A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

Abstract

We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17, 18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation.

Keywords: Spectral methods, Boltzmann Equation, Landau-Fokker-Planck equation, grazing collisions.

1 Introduction

The initial focus of this manuscript was the study of simulating the Boltzmann equation with anisotropic, singular angular scattering cross sections by spectral methods. However, while attempting to verify numerical results based on this method by examining the grazing collision limit of the Boltzmann operator, we found an analytical argument that not only gives an explicit representation of the effect of angular averaging for a family of singular grazing collision angular cross sections, but also the rate of convergence of the grazing collision limit of the the Boltzmann operator to the Landau operator. The bulk of this manuscript will address both the numerical and analytical aspects of this grazing collision limit of the Boltzmann equation in physically relevant regimes, which includes the case of Coulombic intermolecular potential scattering mechanisms.

While numerical methods for solving the Boltzmann equation generally use the assumption of spherical particles with ‘billiard ball’ like collisions, a more physical model is to assume that particles interact via two-body potentials. Under this assumption the Boltzmann equation can be formulated in a very similar manner [11], but in this case the scattering cross section is highly anisotropic in the angular variable. In many cases, such as the physically relevant case of Coulombic interactions between charged particles, the derivation of the Boltzmann equation breaks down completely due to the singular nature of this scattering cross section. Physical arguments by Landau [21] as well as a later derivation by Rosenbluth et al. [27] showed that the dynamics of the Boltzmann equation can approximated by a Fokker-Planck type equation when grazing collisions dominate, generally referred to as the Landau or Landau-Fokker-Planck equation. Later work [5, 13, 12, 30, 2] more rigorously justified this asymptotic limit.

Many numerical methods have been developed for solving the full Landau equation, some stochastic [29, 22] and some deterministic [25], however very few methods have been developed to compute the Boltzmann equation near this grazing collision limit. The small parameter used to quantify this limit is related to the physical Debye length, which quantifies the distance at which particles are screened from interaction, and a heuristic minimum interaction distance for the grazing collisions assumption to hold. Other non-grazing effects with the Boltzmann equation may remain relevant [15] which makes development of numerical methods based on the Boltzmann equation itself relevant for plasma applications. To our knowledge the only numerical method that makes this distinction explicit is the recently proposed Monte Carlo method for the Landau equation of Bobylev and Potapenko [7], which grew out of the work of Bobylev and Nanbu [6]. Pareschi, Toscani, and Villani [26] showed that the weights of their spectral Galerkin method for the Boltzmann equation converged to the weights of a similar method for the Landau equation, but neither estimates nor computations were done for the Boltzmann equation near the grazing collisions limit. This work seeks to bridge that gap using the conservative spectral method for the Boltzmann equation developed by Gamba and Tharkabhushanam [17, 18].

There are many difficulties associated with numerically solving the Boltzmann equation, most notably the dimensionality of the problem and the conservation of the collision invariants. For physically relevant three dimensional applications the distribution function is seven dimensional and the velocity domain is unbounded. In addition, the collision operator is nonlinear and requires evaluation of a five dimensional integral at each point in phase space. The collision operator also locally conserves mass, momentum, and energy, and any approximation must maintain this property to ensure that macroscopic quantities evolve correctly.

Spectral methods are a deterministic approach that compute the collision operator to high accuracy by exploiting its Fourier structure. These methods grew from the analytical works of Bobylev [8] developed for the Boltzmann equation with Maxwell type potential interactions and integrable angular cross section, where the corresponding Fourier transformed equation has a closed form. Spectral approximations for these type of models where first proposed by Pareschi and Perthame [23]. Later Pareschi and Russo [24] applied this work to variable hard potentials by periodizing the problem and its solution and implementing spectral collocation methods.

