Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Volume-preserving algorithms for charged particle dynamics Yang He University of Science and Technology of China March. 2015, Hefei Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Single particle model Newton–lorentz equation x˙ = v, p˙ = q (E(x, t) + v × B(x, t)) p = mv in non-relativistic case, p= p m0 v 1 − v 2 /c2 in relativistic case. Numerical algorithms Due to the multi-scale nature of plasma dynamics, numerical algorithms need to have long term accuracy. Geometric integrators: to preserve conservative volume in phase space: det ∂(x(t),p(t)) ∂(x0 ,p0 ) constants of motion (energy, momentum, etc.), etc. Yang He Volume-preserving Lorentz force equation = 1, Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Splitting method Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Two second order symmetric methods F2 F1 1 2 3 RG2h := φF ◦ φF ◦ φF h ◦ φh/2 ◦ φh/2 ; h/2 h/2 xk+ 1 = xk + 2 h vk , 2 F2 F1 1 2 3 CG2h := φF ◦φF ◦ΦF h ◦φh/2 ◦φh/2 ; h/2 h/2 xk+ 1 = xk + 2 h vk , 2 hq p− = pk + E(xk+ 1 , tk ), 2 2 h + ˆ , t ) p− , p = exp B(x 1 1 k+ 2 k+ 2 m0 γ(p− ) hq pk+1 = p+ + E(xk+ 1 , tk+1 ), 2 2 hq E(xk+ 1 , tk ), 2 2 h p+ + p− + ˆ p = × B(x , tk+ 1 ), k+ 1 2 2 2m0 γ(p− ) hq pk+1 = p+ + E(xk+ 1 , tk+1 ), 2 2 vk+1 = pk+1 /(m0 γ(pk+1 )), vk+1 = pk+1 /(m0 γ(pk+1 )), xk+1 = xk+ 1 + 2 h vk+1 . 2 r where γ(p) = 1+ p2 2 m2 0c p− = pk + xk+1 = xk+ 1 + 2 h vk+1 . 2 in relativistic case, and γ(p) is replaced by 1 in non-relativistic case. Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Contents of this talk Analyze the performances of the two volume-preserving methods, in linear stability, accuracy, cost, preservation of the invariants. Optimizing the methods by designing higher order volume-preserving methods. Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Linear stability: how to choose the proper step size Normalized relativistic Lorentz force equation x˙ = p , γ(p) p˙ = E(x) + where v = p/γ(p), p × B(x), γ(p) γ(p) = p 1 + p2 The test linearized equation x˙ = γ0 = √ 1 p, γ0 p˙ = Λx + ω p×b γ0 1 + s2 > 1, where 0 ≤ s ≤ p(t). Λx = −(λ2x xex + λ2y yey + λ2z zez ), λ > 0. b = [0, 0, 1]. λ2x corresponds to kx E(x0 ), ω corresponds to B(x0 ). Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Linear stability The test equation x˙ = 1 p, γ0 p˙ = −λ2 x + where = ±1, x = [x, y], v = [vx , vy ], J = 0 1 −1 0 ω Jp γ0 , λ2 ∼ kE(x0 ), ω ∼ B(x0 ). The numerical solution γ0 x hp !n 2 = [Mm ((hλ) , hω)] n γ0 x(0) ! hp(0) Proposition For the n-dimensional source-free system, a volume-preserving method is linear stable iff ρ(Mm ) = 1 and Mm is diagonalizable. Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Linear stability domain (a) RG2h (b) CG2h 0) In relativistic case: λ2 ∼ k E(x ,ω∼ B0 c B(x0 ) , B0 γ0 = √ 1 + s2 , 0 ≤ s ≤ p(t) . m0 c 0) 0) In non-relativistic case: λ2 ∼ k E(x , ω ∼ B(x , γ0 = 1. B0 v0 B0 √ The two schemes are stable for hλ/ γ0 < 2. To simulate accurately the Larmor cyclotron it is needed that hω/γ0 ≤ π. Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Other properties of the two volume-preserving methods Accuracy They are of second order, i.e. |x1 − x(0)| ≤ Ch2 . Cost They are explicit, only needs 1 evaluation of E(x) and B(x) per step. Preservation of invariants They can preserve the kinetic energy of the particle exactly with static magnetic field; They can bound the error of invariants up to O(h2 ). Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Higher order methods Two fourth order method designed by composition 3-stage (Yoshida, 90): G4h = Φa1 h ◦ Φa2 h ◦ Φa1 h where a1 = (2 − 21/3 )−1 , a2 = 1 − 2a1 ; 5-stage (Suzuki, 92): G4h = Φb1 h ◦ Φb2 h ◦ Φb3 h ◦ Φb2 h ◦ Φb1 h where b1 = b2 = (4 − 41/3 )−1 , b3 = 1 − 2(b1 + b2 ); A fourth order method designed by processing(Blanes, 99) G4h = χh ◦ Φh ◦ χ−1 h 1 23 1 23 1 The kernel: Φh = φ1a1 h ◦ φ23 b1 h ◦ φa2 h ◦ φb2 h ◦ φa2 h ◦ φb1 h ◦ φa1 h The processor: χh = 5 Y φ1zi h ◦ φ23 yi h i=1 a1 = 0.6762, Yang He b1 = −0.175, a2 = −0.1762, Volume-preserving Lorentz force equation b2 = 1.35 Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Linear stability domain S U S S (c) BG2h S U U U (d) CG2h S U S U (e) by Yoshida composi- (f) by Suzuki composition (g) by processing(only for tion of BG2h of BG2h non-relativistic case) Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Accuracy and cost Truncation error Error coefficients Evaluations of E(x) after N steps Effective error constants G4h Y C1 h4 3.92e-2 3N G4h S C 2 h4 6.86e-4 5N G4h P C3 h4 1.3e-3 3N + 10 1.3349 0.8092 0.5696 Table: G4h Y denotes the Yoshida’s composition method and G4h S denotes the Suzuki’s composition method. G4h P denotes the processing method for the non-relativistic motion equation. Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Numerical experiment 1: non-relativistic case Static electromagnetic field B = B0 ez , E = El ( x y ex + ey ), R0 R0 where B0 = 3T , E0 = 3V /m, R0 = 1m. −4 1.5 10 1 10 −5 −6 10 ∆ H/H0 y*ωce 0.5 0 −0.5 −7 10 −8 10 −9 10 −10 −1 10 −1.5 −1.5 10 2nd order 4th order −11 −1 −0.5 0 x*ωce 0.5 1 1.5 0 1000 2000 3000 4000 5000 6000 7000 8000 Steps Figure: Take the initial momentum as vk (0) = 1v0 , v⊥ (0) = 5v0 , v0 = 0.1m/s. Choose the step size as h = 0.1π, run for 104 steps. Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Ex1: relative errors of the solution 0 −1 10 10 0.067h2 −2 log(phase error) at the time 39.17 relative error of (x,v) −2 10 −4 10 G2h I −6 10 Suzuki processing Yoshida −8 10 4 6.8e−4h 10 −3 10 −4 10 G2hI −5 10 Suzuki of G2hI −6 10 processing of G2hI −7 Yoshida of G2hI 10 −10 10 1 10 0 −1 10 10 −2 10 time step h −8 10 1 10 2 3 10 10 4 10 log (effort) (a) as functions of h (b) as functions of efforts (evaluations of E(x)) Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Numerical experiment 1: relativistic case Static electromagnetic field B = B0 ez , E = El ( x y ex + ey ), R0 R0 where B0 = 3T , E0 = 3V /m, R0 = 1m. −3 2 x 10 −15 10 t=0s 1.5 −16 10 1 −17 ∆ H/H y 0 0.5 0 −0.5 −1 t=6e−8 s 10 −18 10 2nd order 4th order Yoshida 4th order Suzuki −19 10 −1.5 −20 −2 1.05 1.0505 1.051 1.0515 1.052 1.0525 1.053 1.0535 1.054 x 10 0 1 2 3 Steps 4 5 6 Figure: Take the initial momentum as pk0 = 1m0 c, p⊥0 = 5m0 c. Choose the step size as h = 0.1π, run for 105 steps. Yang He 4 x 10 Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Ex1: relative errors of the solution 2 2 10 10 CG2h 0 10 0 10 4th order Yoshida 4th order Suzuki −2 relative error of x/m relative error of x/m 10 −4 10 −6 10 −8 10 CG2h −6 10 10 −10 −12 10 −4 10 −8 4th order Yoshida 4th order Suzuki −10 10 −2 10 1 10 0 10 −1 10 time step h −2 10 −3 10 (a) as functions of h 10 1 10 2 10 3 10 log(effort) 4 10 5 10 (b) as functions of efforts (evaluations of φ3h ) Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Numerical experiment 2: non-relativistic case Axi-symmetric tokamak geometry B0 R0 B0 r B=∇×A= eξ + eθ , R qR where q = 2 and R0 = 1m. ϕ(x) = 0, −14 0.08 0.06 relative error 0.04 z 0.02 0 −0.02 −0.04 −0.06 −0.08 1 1.02 1.04 1.06 1.08 1.1 R (c) Banana Orbit x 10 8 7 6 5 4 3 2 1 0 0 2 volume−preserving methods 4 6 8 10 time 12 14 16 4 x 10 (d) Relative error of energy Figure: Take the initial momentum as vk0 = 1m0 v0 , v⊥0 = 5m0 v0 . Choose the step size as h = 0.1π, run for 2 × 106 steps. Yang He Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Numerical experiment 2: relativistic case Axi-symmetric tokamak geometry B=∇×A= B0 R0 B0 r eξ + eθ , R qR ϕ(x) = ϕ0 R0 , R where q = 2 and R0 = 1m. −16 0.08 20 x 10 2nd order 4th order 0.06 15 0.04 ∆ H/H0 z 0.02 0 −0.02 −0.04 10 5 0 −0.06 −0.08 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 −5 0 R 2 4 6 Steps 8 10 12 Figure: Take the initial momentum as pk0 = 5m0 c, p⊥0 = 1m0 c. Choose the step size as h = 0.1π, run for 105 steps. Yang He 4 x 10 Volume-preserving Lorentz force equation Introduction Geometric properties Volume-preserving integrators Linear stability of the volume-preserving methods Numerical Exp Summary We analyze the performances of two volume-preserving algorithms when simulating charged particle trajectories in both relativistic and non relativistic cases. Higher order methods are constructed by composition and by processing. Numerical analysis and experiments show that the two higher order methods are more efficient than the second order methods for more strict error restriction. Ongoing work To construct more efficient higher order methods for simulating relativistic dynamics of charged particles. To analyze the linear stability of the higher order methods for relativistic case. Yang He Volume-preserving Lorentz force equation

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