# Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants

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## Abstract

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. Unlike for the Boltzmann equation, for DVMs there can appear extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and hence, without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species, but also for binary mixtures and recently extensively for multicomponent mixtures. In this paper, we address ways of constructing normal DVMs for polyatomic molecules (here represented by that each molecule has an internal energy, to account for non-translational energies, which can change during collisions), under the assumption that the set of allowed internal energies are finite. We present general algorithms for constructing such models, but we also give concrete examples of such constructions. This approach can also be combined with similar constructions of multicomponent mixtures to obtain multicomponent mixtures with polyatomic molecules, which is also briefly outlined. Then also, chemical reactions can be added.

## Keywords

Boltzmann equation Discrete velocity models Collision invariants Polyatomic molecules## Mathematics Subject Classification

82C40 35Q20 76P05## 1 Introduction

We consider the Boltzmann equation for polyatomic molecules [19, 25, 29], here represented by that each molecule has an internal energy that can change during collisions. In particular, we study discrete velocity models (DVMs), i.e., we assume that the velocity variable only can take a finite number of different given (vector) values. The Boltzmann equation can be approximated by DVMs up to any order [17, 20, 26, 32], and these discrete approximations can be used for numerical methods, e.g., see [20, 31] and references therein. However, in the construction of DVMs there is a classical question of having the correct number of collision invariants [27]. Unlike in the continuous case, there can be additional collision invariants to the physical ones; mass momentum, and energy, for DVMs. DVMs, without additional collision invariants, for which the collision invariants are linearly independent, are called normal. The construction of normal DVMs for single species as well as for binary mixtures has been well studied, see for example [15, 16, 18, 23, 24, 35, 36, 37], and recently also for multicomponent mixtures [10]. We like to point out, that in [16] main ideas of a general approach to the construction of normal discrete kinetic models, including a brief discussion on the application to DVMs for inelastic collisions, is presented. Still in particular cases, more detailed studies are needed. We consider here the problem of constructing DVMs for single species of polyatomic molecules with the right number of collision invariants and outline the extension to DVMs for mixtures of polyatomic molecules (cf. [7]). We like to stress that we thereby also can handle the important case of mixtures of several monatomic and polyatomic gases that, e.g., is of importance during the reentry of space shuttles in the upper atmosphere [1].

Unlike for single species, each polyatomic molecule has an internal energy, to account for non-translational energies. We assume that the set of allowed internal energies is finite and add to each internal energy a finite set of allowed velocities. In this way we obtain pairs of allowed velocities and internal energies, cf. [21], where two pairs can have the same velocity, but different internal energies, or vice versa. In correspondence with the cases of binary mixtures [16] and multicomponent mixtures [10] we introduce the concepts of semi-supernormal and supernormal DVMs for polyatomic molecules. We present algorithms for constructing such DVMs and give some concrete examples of such constructions. We also prove that for any given (finite number of) multiples of an internal “basic” energy we can construct a supernormal DVM. Our constructed DVMs can always be extended to larger DVMs by the method of one-extensions [14, 15, 35]. A one-extension is obtained in the following way: if three out of four velocities, involved in a possible (i.e., such that the physical quantities are conserved) collision, already are in an existing normal DVM, one adds also the fourth velocity to the DVM; in this way a new normal DVM, with one more velocity, a one-extension, is obtained. We like to stress that it is always possible to extend our constructed DVMs to DVMs symmetric with respect to the axes by the method of one-extensions.

Another important issue is the one of approximating the full Boltzmann equation by DVMs, which have been addressed for single species of polyatomic molecules in e.g., [21] and for mixtures with polyatomic molecules in e.g., [28]. For simulations, it is important to have the right number of collision invariants. Our results concerning the correct number of collision invariants are independent of the modeling of the collision coefficients as long as the collision coefficients for a maximal set of linearly independent collisions (i.e., collisions that cannot be obtained by combining the other collisions, including corresponding inverse collisions, in the set) are nonzero, which will be implicitly assumed below. In [21] it is proven, that for both of the (families of) models considered there, the collision invariants are the physical ones, and only those. However, our results are more general and can be applied to a much larger family of DVMs, including those in [21]. We want to stress that our intention is not at all to discuss different implementations, but instead to make a more careful investigation of normality for DVMs in the case of polyatomic molecules.