These methods require operations per evaluation of the collision operator, where is the total number of velocity grid points in each dimension. While convolutions can generally be computed in operations, the presence of the convolution weights requires the full computation of the convolution, except for a few special cases such as hard spheres in 3D (and Maxwell molecules in 2D) which can be done with in 3D. Spectral methods have advantages over Direct Simulation Monte Carlo Methods (DSMC) in many applications, in particular time dependent problems, low Mach number flows, high mean velocity flows, and flows that significantly deviate from equilibrium. In addition, deterministic methods avoid the statistical fluctuations that are typical of particle based methods.

Inspired by the work of Ibragimov and Rjasanow [20], Gamba and Tharkabhushanam [17, 18] observed that the Fourier transformed collision operator takes the form of a weighted convolution and developed a spectral method based on the weak form of the Boltzmann equation that provides a general framework for computing both elastic and inelastic collisions. Macroscopic conservation is enforced by solving a numerical constrained optimization problem that finds the closest distribution function in in the computational domain to the output of the collision term that conserves the macroscopic quantities. This optimization problem is the approximation of the projection of the Boltzmann solution to the space of collision invariants associated to the corresponding collision operator [3]. In addition these methods do not impose periodization on the function but rather assume that solution of the underlying problem on the whole phase space is obtained by the use of the Extension Operator in Sobolev spaces. They are also shown in the space homogeneous, hard potential case, to converge to the Maxwellian distribution with the conserved moments corresponding to the endowed collision invariants. Such convergence and error estimates results heavily rely on the discrete constrained optimization problem (see [3] for a complete proof and details.)

The proposed computational approach is complemented by the analysis of the approximation from the Boltzmann operator with grazing collisions to the Landau operator by estimating the -difference of their Fourier transforms evaluated on the solution of the corresponding Boltzmann equation, as they both can be easily expressed by a weighted convolution structure in Fourier space. We show that this property holds for a family of singular angular scattering cross sections with a suitably cut-off Coulomb potential. The parameter corresponds to the strength of the singularity in the angular cross section and the parameter corresponds to the angular cutoff, which gives the grazing collision limit as .

The case when the parameter corresponds to the classical Rutherford scattering cross section [28], and includes an inverse logarithmic term in that ensures the limit. This value of is critical to obtain the grazing collision limit in the following sense: if then the error between the Boltzmann and the Landau operators will not necessarily converge to zero in . In addition, for any other value , the rate of convergence of the Boltzmann to the Landau operator is faster in . In these sense we can assert that the Rutherford scattering cross section [28] is the one that contains the weakest possible singularity in the angular cross section for which one can achieve a grazing collision limit to the Landau equation.

These results are shown in Theorem 3.1 and are written for the three dimensional case. There we prove that the -difference of the Fourier transforms between the Landau operator and the Boltzmann operator for this family of cross sections , both acting on solutions of the Boltzmann equation, converges to zero in with rates depending on . This requires that the solution satisfies the regularity and decay condition with and , uniformly in . Our analysis shows the convergence rate in explicitly depends on the parameter that quantifies the strength of the non-integrable singularity associated to the collision angular cross section.

In addition, we examine the consequences of this theorem by numerically studying the differences between Rutherford scattering cross section (), which has logarithmic error in approximating Landau, and the cross section corresponding to , which better approximates Landau. These numerical results clearly exhibit the speed of the approximation and decay rate of the entropy functional for homogeneous solutions of the Boltzmann equation with cut-off Columbic interactions and are benchmarked with the solution to the Landau equation, which is independent of the and parameters.

