Kinetic implicit fluid methods E. Alec Johnson, Stefano Markidis, and Giovanni Lapenta Nov 26, 2013 Abstract: Fully explicit discretizations must resolve all three processes that ideal MHD assumes are instan- taneous: oscillations, collisions, and light waves. Asymptotic-preserving discretization requires stepping over these processes. Fully implicit discretization allows stepping not only over these three processes but also over MHD waves, but is expensive because one must repeatedly re-push particles with successive iterations of the implicit field solver. We therefore present semi-implicit dis- cretizations of Maxwell’s equations that step over subsets of these processes. Discretizing implicitly in the source term steps over plasma oscillations and collisions and allows asymptotic-preserving agreement with (quasi-)relativistic MHD. Discretizing the flux terms of Maxwell’s equations implicitly steps over light waves and allows asymptotic-preserving agreement with classical two-fluid MHD. To facilitate asymptotic-preserving agreement with fluid models and conservation of physical invari- ants, we consider kinetic fluid closure. Johnson Implicit kinetic plasma Nov 26, 2013 1 / 31
kinetic-Maxwell (the “truth”) where σ ( x ) := � particle evolution: p S p ( x ) q p is charge density and J ( x ) := � p S p ( x ) q p v p is d t x p = v p , current density; here S p ( x ) = S ( x − x p ) is d t u p = q p m p ( v p × B ( x p ) + E ( x p )) , the shape function of particle p . To abbreviate, we drop the particle summation γ 2 p := 1 + ( u p / c ) 2 , index p and the independent variable x and v p := u p /γ p . write σ := � qS (charge) , electromagnetic field: ρ := � mS (mass) , J := � v qS ∂ t B + ∇ × E = 0 , (current) , M := � v mS − c − 2 ∂ t E + ∇ × B = µ 0 J , (momentum) , E := � 1 c − 2 ∇ · E = µ 0 σ, ∇ · B = 0 , 2 | v | 2 mS (energy) . Johnson Implicit kinetic plasma Nov 26, 2013 2 / 31
Outline Modeling and moment evolution 1 Semi-implicit methods 2 The Implicit Moment Method (IMM) 3 Conforming fluid-Maxwell 4 Consistent kinetic closure 5 Porting of iPic3D to DEEP 6 Johnson Implicit kinetic plasma Nov 26, 2013 3 / 31
Moment evolution Fields evolve in response to charge Shape motion: moments, and mass, momentum, and energy are conserved, motivating the use ∂ t S = − v · ∇ S , (1) of fluid models. Kinetic closure allows kinetic algorithms to transition efficiently where we have used the chain rule. and smoothly to fluid models. Lorentz force. Moment definitions are of the form � χ S . u = q ˙ m ( E + v × B ) ; To derive fluid equations, we differentiate the moment definition with respect to time and use the basic derivatives to the right. Energy change. Charge density evolution. u = q γ = v · ˙ ˙ m v · E , (2) ∂ t σ + ∇ · J = 0 , because γ 2 = 1 + ( u / c ) 2 , so γ ˙ u / c 2 . γ = u · ˙ using ∂ t σ = � ˙ qS + � q ∂ t S , Velocity change. ∂ t S = − v · ∇ S , and ∇ v = 0. � � General moment evolution q E − vv v = c 2 · E + v × B , ˙ γ m � � � ∂ t χ S + ∇ · v χ S = χ S , ˙ which follows from differentiating u = γ v , to χ = ∂χ u = ∂χ get ˙ u = ˙ γ v + γ ˙ v , i.e. γ ˙ v = ˙ u − vv · ˙ u . ˙ ∂ u · ˙ ∂ v · ˙ v . Johnson Implicit kinetic plasma Nov 26, 2013 4 / 31
Relativistic current evolution (Ohm’s law) Current evolution (Ohm’s law, χ = q v). ∂ t J + ∇ · P = � S q 2 � � · E + � S q 2 I − vv γ m × B , γ m c 2 where P := � q vv . Classical current evolution for species s . ∂ t J s + ∇ · P s = q s m s ( σ s E + J s × B ) , where P s restricts to species s . Johnson Implicit kinetic plasma Nov 26, 2013 5 / 31
Mass moment evolution Relativistic case. Classical case. Mass density ( χ = m ) . Mass density ( χ = m ) ∂ t ρ + ∇ · � m v S = 0 . ∂ t ρ + ∇ · M = 0 , Momentum density ( χ = m u ) . Momentum density ( χ = m v) ∂ t M + ∇ · � m vu S = σ E + J × B , ∂ t M + ∇ · � m vv S = σ E + J × B , Energy density ( χ = mc 2 γ ) . Energy density ( χ = 1 2 m | v | 2 ) ∂ t E + ∇ · M = J · E , � 1 2 m v v 2 S = J · E , ∂ t E + ∇ · Johnson Implicit kinetic plasma Nov 26, 2013 6 / 31
