M. Nunami

R. Kanno

S. Satake

National Institute for Fusion Science (NIFS-Repository)

We develop a new Monte Carlo simulation code forovercoming diﬃculty in conventional Monte Carlo meth-ods, where the diﬃculty is caused by nonlinear terms inﬂuid equations. To conﬁrm the computational principleof the new method, we solve Dirichlet problems in one di-mensional (1D, i.e., radial) coordinate space in the ﬁrsttrial. The code is called DIPS-1D (DIrichlet ProblemSolver in 1D coordinate space).In general, the ﬂuid equation expressed in the formof the Fokker-Planck type equation can be rewritten asthe following initial-boundary value problem (t is re-placed by t1 − t):(L+ η∗)u+∂u∂t= h∗(t,x) in Q, (1)u(t1,x) = Φ(x) on M, (2)u(t,x) = G(t,x) on S, (3)where M is a bounded domain with the boundary ∂M,Q =M× [0, t1), S = ∂M× [0, t1), andLu ={12Dij ∂2∂xi∂xj+ U i∗∂∂xi}u, (4)U i∗ = −U i +12∂Dij∂xj, (5)η∗ = −η −∂U i∂xi, (6)h∗ = −h. (7)If Φ, G, h, η, D and U are assumed to be given-smooth-functions, the solutions of Eqs. (1)-(3) are known to bedescribed asu(t,x ;Φ, G, h∗, η∗,D,U∗)= Et,x[Φ(X(t1)) exp{∫ τtη∗(s,X(s))ds}χτ=t1]+Et,x[G(τ,X(τ)) exp{∫ τtη∗(s,X(s))ds}χτ<t1]−Et,x[∫ τth∗(s,X(s)) exp{∫ stη∗(ϑ,X(ϑ))dϑ}ds],(8)where Et,x is the expectation operator given by the dif-fusion process X(s) in coordinate space:dXi(s) = σij(t,X(s))dW j(s) + U i∗(t,X(s))ds (9)Fig. 1: Radial energy transport for electrons. Dottedline represents the initial guess Φ(r). Dashed line repre-sents the solution found by DIPS-1D at the temporarystep, and solid line represents the solution at the ﬁnalstep.satisfying X(t) = x. Here Dij = σikgkℓσjℓ , gkℓ is themetric,W (s) is a Brownian process, χA is the indicatorfunction of a set A (e.g., χτ<t1 = 1 if τ < t1, and χτ<t1 =0 otherwise), and τ is deﬁned asτ =the ﬁrst time ϑ ∈ [t, t1) that X(ϑ) leaves Mif such a time exists,t1 otherwise.Using the Monte Carlo simulation code DIPS-1Dbased on Eq. (8), we solve iteratively the following radialenergy transport equation for electrons.∂∂t(32nTe)+ 1r∂∂r(rκgBe∂Te∂r)− 32⟨σv⟩rren2Te+S = 0,(10)where the initial condition is given as Φ(r) = T (0)ax −{T (0)ax − T (0)edge}r/a with T (0)ax = 2 keV and T (0)edge = 200eV. The boundary conditions are G(t, r0) = Te(t, r0) =Te(t, r0 + δr) − δr ∂Te/∂r(t, r0) and G(t, a) = Te(t, a) =200 eV with suﬃciently small δr and r0 (i.e., 0 <δr/a, r0/a ≪ 1), and ∂Te/∂r(t, r0) = −1.8 keV/m. Teis the electron temperature, and the constant densityis assumed to be n = 1 × 1019 m−3. κgBe is the gyro-Bohm thermal conductivity, a = 1 m is the minor ra-dius, ⟨σv⟩rre is the radiative recombination rate coeﬃ-cient, and S is the heat source. The steady-state solu-tion of Eq. (10) is given in Fig. 1. Details are shown inRef. [1].1) R.Kanno, S. Satake, M.Nunami, Plasma Fusion Res.(2011), in press.398§26. Monte Carlo Simulation Code for Solving Radial Fluid EquationsKanno, R., Satake, S., Nunami, M.

Monte Carlo Simulation Code for Solving Radial Fluid Equations

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