M. Okamoto

N. Nakajima

W.X. Wang

National Institute for Fusion Science (NIFS-Repository)

§24. General and Accurate Weighting Scheme for Collisional c)f Particle Simulation Wang,W.X., Nakaji1na,N., Okamoto,M. Let us solve linearized drift kinetic equation ~t/1 = -vd · \l !o + C(fo, fl) (1) from the dynamics of a finite number of particles (markers), where the notation D / Dt(f) denotes ~/ = ~~+(vii+ vd) · \7 f- C(f, fo). (2) The distribution function f 1 is represented as !1(i,v,t) = Lwj8(i-ij(t))8(v-vj(t)). (3) j The particle weights w are calculated from W = ~ [- j wSMdw- vd · \lfo + C(fo,Jd], (4) where S M is the marker source used to con-trol marker population during simulation and 9 is marker density[1]. The calculation of particle weight takes a central position in a 8 f simulation. It is demonstrated that valid results essentially rely on the correct evaluation of marker density 9 in weight calculation. Previous weighting schemes including the nonlinear weighting scheme[2] em-·ploy an assumed 9 in weight equation for advanc-ing particle weights. Such a sche1ne is ineffective or inaccurate to solve drift kinetic equation because of a severe constraint the real marker distribution must be consistent with the assumption during a simulation. Instead using an approximation of 9 for advancing particle weights, we solve 9 directly fr01n its kinetic equation .E_9 = j SMdw. Dt Setting 9 = fo + 91, we obtain (5) ~tg1 = -vd · \lfo + j SMdw. (6) We e1nploy a new weight function w and a marker distribution function G M(i, v, w, t), and then solve Eq.(6) using the idea of 8f 1nethod. Let GM obey the kinetic equation ~tGM + :w (wGM) = OM(i,v ,w, t) (7) with GM(t = 0) = fo8(w) , and relate to 9 1 through 91 (x, v, t) = J wG Mdw. (8) 182 Here a source 0 M, like S M, is introduced to control n1arker population. Fr01n the requirement that Eqs.(6)-(8) are consistent, the equation for w is determined as where distribution function h(i, v, t) satisfies (10) Taking (11) Then Eqs.(10) and (5) become the same. We have h(i, v, t) = 9(X, v, t). For a marker of index j, we have which gives 1 9· = --fo·. 1 1-w· 1 J (12) (13) (14) Substituting Eq.(14) into Eqs.( 4) and (9), we ob-tain weight equations for a 1narker as follows W = 1 Jo w [- j wSMdw- vd · \7 fo + C(/o,Jtl], (15) W= 1 j0w [- j wf!Mdw-vd·\lfo+ j SMdw]. (16) Now each simulation particle is assigned two weights rather than one, the second weight w is introduced to effectively evaluate 9 (exactly say-ing, 91)., and we can solve Eqs.(1) and (6) si-multaneously using one set of markers. Equa-tions (15) and (16) represent a general and accu-rate weighting scheme. For practical application it is convenient to take SM = v(t)s(f)fo8(w) and OM= v(t)s(1"~)fo8(w). This choice means that new Maxwellian markers with w = w = 0 are added in terms of rate v(t) and spatial distribution s(f). Then the weight equations are silnplified to W = 1 Jo w [-vd · \7 fo + C(/o,Jd], (17) 1-w w =--[-tid· \l !o + vsfo]. (18) !o References 1) Chen,Y. and White,R.B. Phys. Plasmas i (1997) 3591. 2) Lin,Z., Tang,W.M. and Lee,W.W., Phys. Plasmas ~ (1995) 2975. 

General and Accurate Weighting Scheme for Collisional ホエf Particle Simulation

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