The construction of the DVMs is made such that for half-space problems [3], as the linearized Milne and Kramers problems [2], but also nonlinear ones [34], one obtains similar structures as for the classical discrete Boltzmann equation for one species [4, 5, 8, 13], and can extend corresponding results, cf. [10] for multicomponent mixtures. The same is also true for the analytically difficult problem of existence of shock profiles [22, 30], where one also obtains a similar structure as for the classical discrete Boltzmann equation for one species [9] and can extend corresponding results, see [7].

The remaining part of the paper is organized as follows. Section 2 concerns DVMs for polyatomic molecules; the concepts of semi-supernormal and supernormal DVMs, respectively, for polyatomic molecules are introduced in Sect. 2.1, and algorithms for constructing such models is presented in Sect. 2.2. Section 3 concerns concrete examples of such DVMs. It is also proven that for any given (finite number of) multiples of an internal “basic” energy we can construct a supernormal DVM in Sect. 3. In Sect. 4 we outline the extension to multicomponent mixtures with polyatomic molecules and state some corresponding results.

## 2 DVMs for Polyatomic Molecules

*s*different internal energies \(E^{1},\ldots ,E^{s}\) that can be associated with the particles, which either can be considered as that there only is a finite number of different (internal) energy states or that one (in some way) have modelled a continuous internal energy variable by discretizing it, cf. [21]. We model the Boltzmann equation by discretizing the continuous velocity variable, i.e., we assume that the velocity only can take a finite number of different vector values. In order to do the discretization, we fix a set of velocity vectors \(V_{i}=\{ \mathbf {\xi }_{1}^{i},\ldots ,\mathbf {\xi }_{n_{i}}^{i}\} \subset \mathbb {R }^{d}\) (in applications \(d=2,\,3\)) for each of the internal energies \(E^{i}.\) Now there are \(n=n_{1}+\cdots +n_{s}\) different pairs, being composed of a velocity vector and an internal energy, contained in the set

*E*and velocity \(\mathbf {\xi }\) at time \(t\in \mathbb {R}_{+}\) and position \(\mathbf {x}\in \mathbb {R}^{d}.\)

*E*), we identify

*h*with its restrictions to the pairs \(( \mathbf {v}_{i},\,E_{i}) \in \mathscr {V},\) i.e.,

Note also that under the assumptions above we will have an *H*-theorem as usual, cf. [8].

### 2.1 Supernormal DVMs for Polyatomic Molecules

For DVMs for polyatomic molecules, one can, as in the case of DVMs for mixtures cf. [10, 16] have different kinds of normality. Similarly as in the case of mixtures in [10], we introduce different kinds or levels of normality. We start with the usual definition of normality.

### Definition 1

Note that for normal DVMs, the \(d+2\) linearly independent collision invariants in Definition 1 will be linear combinations of the \(d+2\) trivial collision invariants (8).

A drawback with Definition 1 is that if we look separately on the restriction to a specific energy level, the reduced model does not have to be normal. Therefore, we extend the definition above.

### Definition 2

A DVM (11), with internal energies \(\{ E^{1},\ldots ,E^{s}\}, \) is called semi-supernormal if it is normal and the restriction to each velocity set \(\mathrm {V}_{i},\) \(1\le i\le s,\) is a normal DVM.

However, still if we consider subsets of energy levels, the restrictions to those energy levels do not have to be normal. Therefore, we make a further extension.

### Definition 3

Depending on what we are interested to study we can be satisfied with different levels of normality, where normal is the lowest level and supernormal the highest one (including the other ones).

As we construct semi-supernormal DVMs we can be helped by the following theorem.

