Lecture 4: The particle equations (1)
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1 Lecture 4: The particle equations (1) Presenter: Mark Eric Dieckmann Department of Science and Technology (ITN), Linköping University, Sweden July 17, 2014
2 Overview We have previously discussed the leapfrog method for solving hyperbolic partial differential equations. We have applied this scheme to the normalized Maxwell s equations. We investigated the dispersive properties of the numerical grid: The propagation of electromagnetic waves is good provided that their wavelength is substantially larger than the grid cell size. We will now discuss how PIC codes represent the plasma. We will then perform our simulations with particles in it using the Epoch code. We shall use again Octave to look at the wave and plasma properties.
3 Let s bring back the equations The normalized Vlasov-Maxwell set of equations is: t f n(x,v,t)+v x f n (x,v,t) q n m n (E+v B) v f n (x,v,t) = 0, E(x,t) = B(x,t) J(x,t) (Amperes law), t B(x,t) = E(x,t) (Faradays law), t E(x,t) = ρ T (x,t), B(x,t) = 0, [ ] [ ] ρ T (x,t) = q n f n (x,v,t) dv, J T (x,t) = q n f n (x,v,t) vdv. n We have learned how we can advance in time the field equations. The next steps are to understand (1) how we represent the plasma and (2) how we connect the plasma to the field equations. n
4 (1) The plasma definition Each species j has a phase space density distribution f j (x,v,t). Consider one species and a 1D geometry f j (x,v,t) f(x,v). The function f(x,v) gives the probability with which we find a particle at the position x and the velocity v. 1 Left: spatially uniform plasma with a Maxwellian velocity distribution f(x,v) = exp( v 2 /2v 2 th ). f e (x,v x ) Speed v x x position The thermal speed is v th = k B T/m. k B : Boltzmann s constant, T: temperature in Kelvin, m: mass in kg. f(x,v) can be represented and evolved directly Vlasov codes. Low noise levels, limited dynamical range for velocities and costly in more than 1D. Vlasov code paper: C.Z. Cheng and G. Knorr, J. Comput. Phys. 22, , 1976.
5 (2) The plasma definition Particle-in-cell codes resort to a Lagrangian description of the plasma. The phase space density distribution f(x,v) is approximated by an ensemble of computational particles (CPs). We distribute particles uniformly along x and we want a Maxwellian speed distribution. Step 1: Place N CPs along an interval with the length x max x min = L. Step 2: Compute one speed value v j for each CP. This value is computed with a random number generator that uses a Gaussian probability distribution. We approximate f(x,v) N j=1 δ(x jl/n,v j). This may suggests that we are back to a Liouville description of the plasma. This is not the case because we shall ignore binary interactions.
6 (3) The plasma definition A phase space density distribution f(x,v) in the Lagrangian form will look like in the figure below. Speed v x x position Each dot represents a computational particle (CP). The core population with v x 0 is represented well. The high energy tail of the distribution is poorly represented. Processes that involve energetic CPs are not represented well. Consider the interval 0.4 x 0.6. As time progresses, CPs enter and leave this interval. The number of CPs in the interval fluctuates. Fluctuations yield electric and magnetic field noise.
7 The particle equations of motion The collective electromagnetic fields are the electric E(x, t) and magnetic B(x,t) fields in the 1D geometry. Binary interactions between CPs are neglected in the Vlasov approach. The only force that acts on the k th CPs is the electromagnetic field at the particle position x k at the time t. The particle velocity v k = (v k,x,v k,y,v k,z ) is updated via the Lorentz force equation. Epoch and most other codes use the relativistic Lorentz force equation. We consider for simplicity the non-relativistic one. d dt v k = q k m k (E(x k,t)+v k B(x k,t)). The position of this CP is updated through d dt x k = v k,x.
8 The particle equations of motion The nonrelativistic equations of motion for the j th CP are: d dt v k = q k m k (E(x k,t)+v k B(x k,t)), d dt x k = v k The electric and magnetic fields are advanced in time at steps t. q k and m k are normalized to e and m e. We place E y (j,n) on the grid with integer positions j x and times n t and B z (j +1/2,n+1/2) on a grid shifted by x /2, t /2. The usage of shifted grids improved the accuracy and stability. We are faced with time-derivatives of the particle equations and we wish to have the accuracy of centred schemes. Ideally one update cycle for the fields corresponds to one update cycle for the CP s We define x k and v k at times shifted by t /2.
