Nonlinear Fokker-Planck collision operator in Rosenbluth form for gyrokinetic simulations using

The formulation of Coulomb collisions has been developed mathematically in two different ways: 1) an integro-differential form by Landau [1]. Due to the approximation, it was noted that the operator friction for high-energy particles increases with velocity [6, 9].

Backgrounds

R2cB2 where v∥ is the velocity parallel to B and Rc is the radius of the center of curvature. In this case, heat diffusion by ion-ion collision is the largest, due to the difference in mass between ions and electrons.

Figure 1: Schematic of a tokamak geometry [20].

Motivation and objective

The usefulness of the DG method in those stiff gradient regions is demonstrated in the previous gyrokinetic SOL simulations [ 33 , 36 ]. Careful and consistent numerical treatments are required for the derivatives still present in the weak form.

Thesis outline

There has been growing interest in modeling the entire gyrokinetic tokamak device using DG methods [33, 35]. To name just a few, the methods allow flexible selection of basis functions to represent numerical solutions, which can vary greatly depending on the simulation region. In particular, in the edge or shear layer (SOL) regions of a tokamak, density or temperature gradients can be extremely stiff to be treated at typical grid resolution.

The methods also allow for discontinuities of numerical solutions and enable highly localized computing, which can be exploited for efficient parallelization of simulation. For example, the core-to-core communication required for the equation of motion based on DG methods is limited to the local data exchange between adjacent grid cells, rather than the global communication that can be numerically expensive. However, because DG methods are based on the weak form of gyrokinetic equation, terms with derivatives higher than first order are not easy to handle in standard DG methods.

Since nonlinear Coulomb collisions are an essential component in a comprehensive tokamak plasma WDM, it is highly desired to develop a numerical formulation and scheme for nonlinear collisions using a DG method.

The gyroaveraged RMJ collision operator

In this part, the last term of Eq. 20) is neglected assuming negligible FLR effects, i.e. (vT/B0). This is consistent with previous work showing that the linearized collision operator in the gyrokinetic coordinate can be written in a divergent form when the FLR effect is neglected [ 39 ]. Numerical conservation of the gyrocenter density can be ensured more easily due to this property.

In addition to the RMJ operator, two simpler models (i.e., the test particle collision and the Dougherty model) are also implemented for comparison.

The weight evolution equations with DG basis

The corresponding basis function space forΩ[j,k] is given as. 21) from an arbitrary function W(⃗z) and getRΩ[j,k]×, the following equation is obtained. and do it again and then apply Eq. 31), the weighted equation is taken as Z. For the calculation of those surface terms, we mainly follow the method presented in [32] with some minor changes. For completeness, the models used in this work to calculate the surface terms are presented in the rest of this section.

On the other hand, non-physical accumulation of particles is observed when F[j. v∥ are set to zero at the outermost boundaries. If we set W =ζl[j,k] and replace Eq. 34), the weight evolution equations of ˆfl[′j,k](t) are obtained as. For the temporal discretization, the Runge-Kutta method of the third order SSP (Strong Stability Preserving) is used [41].

As shown in [32], the numerical robustness of the DG collision operator with SSP is strongly affected by the grid size and the order of the basis functions.

Figure 2: Single mesh cell Ω [ j,k] in the phase space.

Conservation of physical quantities

From these moments the parallel fluid velocity U∥and the thermal velocity vT can be obtained as. 45) represents the total number of particles and ∂M0/∂t can be evaluated from Eq. u is set to zero at the outer limits. Similarly, the energy conservation constraint, i.e. ∂M2/∂t=0, is formulated from Eq. 52) Since it is not guaranteed that Eq. 51) and (52) are automatically satisfied, additional numerical operations are required for the conservation of M1 and M2. The 1st term of the right-hand side (RHS) of Eq. 58) is an advection operator inv∥ direction and the 2nd term of RHS is a diffusion operator. Since LA is in the divergence form, the particle conservation is little affected by the introduction of LA.

Onceβ1 and β2 are calculated from equations. 62) and (63), the weight evolution equations are obtained from Eq. Unlike the Dougherty model, the collision of test particles does not satisfy Eqs. J/∂v∥ and the outer boundary terms are neglected. Moreover, γn's are free parameters used to enforce conservation, as in the case of LA. MetLF is the weighted equation, Eq. 70), the conservation constraints are given as follows.

The pasiγn's are calculated from Eq. 71), the weight evolution for fd[j,k] can be calculated from Eq.

FEM solver for the Rosenbluth potential

Unlike the case of LA presented in the previous section, the implementation of ∂M0/∂t=0 is important for LF to guarantee the numerical robustness of the simulations. To distinguish [test particle collision +LF] from [test particle collision +LA], we refer to the former as the “Linearized Collision Operator”, while the latter is simply called the “Test Particle Collision Operator” in Sect. other part of this thesis. must be sufficiently higher to deal with the order of the derivative required for the coefficients αi in Eqs. When we replace those third-order derivatives ofg with first-order derivatives ofh, Eq. 75) Detail of deriving the equation.

