (weakly) gauge invariant [ ] [ ] Ah x, v, Ah + ∇ψh , ϕh = Ah x, v, Ah , ϕh + boundary terms Eh = −∇ϕh −
corresponding conservation law: charge conservation ∂ρh + ∇ · jh = 0 ∂t
Gauge Invariance of the Discrete Action time discretisation (e.g., Lagrange polynomials) s ∑ yh (t) [tn ,tn+1 ] = Yn,m φm n (t),
( ) m φm (t − tn )/(tn+1 − tn ) n (t) = l
m=1
variations of fully discrete action t∫n+1
t∫n+1
dt Ah (t, xh (t)) · x˙ h (t) =
δ tn
dt t∫n+1
dt t∫n+1
dt
=
Ah (t, xh (t)) · δXn,m φ˙ m n (t) + ...
s ∑
δXn,m · ∇Ah (t, xh (t)) · Xn,l φ˙ ln (t) φm n (t)
l,m=1
tn t∫n+1
dt tn
s ∑ m=1
tn
−
δXn,m · ∇Ah (t, xh (t)) · Xn,l φ˙ ln (t) φm n (t)
l,m=1
tn
+
s ∑
s ∑ l,m=1
Xn,l · ∇Ah (t, xh (t)) · δXn,m φ˙ ln (t) φm n (t) + ...
Gauge Invariance of the Discrete Action time discretisation (e.g., Lagrange polynomials) s ∑ yh (t) [tn ,tn+1 ] = Yn,m φm n (t),
( ) m φm (t − tn )/(tn+1 − tn ) n (t) = l
m=1
variations of fully discrete action t∫n+1
t∫n+1
dt Ah (t, xh (t)) · x˙ h (t) =
δ tn
dt t∫n+1
dt t∫n+1
dt tn
s ∑
Ah (t, xh (t)) · δXn,m φ˙ m n (t) + ...
m=1
tn
=
δXn,m · ∇Ah (t, xh (t)) · Xn,l φ˙ ln (t) φm n (t)
l,m=1
tn
+
s ∑
s ∑ l,m=1
δXn,m · Bh (t, xh (t)) · Xn,l φ˙ ln (t) φm n (t) + ...
Summary and Outlook discrete electrodynamics discrete differential forms (discrete exterior calculus, mimetic discretisation): splines, mixed finite elements, mimetic spectral elements, virtual elements stability: exactness and compatibility of the finite element deRham complex
discrete Poisson brackets and variational integrators Poisson structure is retained at the semi-discrete level splitting methods or variational integrators for time integration gauge invariance guarantees charge conservation
variational integrators for degenerate Lagrangians multi-step methods featuring parasitic modes or one-step methods for an extended system drifting off the constraint submanifold projection of variational integrators for the unconstrained extended system very good long-time stability and conservation of energy and momentum maps
ongoing and future work application to the Hamiltonian Gyrokinetic Vlasov–Maxwell System (Burby et al., Physics Letters A, 379, pp. 2073–2077, 2015) extension towards discrete metriplectic and double brackets for dissipative systems
Guiding Centre Dynamics charged particle phasespace Lagrangian L(x, x˙ , v, v˙ ) = (A(x) + v) · x˙ − 12 v2 coordinate transformation (xi , vi )
→
(Xi , θ, u, µ)
with ρ = b × v⊥ / |B| and ˙ u = b · X,
v⊥ = v − ub,
µ = v2⊥ /2 |B| ,
B = ∇ × A,
b = B/ |B|
so that the Lagrangian becomes ( ) ( ) ˙ + ρ˙ + µθ˙ − 1 u2 − µB(X + ρ) L(q, q˙ ) = A(X + ρ) + ub(X + ρ) · X 2 strong magnetic fields: neglect finite gyroradius effects guiding centre Lagrangian (q = (Xi , u) and µ a parameter) ˙ − 1 u2 − µB(X) L(q, q˙ ) = (A(X) + ub(X)) · X 2
Variational Guiding Centre Integrators guiding centre Lagrangian ˙ − 1 u2 − µB(X), L(q, q˙ ) = (A(X) + ub(X)) · X 2
q = (Xi , u)
is degenerate (linear in velocities), that is ∂2L =0 ∂ q˙ i ∂ q˙ j and therefore leads to first order ordinary differential equations straight-forward application of the discrete action principle leads to multi-step variational integrators D2 Ld (qk−1 , qk ) + D1 Ld (qk , qk+1 ) = 0
we need two sets of initial data even though we have first order ODEs support parasitic modes, not long-time stable
Variational Guiding Centre Integrators use discrete Legendre transform to obtain position-momentum form pk
= −D1 Ld (qk , qk+1 ),
pk+1 =
D2 Ld (qk , qk+1 )
use continuous Legendre transform to obtain the second initial condition p0 =
∂L (q0 ) = α(q0 ), ∂ q˙
α(q) = A(X) + u b(X)
one-step method for an extended dynamical system (p, q) whose dynamics is constrained to a subspace defined by ϕ(p, q) = p − α(q) = 0
(Dirac constraint)
variational integrators will in general not satisfy the constraint geometric interpretation for appearance of parasitic modes
Passing Particle 2D, h =
Explicit Runge-Kutta-4
τb 50 ,
nb = 10.000
Variational Runge-Kutta-2
symmetric method in the second step, and the same vector µ in t projection steps.