Outline Modeling and moment evolution 1 Semi-implicit methods 2 The Implicit Moment Method (IMM) 3 Conforming fluid-Maxwell 4 Consistent kinetic closure 5 Porting of iPic3D to DEEP 6 Johnson Implicit kinetic plasma Nov 26, 2013 7 / 31
Field discretizations The Implicit Moment Method is an example of a semi-implicit method. Semi-implicit methods are used to step over fast processes such as plasma oscillations (implicit source) and light waves (IMM) without having to step over all processes. For plasma simulations, one of four types of discretization is generally used: Discretization must resolve. . . must iterate. . . 1. Explicit everything (e.g. ω p ) [no iteration] 2. Implicit Source light waves classical: no iteration relativistic: source 3. Implicit Moment (IMM) electron sound waves fields 4. Fully implicit [no restriction] particles [everything] The Implicit Source discretization naturally suits an asymptotic-preserving tran- sition to relativistic MHD, since ideal relativistic MHD takes the electron charge (gyrofrequency) to infinity. The Implicit Moment Method naturally suits an asymptotic- preserving transition to two-fluid MHD, which takes light speed to infinity. Johnson Implicit kinetic plasma Nov 26, 2013 8 / 31
Explicit discretization Start with the basic equations: In the relativistic case, time-split the velocity update for a symplectic method. In ∂ t B + ∇ × E = 0 , full detail: ∂ t E − c 2 ∇ × B = − J /ǫ 0 , B 2 = B 0 − ∆ t ∇ × E 1 , ∂ t x p = v p , � � E 1 = E − 1 + ∆ t c 2 ∇ × B 0 − J 0 /ǫ 0 , ∂ t u p = q p m p ( E ( x p ) + v p × B ( x p )) . p + q p ∆ t u ∗ p = u 0 ( u ∗ p + u 0 p ) × B 1 ( x 1 p ) , 2 m p γ 0 For a second-order discretization, p use a leapfrog discretization: p + ∆ t q p p = u ∗ u 2 m p E 1 ( x 1 p ) , B does not need particle where B 1 := 1 2 B 0 + 1 2 B 2 . velocities to advance, but E does, so particle velocities u p and current J should be staggered relative to E . ∂ t X = ( X 2 − X 0 ) / ∆ t ∂ t Y = ( Y 3 − Y 1 ) / ∆ t Johnson Implicit kinetic plasma Nov 26, 2013 9 / 31
IMEX discretization (implicit source) [KumarMishra11] Use initial values for flux terms. We designate n = 0 as initial time, and implicit values in stiff source. n + 1 = 1 as final time, and time discretization as Fields: ∂ t Q → ( Q 1 − Q 0 ) / ∆ t , ∂ t B + ∇ × E = 0 , ∇ F → F 0 , ∂ t E − c 2 ∇ × B = − J /ǫ 0 . X = X 1 . Particles: � � Classical case: No source term iteration ∂ t u p = q p E ( x p ) + v p × B ( x p ) , happens to be needed, because v is linear m p in E . Can sum the response over all ∂ t x p = v p , particles to eliminate J = � J + A · E in favor ∂ t σ s + ∇ · J s = 0 . of E . Cassical current: Relativistic case: Must iterate particle � � velocity advance, but positions need not be ∂ t J s + ∇ · P s = q s σ s E + J s × B . m s advanced, so iterative solve involves no communication between mesh cells. High-order accuracy: Use an IMEX Runge-Kutta solver. Johnson Implicit kinetic plasma Nov 26, 2013 10 / 31
Fully implicit method Modify the IMM discretization by making all Remarks. terms implicit: Particle advance must be redone with successive ∂ t B + ∇ × E = 0 iterations of the field solver. ∂ t E − c 2 ∇ × B = − J /ǫ 0 This discretization is fully � � ∂ t u p = q p σ p E ( x p ) + v p × B ( x p ) , symmetric in time, so by m p Noether’s theorem ∂ t x p = v p , conserves energy if ∂ t σ s + ∇ · J s = 0 , iterated to convergence. � � ∂ t J s + ∇ · P s = q s σ s E + J s × B . m s Johnson Implicit kinetic plasma Nov 26, 2013 11 / 31
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