In [45] (h,g) is split into Maxwellian and non-Maxwellian parts to preserve the Maxwellian distribution function exactly at the equilibrium. Here (hM,gM) are the analytical solutions of the following equations. hM,gM) are introduced to improve the numerical resolution when f is close to fM and their specific forms are given in Eq. Since the maximum derivative order of Eq. 75) is the second order, cubic B-splines are chosen as basis functions for (δh,δg).

Here, K[k]¯ and E[k]¯ are complete elliptic integrals of the first and second kind, respectively.

Numerical convergence test with the initially loaded Maxwellian distribution

Boundary conditions for δhandδg' is the time required to calculate the boundary conditions for the Rosenbluth potentials, i.e. Eq. Although Nratio3/2 is an expected scaling law for Eq, there have been several studies to improve the scaling. As mentioned in Section 2.4, a direct LU factorization with the Intel MKL PARDISO library is used for the FEM solver. Field particle collision model' is the calculation of the conservation model for the linearized collision operator.

The most time-consuming part is the calculation of the 'Source terms for the weight evolution', which represents the calculation for the right-hand side of Eq. The computational cost of this part for linearized and nonlinear operators is about 60%–80% higher than for the Dougherty operator, since more computation is required to evaluate the coefficients of the linear and nonlinear models. Solving for the weight development' is the procedure for solving Eq. 44) when the source conditions have been evaluated.

The total computation time of the nonlinear collision operator is about 70% ~ 90% higher than that of the Dougherty operator.

Figure 3: The residual of C( f M ) with varying grid sizes. ∆v is the grid size of each velocity cell for both of v ∥ and u directions.

Relaxation of the bump-on-tail distribution

The time unit is normalized by the total computation time of the Dougherty collision model with [Nv∥,Nu] = [20,10]. Here the entropySis is defined as S≡ −∑. 86) The Dougherty operator shows the fastest time scale and the nonlinear operator shows the slowest time scale for the relaxation. Despite the different relaxation rate for each model, the entropy increases monotonically in time and converges to the same value for all cases, as expected.

Here, nN, U∥, N, and TN correspond to the density, mean parallel velocity, and energy, respectively. Both cases show excellent density conservation since the collision operator is in divergence form. Note that it is possible to further improve density conservation if a more free parameter β3f is added to Eq.

Momentum and energy from simulations without LA increase gradually in time, while the case with LA maintains conservation properties.

Figure 4: Relaxation of the bump-on-tail distribution function f with the nonlinear collision operator.

Anisotropic temperature relaxation

On the other hand, the relaxation rate of the Dougherty operator is about 2.5 times larger than the analytical formula. Numerical stability with different time step sizes is also tested for each collision operator. The maximum time step ∆tmax,theo for the Dougherty collision operator with SSP3 time integration can be estimated as [32]. where ∆v∥ and ∆u are grid sizes in v∥ and directions. Likewise, ∆p,Cadv,pandCdi f,pairs functions of base order p. 92) and ∆tmax from the simulations shown in fig.

In this section, we present neoclassical benchmark results for the various collision models introduced in the preceding sections. A set of gyrokinetic equations [38] without the FLR effect is used for the collisionless part of the simulation. Although these equations are not suitable for the quantitative micro-turbulence study, they are sufficient to investigate neoclassical physics in the drift kinetic limit.

The third-order SSP3 Runge-Kutta method [41] is used for the time integration of gyrokinetic equations, as well as for the collision operator.

Table 4: The maximum time step ∆t max from analytical estimation of Eq. (92) and anisotropic tempera- tempera-ture relaxation simulations when ∆v ∥ = ∆u = 0.5v T and max(|v ∥ |) = max(u) = 5v T .

Neoclassical radial heat diffusivity

Finally, the nonlinear operator exhibits slightly smaller heat diffusivityχ than the linear operator as observed in [50].

Neoclassical poloidal flow

Damping of residual potential

Collisional effects on GAM frequency and damping rates

In order to check whether these behaviors can be reproduced by the crash model of this work, the results of the GAM crash damping benchmark are presented in this subsection. The conservation properties of collision models are important for the stability and reliability of long-term gyrokinetic simulations. Due to the divergent structure of the formulation, density conservation is ensured numerically.

By using LA, the conservation of the momentum and energy is enforced to the precision of the machine. In the case of the linear collision operator, the linearized field particle collision operator is adopted as another way to enforce the conservations. Several benchmark problems are solved to test the numerical properties of the developed collision models.

On the other hand, the damping rates from simulations for ν∗=1.0 are about 50% of the theoretical prediction. This indicates that the contribution of the cross diffusion terms in the small velocity limit is O(x2) smaller than the diagonal diffusion terms in Eq. With the coefficients of the RMJ collision operator, V[j,k] of the weighted equation, i.e. Eq. 74), contains the third-order derivative afg.