Computational plasma physics – extending legacy codes, computing functionals and other ideas Monash Workshop on and Applications 2020 Markus Hegland, ANU February 2020 computational plasma physics 1 / 40
Introduction Introduction computational plasma physics 2 / 40
Overview Introduction Approximation 1: PDEs approximating ODEs Approximation 2: gyrokinetics Approximation 3: Lie perturbation Approximation 4: numerics Approximation 5: sparse grids Other approximations Introduction computational plasma physics 3 / 40
challenges in computational science and engineering ◮ exascale computing ◮ faults ◮ synchronisation and communication ◮ new approximations ◮ assimilating data with computational solutions of PDEs ◮ including extensive computations in control ◮ uncertainty in models, data and computations ◮ managing very complex computational codes ◮ focus on quantities of interest and dual problems ◮ inverse problems and optimisation Introduction computational plasma physics 4 / 40
role of mathematics ◮ enhance understanding of assumptions and observations used in code development ◮ approximation errors in legacy and new code ◮ complexity ◮ properties of models (e.g. PDE existence and uniqueness theorems) ◮ error propagation ◮ understanding the nature of collaborations and role of different disciplines ◮ people are interdisciplinary Introduction computational plasma physics 5 / 40
our project ◮ code base: GENE – development lead by Frank Jenko, IPP Munich ◮ highly scalable, tested on various HPCs ◮ international user base ◮ under constant development ◮ our aim: extending the capability of GENE without changing the core ◮ approach: numerical extrapolation based on multiple simulations with different grid parameters ◮ applications: solve larger problems, parameter optimisation, uncertainty quantification ◮ resources: 4 PhD students, ARC Linkage project with Fujitsu Europe and collaboration with TU Munich through DFG excellence initiative ◮ so far: fault tolerant sparse grids ◮ target: mathematics behind GENE computations ◮ in this talk: explore approximations used Introduction computational plasma physics 6 / 40
GENE ◮ open source plasma research code ◮ state of the art, highly optimised for high performance computers ◮ our work: utilise sparse grids to improve performance and fault tolerance Introduction computational plasma physics 7 / 40
Collaborators This talk is based on past and current collaborative research with Yuancheng Zhou (ANU), Christoph Kowitz (formerly TU Munich), Brendan Harding (UoA), Peter Strazdins (ANU), Peter Vasiliou (ANU), Matthew Hole (ANU), Stuart Hudson (PPL Princeton), Frank Jenko (MPI Garching) and Dirk Pfluger (Uni Stuttgart) Introduction computational plasma physics 8 / 40
Approximation 1: PDEs approximating ODEs Approximation 1: PDEs approximating ODEs computational plasma physics 9 / 40
dynamics of a single particle ◮ Newton’s equations for charged particles � � � � d x v = 1 v m F ( x , v ) dt ◮ Lorentz force F ( x , v ) = q ( E + v ∧ B ) ◮ Hamilton’s equations � � � � d x 0 I = ∇ H p − I 0 dt c A � 2 + φ ( x ) 2 m � p − q 1 ◮ Hamiltonian H ( x , p ) = ◮ fields E = ∇ φ − ∂ A ∂ t and B = ∇ ∧ A ◮ momentum p = v + q c A Approximation 1: PDEs approximating ODEs computational plasma physics 10 / 40
Maxwell’s equations ∂ E ∂ t = c 2 ∇ ∧ B − j ǫ 0 ∂ B ∂ t = −∇ ∧ E and ∇ · E = ρ ǫ 0 ∇ · B = 0 Approximation 1: PDEs approximating ODEs computational plasma physics 11 / 40
solutions E = −∇ φ − ∂ A ∂ t B = ∇ ∧ A where � ρ ( ξ, t − r / c ) φ ( x , t ) = d ξ 2 πǫ 0 r � j ( ξ, t − r / c ) A ( x , t ) = d ξ 2 πǫ 0 r and r = � x − ξ � ◮ GENE solves Poisson-Ampere equations Approximation 1: PDEs approximating ODEs computational plasma physics 12 / 40
Vlasov equations ◮ let X ( t ) and V ( t ) solve Newton’s equations ◮ let µ 0 ( x , v ) be continuously differentiable and µ ( x , v ; t ) = µ 0 ( x − X ( t ) , v − V ( t )) ◮ then µ satisfies ∂µ ∂ t = ˙ X T ∇ x µ + ˙ V T ∇ u µ ◮ eliminate the derivatives of X and V using Newton’s equations ∂µ ∂ t = V · ∇ x µ + F ( X , V ) · ∇ u µ m ◮ Vlasov equations ∂µ ∂ t = v · ∇ x µ + F ( x , v ) · ∇ u µ m approximation if supp( µ 0 ) small Approximation 1: PDEs approximating ODEs computational plasma physics 13 / 40