9 Defining the particle positions and speeds We define the particle positions x k at times n t. The particle velocities v k are defined at times (n+1/2) t. We have to compute the force at n t and update with it the velocity v k,n 1/2 to v k,n+1/2. The discretized equation is v k,n+1/2 v k,n 1/2 t = q ( k E(x k,n )+ v ) k,n+1/2 +v k,n 1/2 B(x k,n ) m k 2 The magnetic field is computed on the shifted grid, while we need it at the time step n: B j+1/2,n = (B j+1/2,n+1/2 +B j+1/2,n 1/2 )/2. We want an explicit equation that separates v k,n+1/2 and v k,n 1/2. Boris, J.P. Relativistic plasma simulation-optimization of a hybrid code. Proceedings of the 4th Conference on Numerical Simulation of Plasmas. Naval Res. Lab., Washington, D.C. pp. 367, 1970.
10 The Boris scheme The velocity update for the particle at time step n involves electric and magnetic forces. Both are computed at the time step n. The force components are evaluated sequentially: 1. An electric field push during the time t /2. 2. The magnetic field push for a full time step. 3. A second electric field push during the time t /2. We shall discuss this scheme in the nonrelativistic limit. The relativistic version can yield spurious forces: J.L. Vay, Simulation of beams or plasmas crossing at relativistic velocity, Phys. Plasmas 15, , 2008
11 The Boris scheme We start with the velocity v k,n 1/2. v k = v k,n 1/2 + q k t E n (x k ) 2m k v k = v k +v k B(x k ), v + k = v k +v k 2 B 1+ B B B(xk ) = q k t 2m k B(x k ). v k,n+1/2 = v + k + q k t E n (x k ) 2m k After this sequence of operations we obtain the velocity v k,n+1/2. Remaining problem: We initialize particle positions at the simulation start n = 0 (and thus t = n t = 0). We initialize the particle velocities at n = 0, but we need them at n = 1/2.
12 The Boris scheme and the initial conditions We have initialized the positions and velocities of the CP s at the time n t = 0 but we need the velocities at the time n = 1/2. We compute those particle velocities at the time (n 1/2), which give the ones at n = 0 if the forces at n = 0 are taken into account. we go one half-step back in time with the Boris scheme. v k = v k,n=0 +v k,n=0 B(x k ), B(xk ) = ( 1/2) q k t 2m k B(x k ). v + k = v k +v k 2 B 1+ B B v k, 1/2 = v + k q k t E n (x k ) 2m k What remains to be understood is how E n (x k ) and B n (x k ) are related to the fields on the grid.
13 The particle s shape function Speed v x x position The plasma in PIC simulations is a collection of Lagrangian markers that move in space and time. The computational particles in PIC simulations can not be points. They must have a spatial extent in order to communicate with the electromagnetic fields, which are defined on a grid. If we place the field nodes at x = 0,1 in the figure above, the particles need a shape function with a width of at least 1. A computational particle can still be a point along v x, because there is no grid along this direction.
14 Interpolation particle-grid: NGP scheme The electric field E x is defined on the original grid x j and we want a centred scheme for d dx E x = ρ/ǫ 0 (electrostatic field). We interpolate the micro-charge from each CP to the nodes x j+1/2 of the grid, on which the macroscopic charge density is defined. The k th particle is centered at the position x k. The width of its shape function is x. Here we have x j+1/2 x k < x /2. We assign the charge of the particle k to the node x j+1/2. Xj 1/2 Xj+1/2 Xj+3/2 x k x x This charge assignment is called the next grid point (NGP) interpolation. It is fast but noisy for low numbers of particles per cell
15 Interpolation particle-grid: CIC scheme This interpolation is less noisy than the NGP charge assignment, because there are no jumps in the charge density as the CP goes from one cell to the next. Particle k is centered at x k. Its width is 2 x. It is closest to the center of the cell with index j. Its charge is mainly assigned to the cell x j. A smaller fraction is here assigned to the cell x j 1. x xk X X X j 1/2 j+1/2 j+3/2 The charge density of CP k is interpolated to the nodes x j 1/2 and x j+1/2 as ρ j 1/2 +ρ j+1/2 = ρ k and ρ j+1/2 /ρ k = (x k x j 1/2 )/ x and ρ j 1/2 /ρ k = (x j+1/2 x k )/ x. Bi-linear interpolation is called the cloud-in-cell method.
