Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Volume-Preserving Algorithm for Secular Relativistic Dynamics of Charged Particles Ruili Zhang, Joint work with Jian Liu, Hong Qin, Yulei Wang, Yang He and Yajuan Sun Department of Modern Physics University of Science and Technology of China [email protected] Mar 24th, 2015 Ruili Zhang et al. VPA for relativistic charged particles Summary Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Outline 1 Secular relativistic dynamics 2 Systematic method to construct VPAs 3 Numerical experiments 4 Summary Ruili Zhang et al. VPA for relativistic charged particles Summary Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Dynamics of relativistic charged particles serves as the first principle model underlying plasma physics, accelerator physics, astrophysics, space physics, and many other subfields of physics! The long-term relativistic dynamics is critical to describe various multi-timescale problems, such as runaway electrons in tokamaks. Ruili Zhang et al. VPA for relativistic charged particles Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary The relativistic charged particle dynamics is dx p = p 2 , dt m0 + p2 /c2 (1) dp p × B(x, t) = q E(x, t) + p 2 dt m0 + p2 /c2 ! . x and p denote position and momentum vector Ruili Zhang et al. VPA for relativistic charged particles (2) Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Difficulties: Time-dependent electromagnetic fields How to deal with the relativistic factor? Multi-scale problem Long-time simulation Geometric numerical integrator! Ruili Zhang et al. VPA for relativistic charged particles Summary Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Geometric Properties: 6-form is an invariant Liouville’s theorem The solution flow ϕt : z(t0 ) → z(t) preserves the phase space volume V(t), i.e., Volume-preserving algorithm (VPA)! Ruili Zhang et al. VPA for relativistic charged particles Summary Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Volume-preserving algorithm: Φh (xk , pk ) = (xk+1 , pk+1 ) Is there a systematic method to construct VPAs? Ruili Zhang et al. VPA for relativistic charged particles Summary Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary This VPA is constructed using the splitting method consisting of three steps: (1) To split the original system into several incompressible subsystems. (2) To find a VPA for each subsystem. (3) To combine the sub-algorithms into a desired VPA for the original system. Ruili Zhang et al. VPA for relativistic charged particles Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Expressing using Lie-derivative Correspondence between Lie algebras and Lie groups BCH formula exp(tXF ) = exp(tXF1 ) exp(tXF2 ) + O(t) , t t exp(tXF ) = exp( XF1 ) exp(tXF2 ) exp( XF1 ) + O(t2 ) , 2 2 t t t t exp(tXF ) = exp( XF1 ) exp( XF2 ) exp(tXF3 ) exp( XF2 ) exp( XF1 ) + O(t2 ) . 2 2 2 2 Ruili Zhang et al. VPA for relativistic charged particles (3) Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Step 1: To split the original system Step 1: To split the original system into three subsystems. dx p , = p 2 dx = 0 , m0 + p2 /c2 dt S1 := dt S2 := dp dp = qE(x, t) , =0, dt dt dx =0, dt ! S3 := dp p × B(x, t) . dt = q p 2 m0 + p2 /c2 Each subsystem is source-free system! Ruili Zhang et al. VPA for relativistic charged particles (4) (5) Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Step 2: To find a VPA method for each subsystem. Step 2: To find a VPA method for each subsystem. The exact solution of S1 : xk+1 = xk + ∆t p pk , 1 m20 + p2k /c2 ϕ (∆t) : pk+1 = pk . Symmetric VPA of order 2 for subsystems S2 using mid-point rule: xk+1 = xk , 2 ϕ (∆t) : pk+1 = pk + q∆tE(xk , tk + ∆t ) , 2 Time-dependent electromagnetic fields. X Ruili Zhang et al. VPA for relativistic charged particles (6) (7) Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments For S3 , p2 (t) = p20 . The relativistic factor is an invariant γ = Summary p m20 + p20 /c2 S3 is equavalent to dx =0, dt S3 := dp q = p 2 (p × B(x, t)) . dt m0 + p20 /c2 0 (8) Symmetric VPA of order 2 for subsystems S3 using mid-point rule: x = xk , (9a) k+1 ! 3 ϕ (∆t) : pk + pk+1 ∆t p × B(xk , tk + ) (.9b) pk+1 = pk + q∆t 2 2 2 2 2 m0 + pk /c How to deal with the relativistic factor? X Ruili Zhang et al. VPA for relativistic charged particles Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Step 3: Composition the sub-VPAs Step 3:Composition the sub-VPAs Φ(∆t) = ϕ1 ( ∆t ∆t ∆t ∆t ) ◦ ϕ2 ( ) ◦ ϕ3 (∆t) ◦ ϕ2 ( ) ◦ ϕ1 ( ) . 2 2 2 2 The resulting VPA using the combination in Eq. (10) is ∆t pk , xk+ 12 = xk + 2 p 2 m + p2k /c2 0 ∆t p− = pk + q Ek+ 1 , 2 2 ˆ k+1/2 + q∆tB q p− , Φ(∆t) : p = Cay 2 2 − 2 2 m0 + p /c ∆t + pk+1 = p + q Ek+ 1 , 2 2 ∆t pk+1 q x = x + . 1 k+1 k+ 2 2 2 m0 + p2k+1 /c2 Ruili Zhang et al. VPA for relativistic charged particles (10) (11) Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Cayley transform: −1 ˆ 1 p− = Cay(aB ˆ 1 )p− , ˆ 1 I + aB p+ = I − aB k+ k+ k+ 2 2 2 Summary (12) ˆ 1 is an anti-symmetric matrix, and where B k+ 2 q∆t . a= p 2 2 m0 + (p− )2 /c2 (13) It can be manipulated explicitly. p+ = I+ 2a 2a2 ˆ 1 + ˆ2 1 B B k+ 2 2 2 1 + a |Bk+ 1 | 1 + a2 |Bk+ 1 |2 k+ 2 2 Ruili Zhang et al. ! p− . 2 VPA for relativistic charged particles (14) Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Advantages: It preserve phase space volume exactly! It is symmetric! It can be computed explicitly! Ruili Zhang et al. VPA for relativistic charged particles Summary Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Numerical experiments Multi-scale problem: the runaway electron runs at the speed nearly 3 × 108 m/s; the outward neoclassical drift velocity only about 3m/s. B0 B0 R0 B=− eζ − R R0 E = − Et eζ , R p (R − R0 )2 + z2 eθ , 2R (16) where B0 = 3T, R0 = 1.7m and Et = 3V/m. Ruili Zhang et al. (15) VPA for relativistic charged particles Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Figure: The long-term simulation results of runaway electron dynamics in tokamak running 6.28 × 1010 time steps by VPA. Ruili Zhang et al. VPA for relativistic charged particles Summary Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Figure: The simulation result by the Boris algorithm, as a comparison to the VPA. (a) The snapshots of runaway drift orbits calculated by the Boris algorithm at t=0s, t=10−4 s, and t=1.07 × 10−4 s. Multi-scale problem X Long-time simulation X Ruili Zhang et al. VPA for relativistic charged particles Secular relativistic dynamics Systematic method to construct VPAs Numerical experiments Summary Summary A systematic method to construct VPAs for relativistic charged particles dynamics has been given. Relative error of energy can be bounded. Relative error of canonical angular momemtum can be bounded. The VPA enables the discovery of new physics for secular runaway dynamics. High order VPAs can be obtained using the splitting method. Thank you for you attention! Ruili Zhang et al. VPA for relativistic charged particles

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