Fusion plasma is a typical nonequilibrium and nonlinear system with the interaction between different species well described by the VlasovFokkerPlanck VFP equations. The transport of mass momentum energy and temperature relaxation are important issues which are affected by the collision term of VFP even in socalled collisionless plasma domain. Hence nonlinearity and collisions are crucial features in large regime. A successful numerical simulation for nonequilibrium plasma has to be able to conserve mass momentum and energy while to satisfy Boltzmanns Htheorem and higherorder moment convergence. An expansion of the distribution function in spherical harmonics Legendre basis when the velocity space exhibits axisymmetry in angle coordinate and in King basis in speed coordinate of velocity space is well suited to address these requirements. This paper reviews the formulation of the 0D2V VFP equation in terms of spherical harmonics coupled with King function and its solution in our NLVFP code. In this topic review we will introduce the background physics related to the nonlinear VFP simulation then describe NLVFP for 0D2V homogeneous weakly anisotropic plasma with utilization of the Shkarofskys form of FokkerPlanckRosenbluth FPRS collision operator.
['Physics - Plasma Physics', 'Astrophysics - Solar and Stellar Astrophysics', '52.65.Ff', '52.25.Fi', '52.25.Dg', '52.35.Sb']
Higherorder moment convergent method in weakly anisotropic plasma and the NLVFP code for solution of the 0D2V VlasovFokkerPlanck equation
{'Yanpeng Wang ∗': 'School of Nuclear Sciences and Technology,  \nUniversity of Science and Technology of China, Hefei, 230026, China', 'Abstract': "Fusion plasma is a typical non-equilibrium and nonlinear system, with the interaction between different species well described by the Vlasov-Fokker-Planck (VFP) equations. The transport of mass, momentum, energy, and temperature relaxation are important issues, which are affected by the collision term of VFP even in so-called collision-less plasma domain. Hence, nonlinearity and collisions are crucial features in large regime. A successful numerical simulation for non-equilibrium plasma has to be able to conserve mass, momentum and energy, while to satisfy Boltzmann's H-theorem and higher-order moment convergence. An expansion of the distribution function in spherical harmonics (Legendre basis when the velocity space exhibits axisymmetry) in angle coordinate and in King basis in speed coordinate of velocity space is well suited to address these requirements. This paper reviews the formulation of the 0D-2V VFP equation in terms of spherical harmonics coupled with King function and its solution in our NLVFP code. In this topic review, we will introduce the background physics related to the nonlinear VFP simulation, then describe NLVFP for 0D-2V homogeneous, weakly anisotropic plasma with utilization of the Shkarofsky's form of Fokker-Planck-Rosenbluth (FPRS) collision operator.  \nKeywords: Finitely distinguishable independent features hypothesis, Higher-order moment convergence, VlasovFokker-Planck equation, Kinetic moment-closed model, Fokker-Planck-Rosenbluth collision operator, Axisymmetry", 'I. INTRODUCTION': "In fusion plasma relevant to conventional magnetic confinement fusion (MCF) and inertial confinement fusion (ICF), the transport processes between the main-body plasma species, as well as the wave-particles interactions, typically play a dominant role in shaping the evolution of the nonequilibrium and nonlinear system[1]. Transport, since it involves collisions, has traditionally been modelled with Vlasov-Fokker-Planck (VFP) codes. However, simulations for these transport processes usually face challenges to conserve mass, momentum and energy[2]. The fundamental difficulty in simulations lies in the need to include both collisionless fast particles (such as high-energy α particles) and highly collisional thermal particles (fusion fuels) into a consistent simulation at the same time. While hybrid codes[3] separate the two distributions for treatment, they encounter issues such as fast particles escape effects, non-Maxwellian distortion of the thermal electron distribution, and thermal particles being elevated to fast particles in the absorption region of ion cyclotron resonance heating (ICRH) or electron cyclotron resonance heating (ECRH). That is saying for certain problems, it is imperative to treat the particle population as a unified entity rather than separating them. Additional difficulties arise from the disparate thermal velocities[4-6] due to the significant mass discrepancy ( i -e collisions) or energy difference ( i -α collisions), and a linearized model is usually conventionally chosen[7-11].  \nAn effective approach based on the finite volume method[12] (FVM) is employed to solve the 0D-2V nonlinear VFP equation[5, 13-16], incorporating the divergence form of Fokker-PlanckRosenbluth[5] (FPRD) collision operator. This approach has successfully preserved mass, momentum and energy, satisfying Boltzmann's H-theorem [17], while it is still a challenge to achieve the higherorder moment convergence. Other moment convergent kinetic approaches, such as discrete Boltzmann method[18] (DBM), and lattice Boltzmann method[19] (LBM) have demonstrated the importance of higher-order moment convergence in simulating the VFP equation (Boltzmann equation in plasma physics). However, these kinetic approaches typically rely on conventional near-equilibrium assumption, which may limit their accuracy and applicability. The above limitations can be effectively addressed by NLVFP.  \nBuilding upon the previous works by Rosenbluth[20], Bell[10], and Tzoufras[11] in employing the spherical harmonic expansion (SHE) method (Legendre basis will be utilized for axisymmetric velocity space), we have developed a novel higher-order moment convergent method[6, 21] for weakly anisotropic plasma in NLVFP. Our approach utilizes the nonlinear FPRS collision operator and incorporates a King function expansion (KFE) method following SHE. This methodology, encompassing both a moment approach[21, 22] and a meshfree approach[6], is introduced for solving the VlasovFokker-Planck (VFP) equation in scenarios where the plasma is weakly anisotropic[6, 10]. In contrast to conventional methods such as Grad's moment method[23], DBM[18], and LBM[19], which are based on the near-equilibrium assumption, this moment convergent method relies on a finitely distinguishable independent features[24, 25] (FDIF) hypothesis introduced in Ref.[21]. Under this hypothesis, we have presented a series of studies[6, 21, 22]. A general relation model called the kinetic moment-closed model[21, 22] is derived from the VFP equation for homogeneous plasma under spherical symmetry and axisymmetry in velocity space respectively, which can be extended to the particle distribution function (PDF) with 3V velocity space. A conservative, implicit solver for 0D-2V multi-species nonlinear VFP equation in the meshfree approach is introduced in Ref.[6].  \nThe key innovation of this moment approach lies in the introduction of new functions, specifically the King introduced in Ref.[21] and R function introduced in Ref.[22]. This approach enables the achievement of a relaxation model[21] for homogeneous plasma and its extension to a general relaxation model[22]. This general relaxation model allows for the derivations of a general temperature relaxation model and the explicit determination of the general characteristic frequency of temperature relaxation in the presence of spherical symmetry in velocity space. Furthermore, when the velocity space is axisymmetric, an optimal implicit meshfree algorithm also has been introduced for solving the coupled multi-species nonlinear VFP equation in 0D-2V[6]. The key to this meshfree approach is expanding the solutions in terms of the Legendre basis (in polar angle coordinate) and King basis (in speed coordinate). The Legendre polynomial expansion converges exponentially, while the novel King method serves as a moment convergent algorithm that ensures conservation with high precision in discrete form. Both of these moment convergent methods satisfy conservation laws, Boltzmann's H-theorem and higher-order moment convergence, making them suitable for studying nonlinear evolution at multi-time and multi-space scales in weakly anisotropic plasma physics.", 'A. Expansions utilized in NLVFP': "NLVFP, shorts for nonlinear VFP, is currently a zero-dimensional in physics space, multidimensional in velocity space VFP code, employing expansion methods. In NLVFP, the 0D-2V normalized PDF is represented in normalized velocity space ( ˆ v, µ ) by an expansion in Legendre basic[26], P l and King basis[21], Κ l :  \nwhere, C 3 = √ 2 π/π 3 / 2 , l ∈ N , l M ∈ N , s ∈ N + , N K a ∈ N + . n a and v ath are respectively the number density and thermal velocity of species a . Normalized velocity intensity of species a , ˆ v = v/v ath and normalized PDF of species a , ˆ f = n -1 a v 3 ath f . Parameters ˆ u a s and ˆ v ath s are the characteristic parameters of the s th King function, with weight function ˆ n a s . The convergence of King function expansion (KFE) is established by utilizing Wiener's Tauberian theorem[27] and demonstrated in Ref.[21] under the FDIF hypothesis. This hypothesis indicates that N K a will typically be a finite number, usually no more than 20 for a weakly anisotropic plasma.  \nˆ f (ˆ v, µ, t ) = C 3 n a v 3 ath l M ∑ l =0    N Ka ∑ s =1 [ˆ n a s ( t ) Κ l (ˆ v ; ˆ u a s ( t ) , ˆ v ath s ( t ))]    P l ( µ ) , (1)  \nBell et al.[10] have presented a comprehensive overview of the benefits and drawbacks of utilizing SHE in angular coordinate of velocity space coupled with a finite difference method (FDM) in speed coordinate. In addition to the two limitations mentioned by Bell, such as negative of amplitude of distribution function and challenge near the point where ˆ v = 0 , momentum conservation and fully nonlinearity also pose challenges for their algorithm. However, all these obstacles can be overcome by employing the KFE approach.  \n- 1. Some advantages of KFE\n- (i) NLVFP will not be limited by the near-equilibrium assumption, commonly utilized in conventional methods such as LBM/DBM and Grad's moment approach. This is because KFE is based on the FDIF hypothesis[21].