This article is organized as follows. In Section 2, we present the derivation of the spectral formulation of the collision operator for an arbitrary anisotropic scattering cross section. In Section 3 we present the Landau equation and and present also a broad class of singular angular cross sections formulated by Villani [31] and Bobylev [7] satisfying suitable conditions for the study of grazing collisions limits of the anisotropic Boltzmann equation. We then introduce a family of angular primitives parametrized by to define admissible angular cross sections of the scattering angle in order to achieve a grazing collision limit. This family of angular cross sections includes the screened Rutherford cross section for Coulombic interactions. Then, we prove in Theorem 3.1 the estimates for the difference of the Fourier Transforms of the Landau and Boltzmann grazing collisional operators in terms of the and parameters that yields, as shown in Corollary 3.2, the rate of asymptotic convergence of solutions of the Boltzmann collision to approximate solutions to the Landau equation for this family cross sections given some condition on the solution of the corresponding Boltzmann equation as described above. In Section 4, we present the details of the numerical method based on this formulation and provide some practical observations on its implementation. In Section 5, we numerically investigate the method’s performance for small but finite values of , the grazing collision parameter, for the choice of and . We conclude with a discussion of future work in this area.

2 The space homogeneous Boltzmann equation

The space homogeneous elastic Boltzmann equation is given by the initial value problem

(1)

with

where is a probability density distribution in v-space and is assumed to be at least locally integrable with respect to v.

The collision operator is a bilinear integral form in given by

(2)

where the velocities are determined through a given collision rule depending on . The positive term of the integral in (2) evaluates in the pre-collisional velocities that can result in a post-collisional velocity the direction v. The scattering cross section is a given non-negative function depending on the size of the relative velocity and , where in the dimensional sphere is referred to as the scattering direction, which coincides with the direction of the post-collisional elastic relative velocity.

The elastic (or reversible) interaction law written in the scattering direction is given by

(3)

The angular cross section function may or may not be integrable with respect to ; the case when integrability holds is referred to as the Grad cut-off assumption on the angular cross section.

The parameter regulates the collision frequency as a function of the relative velocity . This parameter corresponds to the interparticle potentials used in the derivation of the collisional cross section and choices of are denoted as variable hard potentials (VHP) for , hard spheres (HS) for , Maxwell molecules (MM) for , and variable soft potentials (VSP) for . The case corresponds to a Coulombic interaction potential between particles. If is independent of we call the interactions isotropic, e.g., in the case of hard spheres in three dimensions.

2.1 Spectral formulation for anisotropic angular cross section

The key step in our formulation of the spectral numerical method is the use of the weak form of the Boltzmann collision operator [17]. For a suitably smooth test function the weak form of the collision integral is given by the double mixing operator

(4)

If one chooses

then, (4) is the Fourier transform of the collision integral with Fourier variable :

(5)

where denotes the Fourier transform and

(6)

Further simplification can be made by writing the Fourier transform inside the integral as a convolution of Fourier transforms:

(7)

where the convolution weights are given by

(8)

These convolution weights can be precomputed once to high accuracy and stored for future use. For many collisional models, such as isotropic collisions, the complexity of the integrals in the weight functions can be reduced dramatically through analytical techniques[17, 18]. However unlike previous work, in this paper we make no assumption on the isotropy of and derive a more general formula. We remark that this formulation does not separate the gain and loss terms of the collision operator, which is important for obtaining the correct cancellation in the grazing collision limit below.

We begin with and define a spherical coordinate system for with a pole in the direction of u, i.e. let . We obtain

(9)

For the remainder of this paper, we will work in three dimensions (). We write the unit vector as , where are mutually orthogonal vectors with u. Thus the right hand side of (9) becomes

for to be justified below.

Using the trigonometric identity

for a unique , the integration with respect to the azimuthal angle is equivalent to

where . Finally, let , then by symmetry we obtain

(10)

where is the Bessel function of the first kind (see [1] 9.2.21). Note that for the isotropic case the angular function is constant and thus can be used instead of u as the polar direction for , resulting in an explicit expression involving a sinc function [17].

Next, we take to be the Fourier transform of . Note that by symmetry is real valued, thus this transform is taken on a ball centered at 0 in order to ensure that this symmetry is maintained while computing them numerically.

Then, the convolution weights from (8), written in 3 dimensions, are computed as follows

We now take to be the polar direction for the spherical integration of and use that is real-valued to obtain

(11)

where is the polar angle for the integration.