multiple particles ◮ could use Vlasov equations to define (very) weak solutions of ODEs ◮ here we consider instead multiple particle solutions given by n µ ( x , v ; t ) = 1 µ 0 ( x − X ( i ) ( t ) , v − V ( i ) ( t )) � n i =1 ◮ if all ( X ( i ) ( t ) , V ( i ) ( t )) satisfy Newtons equations one gets the Vlasov approximation as ∂µ ∂ t = v · ∇ x µ + F ( x , v ) · ∇ u µ m if the forces are purely external, i.e., there are no interactions Approximation 1: PDEs approximating ODEs computational plasma physics 14 / 40
multiple particles with interactions ◮ interactions between the particles: approximate F which now depends on µ ∂µ ∂ t = v · ∇ x µ + F ( x , v ; µ ) · ∇ u µ m ◮ interactions between the particles obtained from the charge and current densities ρ and j � ρ q ( x ; t ) = q µ ( x , v ; t ) dv , � j q ( x ; t ) = q v µ ( x , v ) dv ◮ nonlinear (quadratic) system of integro-differential equations ⇒ turbulence Approximation 1: PDEs approximating ODEs computational plasma physics 15 / 40
application and approximation ◮ the Vlasov equations are used to approximate systems of ODEs arising from very large systems of charged particles ◮ Vlasov equations are often solved using particle methods which basically model the dynamics of agglomerates of particles ◮ the accuracy and of the approximations of distributions of discrete particles by densities µ is an area of active research in mathematics especially for the case of Lorentz forces, i.e., the Vlasov-Maxwell equations ◮ MHD based on moments of µ similar to ρ and j Approximation 1: PDEs approximating ODEs computational plasma physics 16 / 40
Approximation 2: gyrokinetics Approximation 2: gyrokinetics computational plasma physics 17 / 40
constant fields mdv dt = q ( E + v ∧ B ) ◮ decomposition into terms parallel and orthogonal to B B = (0 , 0 , | B | ) T v = v � + v ⊥ , E = E � + E ⊥ 0 1 0 B ∧ v = | B | − 1 0 0 v 0 0 0 ◮ differential equations for v dv � dt = q mE � dv ⊥ = q m ( E ⊥ + v ⊥ ∧ B ) dt Approximation 2: gyrokinetics computational plasma physics 18 / 40
solutions of the constant field case v � ( t ) = v � (0) + qE � m t � � cos( τ ) sin( τ ) v ⊥ ( t ) = u ⊥ + ( v ⊥ (0) − u ⊥ ) − sin( τ ) cos τ where partial (constant) solution u ⊥ satisfies E ⊥ + u ⊥ ∧ B = 0 and τ = Ω t where the gyrofrequency is Ω = q | B | m ◮ integrate to get location t 2 x � ( t ) = x � (0) + v � (0) t + qE � m 2 � � sin( τ ) − cos( τ ) + 1 x ⊥ ( t ) = x ⊥ (0) + u ⊥ t + Ω ( v ⊥ (0) − u ⊥ ) cos( τ ) − 1 sin( τ ) Approximation 2: gyrokinetics computational plasma physics 19 / 40
discussion of solution ◮ solution takes the form of a spiral which has two components: 1. movement of centre ◮ in direction of B ◮ drifting from this direction 2. gyration with frequency Ω around centre ◮ Hamiltonian formulation leads to introduction of gyro coordinates and separation of gyro motion from the rest Approximation 2: gyrokinetics computational plasma physics 20 / 40
invariants and dimension reduction ◮ Vlasov equations for Hamiltonian systems ∂ f ∂ t = { H , f } where Poisson bracket is { H , f } = ∇ p H T ∇ x f − ∇ x H T ∇ p f ◮ for particles with charge e and E = ∇ φ and B = ∇ ∧ A H = 1 2 m � p − e c A � 2 + φ ◮ if ∂ H /∂ x i = 0 then Vlasov equations don’t contain ∂ f /∂ p i which allows integration Approximation 2: gyrokinetics computational plasma physics 21 / 40
Example ◮ simple 1D example H = p 2 / 2 then ∂ f ∂ t = pf x with solution f = g ( pt + x ) ◮ constant fields with φ = E T x , B = ∇ ∧ A and Hamiltonian H = 1 2 � p − q 2 c B ∧ x � 2 + E T x � 2 � 2 = 1 � p 1 + q | B | + 1 � p 2 − q | B | + 1 2 p 2 2 c x 2 2 c x 1 3 + E 1 x 1 + E 2 x 2 2 2 ◮ Hamiltonian independent of x 3 Approximation 2: gyrokinetics computational plasma physics 22 / 40
Gyrokinetic equations ◮ approximate one gyrating particle by uniform particle density rotating around gyrocentre ◮ new coordinates: gyrocentre X , parallel velocity v � and magnetic moment µ = | mv ⊥ | 2 (apologies: different µ . . . ) 2 B ◮ ODEs (based on Lorentz force) dX dt = φ 1 ( x , v � , µ ; t ) dv � dt = φ 2 ( x , v � , µ ; t ) d µ dt = 0 ◮ Vlasov equations ∂ f ∂ f ∂ f ∂ t + φ T ∂ X + φ 2 = 0 1 ∂ v � ◮ need to transform Maxwell’s equations too Approximation 2: gyrokinetics computational plasma physics 23 / 40
Approximation 3: Lie perturbation Approximation 3: Lie perturbation computational plasma physics 24 / 40
Recommend
More recommend