### Theorem 1

A DVM (11), with internal energies \( \{ E^{1},\ldots ,E^{s}\}, \) is semi-supernormal if, for each \(2\le j\le s\) there exists \(1\le i<j\le s,\) such that the restriction to the pair \(\{ \{ \mathrm {V}_{i},\,E^{i}\} ,\,\{ \mathrm {V} _{j},\,E^{j}\} \} \) is a supernormal DVM.

### Proof

The restriction to each velocity set \(\mathrm {V}_{i}=\{ \mathbf {\xi } _{1}^{i},\ldots ,\mathbf {\xi }_{n_{i}}^{i}\} ,\) \(1\le i\le s,\) is normal by the supernormality of \(\{ \{ \mathrm {V}_{i},\,E^{i}\} ,\,\{ \mathrm {V}_{j},\,E^{j}\} \} .\) Hence, the collision invariants will be of the form \(\phi =( \phi ^{1},\ldots ,\phi ^{s}), \) where \(\phi _{j}^{i}=a_{i}+m\mathbf {b}^{i}\cdot \mathbf {\xi } _{j}^{i}+c_{i}( m\vert \mathbf {\xi }_{j}^{i}\vert ^{2}+2E^{i}) \) for \(1\le j\le n_{i}\) and \(1\le i\le s.\)

Denote \(a_{1}=a,\) \(\mathbf {b}^{\mathbf {1}}\mathbf {=b},\) and \(c_{1}=c.\) Assume that \(a_{j-1}=a_{j-2}=\cdots =a_{1}=a\), \(\mathbf {b}^{j-1}\mathbf {=b}^{j-2} \mathbf {=\cdots =b}^{\mathbf {1}}\mathbf {=b},\) and \(c_{j-1}=c_{j-2}=\cdots =c_{1}=c\) for some \(2\le j\le s.\) Then there exists \(1\le i\le j-1,\) such that the restriction to the pair \(\{\{ \mathrm {V}_{i},\,E^{i}\} ,\,\{ \mathrm {V}_{j},\,E^{j}\} \} \) is normal and therefore \( a_{j}=a_{i}=a,\) \(\mathbf {b}^{j}\mathbf {=b}^{i}\mathbf {=b},\) and \( c_{j}=c_{i}=c.\) Hence, the collision invariants will be of the form \(\phi =( \phi ^{1},\ldots ,\phi ^{s_{s}}), \) where \(\phi _{j}^{i}=a+m \mathbf {b\cdot \xi }_{j}^{i}+c( m\vert \mathbf {\xi } _{j}^{i}\vert ^{2}+2E^{i}) \) for \(1\le j\le n_{i}\) and \(1\le i\le s.\) \(\square \)

For constructing supernormal DVMs (or checking if existing DVMs are supernormal), the following theorem can be useful.

### Theorem 2

### Proof

The theorem follows directly from the definition of supernormal DVMs and Theorem 1. \(\square \)

### 2.2 Algorithms for Construction of Semi-supernormal and Supernormal DVMs for Polyatomic Molecules

*n*-dimensional vector with 0, \(-1,\) and 1 as the only coordinates, see, e.g., [16, 18], in the way that collision (7) is represented by a vector (with non-zero elements at the positions \( i,\,j,\,k,\) and

*l*)

*Algorithm for construction of semi-supernormal DVMs for polyatomic molecules*

- (1)
Choose a set of velocities \(\mathrm {V}_{1}\) such that it corresponds to a normal DVM for a monatomic species. Here, and in all the steps below, the set should be chosen in such a way, that we can obtain normal models for any mass ratio and/or energy levels we intend to consider; otherwise we might also be able to extend the set(s) later, as we realize that it is needed.

- (2)
Iteration step. For \(j=2,\ldots ,s\):

### Remark 1

Note that in the spirit of Remark 1 we will have satisfactorily many inelastic collisions as long as we have satisfactorily many elastic collisions.

*Algorithm for construction of supernormal DVMs for polyatomic molecules*

- (1)
Choose a set of velocities \(\mathrm {V}_{1}\) such that it corresponds to a normal DVM for a monatomic species. As above, here, and in all the steps below, the set should be chosen in such a way, that we can obtain normal models for any mass ratio and/or energy levels we intend to consider; otherwise we might also be able to extend the set(s) later, as we realize that it is needed.