3 The grazing collisions limit and convergence to the Landau collision operator

3.1 The Landau collision operator

The Landau collision operator describes binary collisions that only result in very small deflections of particle trajectories, as is the case for Coulomb potentials between charged particles [28]. This can be shown to be an approximation of the Boltzmann collision operator in the case where the dominant collision mechanism is that of grazing collisions. The operator is given by

(12)

and the weak form of this operator is given by [26]

where denotes the Hessian of and ’’ is the matrix double dot product.

As done in the Boltzmann case above, we choose to be the Fourier basis functions and obtain after some calculation

(13)

where , the orthogonal component of to u. Thus the weight function in terms of (6) is now given by

(14)

The used in the final computation is the Fourier transform of with respect to u, but we will work with this representation to make the convergence analysis below more clear.

3.2 The grazing collisions limit

To show that the spectral representation of Boltzmann operator is consistent with this form of the Landau operator, we must take the grazing collisions limit within this framework. To obtain this limit, it is enough to assume that the cross section satisfies the following properties.

Let be the small parameter associated with the grazing collisions limit and be a parameter associated with the strength of the singularity at , to be explained below. A family of cross sections represents grazing collisions if [2, 7]:

(15)

These conditions are sufficient show that the collisional integral operator converges to the Landau operator at a rate that depends on the choice of the angular function , independently of and , provided the solution of the Boltzmann equation for grazing collisions (1,2,3) with (3.2) satisfies some regularity and decay at infinity, as it will be shown in Theorem 3.1.

The most significant and perhaps physically meaningful example family of cross sections that satisfy these conditions can be generated from Rutherford scattering, corresponding to a family given by

(16)
  • We note that the logarithmic term that appears here is the Coulomb logarithm originally derived by Landau [21], where is proportional to the ratio between the mean kinetic energy of the gas and the Debye length. As will be observed later, this rescaling of the cross section is required in order to take the limit , as the form of the Landau equation we are using (12) does not have the Coulomb logarithm. Without this rescaling the leading order term of the collision operator would be the Landau equation (12) simply multiplied by , and the remainder terms would also be multiplied by this factor.

    Another angular cross section that satisfies conditions (3.2) is given by

    (17)

    which we will refer to as the -linear cross section. While this cross section is not physically motivated, it is useful for numerical convergence studies. Other angular cross sections that satisfy conditions (3.2) have been used in DSMC methods for computing the Landau equation; for an overview see [7].

    In fact it is possible to identify a large family of possible angular function choices corresponding to two body interaction potentials that includes both the Coulombic case (16) and the -linear one (17), the former being the critical case for the grazing collision limit.

    3.3 A family of angular cross sections for long range interactions

    We next introduce a more general way to define a family of angular cross sections that will satisfy conditions (3.2). For this purpose we introduce the functions and as the primitives of

    (18)

    for a given (non cut off) cross section .

    These are related to the grazing conditions by setting . Indeed, for we have

    (19)

    Note that using this function we have

    We similarly define

    (20)

    for convenience, as it will arise in the proof of the grazing limit.

    In order to satisfy the conditions of the second and third bullets for the grazing limit (3.2), it is sufficient that the angular function is singular enough such that

    (21)

    for some constant depending only on .

    Using these just introduced definitions, the -dependent angular cross section with a short range cut-off can be written in terms of the function from (18) as follows

    (22)

    Note that by construction, this cross section clearly satisfies the third grazing limit condition in (3.2). It also satisfies the first grazing limit condition:

    For the second grazing condition, note that

    thus any result will be less singular than the as .


    A -family of admissible angular singularities: One can see that when the angular function takes the form

    (23)

    then, after introducing the and reference parameters in the notation of the -grazing and -singular angular function , the function can be explicitly computed from the area differential of the angular part of the differential cross section (3.3)

    (24)

    Thus, also equating the last term above to the right hand side of relation (3.3), one can explicitly calculate as the antiderivative of , so it has the form

    (25)

    Similarly, the corresponding function , as defined in (3.3), satisfies

    (26)

    The choice of the exponent must be done in order to satisfy the third bullet condition (3.2), i.e. conditions (21) for both and .