Orthogonal Projection
M y1 y0
y2
y3
y0
Fig. 4.1. Standard projection (left picture) compared to symmetric proj
orthogonal symplectic projection of primary constraint, z = (p, q)
Existence of the Numerical Solution. The vector µ and the num ˜zn+1 = Ψh (zn ) apply variational one-step method tion yn+1 are implicitly defined by −1 T zn+1 = ˜zn+1 + Ω ∇ϕ (zn+1 )λn+1 project #on constraint submanifold $ " yn+1 − Φh yn + G(yn )T µ − G(yn+1 ) 0 = ϕ(zn+1 ) F (h, yn+1 , µ) = g(yn+1 ) with Ω the canonical symplectic matrix Since F (0, yn , 0) = 0 and since ( ) " 0 −1 # Ω= ∂F I −2G(yn )T 1 0 0, yn , 0) = G(yn ) 0 ∂(yn+1 , µ)
Passing and Trapped Particle 2D, h =
Variational Runge-Kutta 2
τb 50 ,
nb = 25.000
Orthogonal Projection
method in the second step, and the same vector µ in the perturbation and steps. Symmetric Projection M y1
y2
y3
M y0
y3
andard projection (left picture) compared to symmetric projection (right)
symmetric symplectic projection of primary constraint, z = (p, q)
T of the Numerical the numerical approxima- perturb initial data ˜zk = Solution. zk + Ω−1The ∇ϕvector (zk )µλand k+1 are implicitly defined by ˜zk+1 = Ψh (˜z#k ) apply variational one-step method $ " % yn+1 − Φh yn +−1 G(yn )TTµ − G(yn+1 )T µ = 0. on(4.2) zk+1 + Ω ∇ϕ (zk+1 )λk+1 project constraint submanifold n+1 , µ) =zk+1 = ˜ g(yn+1 )
0 = ϕ(zk+1 ).
, yn , 0) = 0 and since
" % with∂FΩ the matrix T # canonical symplectic I −2G(y n) 0, yn , 0) = (4.3) 0 ) G(yn ) ∂(yn+1 , µ) ( 0 −1 full rank), an application of the implicit funce (provided Ω that=G(y1 n ) has 0
em proves the existence of the numerical solution for sufficiently small h. The simple structure of the matrix (4.3) can also be exploited for an
Passing and Trapped Particle 2D, h =
Explicit RK4
Variational RK2 (1 stage)
τb 50 ,
Orthogonal Projection
nb = 25.000
Symmetric Projection
Passing and Trapped Particle 2D, h =
Explicit RK4
Variational RK4 (2 stage)
τb 50 ,
Orthogonal Projection
nb = 25.000
Symmetric Projection
Passing Particle 4D, h =
τb 50 ,
nb = 106 , nt = 5 × 107
Energy (Relative Error)
Toroidal Momentum (Relative Error)
Variational Runge-Kutta, 2 stages, order 4, symmetric projection
Passing Particle 4D, h =
τb 50 ,
nb = 106 , nt = 5 × 107 0 1e 3 1 2 3 4 5 6 0
200000
400000
600000
800000
1000000
800000
1000000
Energy (Relative Error)
0 1e 1 1 2 3 4 5 6 0
200000
400000
600000
Toroidal Momentum (Relative Error)
Explicit Runge-Kutta, order 4
Implementation in Gyrokinetic Application Codes first results from conjugate symplectic particle integrators in NEMORB
Explicit RK4
Conjugate Symplectic RK4
Runtime increased by 20% ∼ 30% but no dissipation of energy