16 Current assignment using CIC-interpolation Electromagnetic PIC codes use J rather than ρ to update E and B. The charge density is sometimes used to test if ρ and E computed from Ampere s law fullfills also Gauss law. We solve the normalized Maxwell s equations in 1D: t E y(x,t) = x B z(x,t) J y, t E z(x,t) = x B y(x,t) J z, t E x(x,t) = J x. t B z(x,t) = x E y(x,t). t B y(x,t) = + x E z(x,t). In 1D along x: B = d dx B x = 0 and d dt B x = d dz E y d dy E z = 0. The B x component remains constant in space and time.
17 Current assignment using CIC-interpolation The three equations that use the current are the electromagnetic: t E y(x,t) = x B z(x,t) J y, and the electrostatic t E x(x,t) = J x. The current has to be defined on the grid. t E z(x,t) = x B y(x,t) J z, The current is related to the time-derivative of the electric field J is defined at the same time as B. It is thus defined at the same time as the particle velocities. The current in the electromagnetic equations is added to the spatial derivative of the magnetic field it is defined at the same position as E.
18 Current assignment using CIC-interpolation We have computed the force acting on the CP at its position x k (n t ). The new speed v k ((n +1/2) t ) is available and hence x k ((n +1) t ). x k,n+1 v k,n+1/2 x k,n+1/2 J j,n+1/2 x k,n We obtain from x k (n t ) and n k ((n+1) t ) the position x k ((n+1/2) t ). We use this position to interpolate the micro-current of the k th CP to J j,n+1/2. A loop sums the micro-current over all CP s and we have the macroscopic current update the system with Ampere s law.
19 Interpolation grid-particle / Electric field The shape function S(x k x) of a CP, which is centered at x k, is used to interpolate the electric field to the particle position. Ej 1 Ej Ej+1 The electric field E j,n on the grid nodes j is defined at times n t Xj 1 Xj X k Xj+1 The interpolated electric field (cloud-in-cell method) is We can use the position x k, which is defined at n t, to interpolate E j,n on the grid to E(x k [n t ]). E(x k ) = E j (x j+1 x k ) x +E j+1 (x k x j ) x The electric field at the time n t and at the position x k is the one, which we use to advance the particle velocity (Boris scheme).
20 Interpolation grid-particle / Magnetic field We use the same CIC shape function S(x k x) for the k th CP. The Boris pusher needs B(x k [n t ]). The particle position x k is defined at n t, while B(j +1/2,n+1/2) is defined at half-integer times and positions. j j+1 X k j 1/2 j+1/2 j+3/2 n n 1/2 n+1/2 Faraday s law allows us to compute B j+1/2,n+1/2 from B j+1/2,n 1/2 with no knowledge of ρ,j We have the magnetic field at both times. Let W 1 = S(x k x j+1/2 ) and W 2 = S(x k x j+3/2 ) B(x k [n t]) = (W 1/2) ( B j+1/2,n 1/2 +B j+1/2,n+1/2 ) +(W2/2) ( B j+3/2,n 1/2 +B j+3/2,n+1/2 )
21 Discussion We have introduced the plasma into our simulations in the form of a phase space fluid. Its phase space density distribution is the probability distribution for a random number generator. We use it to compute the initial velocities for the particles. The Boris scheme was discussed as a method to advance the particle velocities. Shape functions connected each CP to the fields on the grid and vice versa. Multiple plasma species: We treat them separately and add up their currents and charges to advance the electromagnetic fields. We know the working principle of a PIC code and now we can play around with it.
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