    The case yields the Rutherford cross section where

    (27)

    These functions satisfy conditions (21), as

    For the case, the and functions become

    (28)

    These two functions also satisfy conditions (21), as

    Finally, notice that the case corresponds to the the -linear cross section (17), as as . In this case we have , and thus , satisfying (21).

    The critical case of corresponds to the Rutherford scattering (16), for which the Landau limit would be possible. Clearly, this case is the smallest value of the exponent in the singularity of the cross section written in negative powers of such that the bullet conditions (3.2) for the grazing collision limit are satisfied. In this sense the Coulombic potential case (16) is the critical case for which the Boltzmann operator can converge to the Landau operator.

    In addition, this approach breaks down when as condition (21) would not be satisfied on . This value of is the critical one at which more terms in the Taylor expansion for the angular cross-section contain singularities, and the next term of expansion of would need a similar treatment for as was done for (see the first terms of the expansions in equations (36) and (37).)


    3.4 The grazing collision approximation Theorem in three dimensions

    In the following theorem we estimate the difference of the grazing collision limit for the Boltzmann solutions evaluated at the collisional integral and Landau operators for a class of cross sections given by the general form of -grazing and -singular angular cross sections satisfying (3.3) and (3.3).

    We begin by taking a look at the grazing collisions limit for angular cross sections satisfying conditions (3.3), and all related conditions for the functions and as defined in the previous section.

    Theorem 3.1

    Assume that satisfies

    (29)

    with uniformly bounded by , constant, and . We also assume that the angular scattering cross section satisfies conditions in (3.3) related to the function in (25) satisfying conditions (3.3) and (21), with and .

    Then the rate of convergence of the Boltzmann collision operator with grazing collisions to the Landau collision operator is given by

    (30)

    From this Theorem, the following corollary follows easily by using the assumption that solves the Boltzmann equation for the family of admissible grazing collision cross sections. Indeed setting , taking their Fourier transforms and replacing into (30) one obtains the following estimate for approximate solutions to the Landau equation.

    Corollary 3.2

    Under the conditions of Theorem 3.1 the following approximation holds

    (31)
    • Assumption (29) means that has at least third order derivatives in as well as strong decay in , and in fact rapidly decreasing functions in the Schwarz class satisfy this assumptions.

      In addition, it is easy to see that the assumption (29) is well satified for being a Maxwellian distribution in v space. Indeed take, for ease of presentation, . Then

      (32)
      • We observe that, as expected from the result of Theorem missing3.1, the decay rate to equilibrium for the Rutherford -logarithmic cross section (16) is much faster than the one for the -linear cross section (17), and the latter one actually mimics the entropy decay rate of the Landau equation. This fact is well illustrated in Section 5 where we show the numerically computed entropy decay associated to the solution of the initial value problem for Boltzmann with Rutherford cross section (16). This is in fact an expected observation, as we explain below.


      • missingof Theorem missing3.1.

      With this angular cross section and , the calculation for the weight function can be computed by Taylor expanding the exponential term in (6) to obtain:

      (33)

      for some such that .

      We then define in terms of the polar and azimuthal angles and , respectively associated to the change of coordinates , with now used in Section 2.1: As a consequence, the following representation for the weight function holds

      (34)

      where is the unit vector in the direction of the part of that is orthogonal to the pole u, which arises from this choice of spherical coordinates, and note that . We stress that this expansion occurs in the convolution weights in this formulation rather than the distribution function as is done in other derivations of the grazing collisions limit.

      Next, we use the form of the angular cross section function as defined through (3.3), (23) and (25); and examine the result arising from the first two terms of the expansion to obtain