M. Okarnoto

N. Nakajirna

S. Murakarni

W.X. Wang

National Institute for Fusion Science (NIFS-Repository)

§ 18. On Linear Monte Carlo Collision Operator Conserving Momentum and Energy \iVang,\iV.X., Okarnoto,M., Nakajirna,N ., Murakarni,S. Usually, a nonlinear Monte Carlo colli-sion operator is very tin1e consurning. This point lin1its its application although it con-serves n1ornenturn and energy accurately and well describes Landau collision integral. A lin-ear Monte Carlo operator is required in n1any plasn1a sirnulations. Recently such linear oper-ators are developed for gyro kinetic b f particle simulations[1 ,2]. The linear operators conserving mon1enturn and energy can also serve full- f particle sin1ula-tion when f is close to a Maxwellian. Consider a general Fokker-Planck equation df af . . af -d =-a + zl_a . = Crp(f) + P(Z)Fm (1) t t Z 1 where Z represents the phase space variables. The test particle operator reads Crp(f) = where the fraction coefficient and diffusion tensor are calculated under Ivlaxwellian back-ground distribution. The source term P(Z)Fm accounts for the n1omentum and energy con-servation by replenishing in a Maxwellian dis-tribution F m the momentum and energy re-moved by Crp(f). To implernent the source term one introduces changing particle weight by setting f = w(Z, t)g(Z, t) . In conventional particle sirnulation the weight is constant in time and uniforn1 in velocity space, for exan1-ple, w('v, t) = 1. Then, Eq.(1) is rewritten into dg dw w[dt - Crp(g)] + g[dt- Crp(w) + wCrp(1) _ aw a ln g . < b:.ih:~iJ >0 ] = P( .... )F av a·v . b:.t v rn. (3) Setting dg dt - Crp(g) = 0, (4) we obtain the equation of evolution for the par-ticle weight as follows Dw dw Dt = dt- Crp(w) + [wCrp(1) _ aw a ln g . < b:. v b:. v > 0 ] = P ( v) F m ( 5) av av · l::lt 9 Equat ion ( 4) is solved by standard particle I\1onte Carlo sirnulation to obtain g = ~ib(Z­Zi)· To understand Eq.(5) and find the rnethod to solve it, consider the Fokker-Planck Equa-tion ·without the source tenn df dt = Crp(f). (6) By the san1e method as for Eq. ( 1) we obtain dg -- Crp(g) = 0 and dt Dw = O. Dt (7) The fact the equation for g has the completely sarne form as Eq.(6) indicates that w = 1 is a solution of equation Dw I Dt = 0. In other words, the test particle operator Cr p does not change the particle weight (it changes only the phase space coordinates of particles) and Dw I Dt can be considered as the tilne deriva-tion of weight along the partiCle trajectories as influenced by the test particle collisions. Now it is found that the particle weight can be solved by si1nply integrating Eq.(5) as follows (g ~ Fm is assumed) b:..wlzi = !:ltP(Z)Izi. (8) The solution for Eq.(1) is then obtained as (9) Finally, we point out that in the previous work the equation for the weight is incorrect (the t\VO tenns in square bracket of Eq.(5) is lost). Reference 1) Dimits,A.M. and Cohen,B.I. Phys. Rev. E 49 (1994) 709. 2) Lin,Z., Tang,W.M. and Lee,W.W., Plasrna Phys 2. (1995) 2975. 199 

On Linear Monte Carlo Collision Operator Conserving Momentum and Energy

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