\n- (ii) King function serves as a one-dimensional shape function with continuous smoothness, which also acts as the smoothing step for calculating the PDF.\n- (iii) KFE successfully navigates the challenge near the point where ˆ v = 0 in SHE, thanks to the smoothness of the King function.\n- (iv) KFE is able to automatically maintain the non-negativity by determining the appropriate truncation for SHE.\n- (v) Coulomb collisions not only make amplitudes of PDF decay exponentially in SHE due to angular scattering[10], but also lead to rapid convergence for the amplitudes in speed coordinate. The KFE is a moment convergent technique that has been demonstrated to achieve up to order 16[6]. Therefore, conservation can be achieved with high precision in a discrete manner, and high-order moment convergence can be easily addressed for any order without special treatment.\n- (vi) Coupling KFE with SHE ensures the symmetry of the collision operator in discrete and accurately satisfies conservation through the application of manifold technique.\n- (vii) NLVFP is able to accurately capture the full nonlinear effects of the VFP equation, attributed to the rapid convergence of SHE and KFE.\n- (viii) NLVFP, employing SHE and KFE, can naturally capture the isotropic Maxwellian state in velocity space, and the resulting solution exhibits robustness.  \n- 2. Some disadvantages of KFE\n- (i) NLVFP retains the full nonlinearity in both the angular and speed coordinate of velocity space, resulting in a certain degree of increase in computational complexity.\n- (ii) The characteristic parameters in KFE should be determined using alternative methods[6, 21], such as solving the characteristic parameter equations as described in Sec. II B 3.\n- (iii) The current version of NLVFP is limited to weakly anisotropic plasma due to the assumption that characteristic parameters in KFE are independent of l , resulting in the code's inability to handle scenarios where the average velocity is comparable to thermal velocity.  \nThe initial two challenges, (i) and (ii), are inherent outcomes of the nonlinearity and have not been observed to pose significant issues individually. The third limitation can be addressed by expanding to a more general version for moderate anisotropic plasma, where the characteristic parameters are dependent on l . This enhancement will be further developed in future studies.  \nExpansion in spherical harmonics and King function might appear complicated, but in fact the equations turn out to be fairly straightforward and robust due to the following three reasons: (A) Many natural statistical systems, including plasma system, are independent and identically distributed systems that often adhere to the central limit theorem [28], allowing their probability density functions to be described using Gaussian functions. (B) Spherical harmonics are a natural basis for angular calculations in 3D, as found in many branches of physics. (C) King function is the form of Gaussian function in spherical coordinate system, which can be obtained by employing SHE.", 'B. The equation solved by NLVFP': 'NLVFP currently solves the VFP equation in zero spatial and two velocity dimensions, which can be extended into three velocity dimensions by defining a more general function called associate King function, Κ m l .', '1. Meshfree approach': 'The 0D-2V NLVFP equations in the meshfree approach[6] are VFP spectrum equation, which can be written in the form  \n∂ ∂t f l (ˆ v, t ) = n a v 3 ath ˆ C l (2)  \nfor the evolution of all amplitudes of distribution function, f l = n a v -3 ath ˆ f l . In contrast to traditional usage of the simplified model of FPRS collision operator[7-11, 29], the l th -order normalized amplitude of collision operator utilized in NLVFP is the full nonlinear form of FPRS, which is described as:  \nˆ C l (ˆ v, t ) = ∑ b n b v 3 bth ˆ Γ ab ˆ C lab . (3)  \nFunction ˆ C lab represents the l th -order amplitude of normalized FPRS collision operator for species a colliding with species b , as given in Ref.[6], and can be quoted as:  \nˆ C lab (ˆ v, t ) = ∑ l M 1 β =1 w µ β P l ( µ β ) ˆ C ab (ˆ v, µ β , t ) . (4)  \nHere, function ˆ C ab represents the normalized FPRS collision operator, which can be calculated directly based on the values of ˆ f l . It has been demonstrated in Ref. [6] that these meshfree methods, utilizing SHE and KFE techniques, exhibit moment convergence.', '2. Moment approach': 'The 0D-2V NLVFP equations in the moment approach is the kinetic moment evolution equation, serving as a transport equation and can be written in the form  \n∂ ∂t Μ j,l ( t ) = ρ a ( v ath ) j ˆ Ρ j,l , j ≥ -2 -l . (5)  \nfor the evolution of all kinetic moments, Μ j,l . Here, the ( j, l ) th -order kinetic moment and normalized kinetic dissipative force of species a are respectively defined as:  \nΜ j,l ( t ) = ρ a ( v ath ) j ˆ Μ j,l = 4 πρ a ( v ath ) j ˆ ∞ 0 ˆ v j +2 ˆ f l dˆ v, j ≥ -2 -l, (6)  \nˆ Ρ j,l ( t ) = ∑ b n b v 3 bth ˆ Γ ab ˆ Ρ abj,l , j ≥ -2 -l, (7)  \nwhere mass density ρ a = m a n a and the normalized kinetic dissipative force between species a and species b  \nˆ Ρ abj,l ( t ) = 4 π ˆ ∞ 0 ˆ v j +2 ˆ C lab dˆ v, j ≥ -2 . (8)  \nThe kinetic dissipative force closure relation, as represented by Eq. (7), is solely dependent on the mass ratio, thermal velocity ratio and characteristic parameters of King function. Its analytical expression has been derived for shell-less spherical symmetric velocity space[21] under the FDIF hypothesis. This expression is then extended to general spherical symmetric velocity space through the introduction of the R function[22]. These results can also be extended to scenarios where the velocity space exhibits axisymmetry and will be published in the future.  \nThe combination of the transport equation (5), kinetic dissipative force closure relation (7), analytical expression of ˆ Ρ abj,l and the character parameter equations (9) constitutes a set of nonlinear equations, which is referred to as kinetic moment-closed model[22]. A general temperature relaxation model[22] for spherical symmetric velocity space has been derived from this general relaxation model. Furthermore, the classic models such as two-temperature thermal equilibrium model, Braginskii model and thermodynamic equilibrium model are special cases of the relaxation model[21].', '3. Characteristic parameter equation': 'Under the FDIF hypothesis, the characteristic parameters ˆ n a s , ˆ u a s and ˆ v ath s in Eq. (1) and kinetic dissipative force closure relation (8) are solely functions of any 3 N K a different orders of normalized kinetic moments. Those parameters can be determined by solving the characteristic parameter equations (CPEs)  \nˆ Μ j,l ( t ) =                C M l j N Ka ∑ s =1 ˆ n a s (ˆ v ath s ) j   1 + j/ 2 ∑ k =1 C k j,l ( ˆ u a s ˆ v ath s ) 2 k   , l ∈ 2 N , C M l j N Ka ∑ s =1 ˆ n a s (ˆ v ath s ) j ( ˆ u a s ˆ v ath s ) l   1 + ( j -1) / 2 ∑ k =1 C k j,l ( ˆ u a s ˆ v ath s ) 2 k   , l ∈ 2 N +1 . (9)  \nHere, the coefficients C k j,l and C M l j are contingent upon the parameters k , j , and l as delineated in Ref.[6]. The CPEs are a system of nonlinear algebraic equations, typically served as constraint equations for the VFP spectrum equation (2) or the kinetic moment evolution equation (5). In Ref.[6], several algorithms are presented for solving CPEs to determine characteristic parameters, and more advanced algorithms are currently being developed.', 'C. NLVFP: further extension and applications': 'The aforementioned equations lie at the heart of NLVFP, a framework that utilizes the expansion method based on the FDIF hypothesis. Generally, the moment approach, resembling an analytical method that does not require field grids of velocity space, proves to be more effective than the meshfree approach which is a semi-analytical and semi-numerical method in velocity space. Both approaches in NLVFP can be extended to general scenarios, involving axisymmetric velocity space where characteristic parameters may dependent on l and a general velocity space. Furthermore, we aim to develop the nD-mV NLVFP code by incorporating the Vlasov parts in our future work.  \nOne strength of expansion method over a particle code is that it produces results without noise and provides a solid platform for the inclusion of further physics, and over an FDM/FVM code is the ease of handling full nonlinearity and achieving higher-order moment convergence. NLVFP has the potential for application to a wide range of plasma types, particularly in scenarios where magnetic field-dominated transport is involved. Its expansion in spherical harmonics makes it well-suited for representing the rotation of particle trajectories with algebraic expressions, without involving differentials in velocity space. This method is especially effective for magnetized particle transport when the distribution is weakly anisotropic, requiring only a small numbers of spherical harmonics and King functions. It is worth to note that expansion methods avoid extensive computer time spent on modeling the isotropic part of the distribution, unlike traditional particle codes. Therefore, NLVFP may find utility in MCF as well as in space and astrophysical plasmas where magnetic fields play a dominant role.', 'III. CONCLUSION': "Anexpansion framework based on SHE and KFE methods has been developed in NLVFP code to solve the 0D-2V nonlinear VFP equation for weakly anisotropic plasma. This framework, operating under the FDIF hypothesis rather than the conventional near-equilibrium assumption, offers significant advantages over traditional methods. A general relation model, known as the kinetic momentclosed model, is presented for homogeneous plasma under spherical symmetry and axisymmetry in velocity space respectively. This provides a general temperature relaxation model and explicitly determines the explicit determination of the general characteristic frequency of temperature relaxation in the presence of spherical symmetry in velocity space. Those models can be extended to PDF with tree-dimension velocity space. 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