- (2)
Iteration step. For \(j=2,\ldots ,s\):

## 3 Construction of a Family of Supernormal DVMs with Internal Energies

This section concerns an approach of the construction of supernormal DVMs with internal energies. We will use an odd-integer grid as our basic universe, instead of the usual integer grid, since in some applications (e.g., boundary layers [4, 5, 6, 8, 13]) it is preferable that the first component of the velocity is non-zero. However, the integer grid and the odd-integer grid are the same up to a shift and a scaling, and, of course, the odd-integer grid is also contained in the integer grid. Hence, if we would like to, we could also use the integer grid as our basic universe. If desirable, it is also possible to find “larger” normal (and symmetric) DVMs that contains the velocity sets for all different energy levels and hence, can be used as a common velocity set for all energy levels. We are concerned with finding \( d+2\) linearly independent (also with respect to the collisions inside the energy levels) collisions between each energy levels. These collisions are not the only ones between each two energy levels, but all collisions between each two energy levels can be obtained by combining (one or more of) those linearly independent collisions (including corresponding reverse collisions) with the collisions inside energy levels, cf. Remark 1 above.

*E*. We start with a set of velocities \(\mathrm {V},\) which includes the six velocities

*m*denotes the mass. More generally, we can use different sets \( \mathrm {V}\) (as long as they contain the necessary velocities) for different internal energies. As noted above, the minimal models are normal DVMs. A drawback of the minimal models is that the maximal total change of energies under a collision is

*E*, i.e., except the elastic collisions, only collisions such that

### Lemma 1

*rE*and

*qE*, where

*r*and

*q*are positive integers, such that \(r<q,\) there is a supernormal DVM

### Proof

*m*denotes the mass. Without any collisions between the different species we will have, since the DVMs are normal, the collision invariants

Note that the sets of velocities used in the proofs of Lemma 1, in no way are unique. Furthermore, there can be sets of velocities that do not contain the velocities assumed in the proof, but still are supernormal for the given internal energies. We have just proven that there exist such sets of velocities for any given two internal energies *rE* and *qE*, where *r* and *q* are positive integers.

### Example 1

*E*and 2

*E*, and let

### Example 2

*E*, 2

*E*, and 4

*E*. If we let

### Theorem 3

## 4 DVMs for Mixtures with Polyatomic Molecules

We can combine the approach for DVMs for single species with polyatomic molecules with the approach for DVMs for multicomponent mixtures [10] in an obvious way to obtain models for mixtures of polyatomic molecules [7]. We note that some (or even all) of the species actually can be monatomic. For that purpose assume that we have *s*, \(s\ge 1,\) different species, labelled with \(\alpha _{1},\ldots ,\alpha _{s},\) with the masses \( m_{\alpha _{1}},\ldots ,m_{\alpha _{s}},\) and that we for each species \(\alpha _{i}\) have \(r_{i},\) \(r_{i}\ge 1,\) different internal energies \( E_{i}^{1},\ldots ,E_{i}^{r_{i}}.\) Note that \(s=r_{1}=1\) will give us the case of single species of monatomic molecules, \(s>1\) with \(r_{1}=\cdots =r_{s}=1\) will generate mixtures of monatomic molecules [10], while \(s=1\) will give us the case of single species of polyatomic molecules discussed above.

*k*and

*l*interchanged in (23) otherwise, i.e., if \(\alpha (i)=\alpha (l).\) In general, the collision obtained by (22) is geometrically represented by a trapezoid in \(\mathbb {R}^{d}\) [a parallelogram if additionally \(\alpha (i)=\alpha (j),\) or more generally if and only if \(m_{\alpha (i)}=m_{\alpha (j)}\)], with the corners in \( \{ \mathbf {v}_{i},\,\mathbf {v}_{j},\,\mathbf {v}_{k},\,\mathbf {v}_{l}\}, \) where \(\mathbf {v}_{i}\) and \(\mathbf {v}_{j}\) (and therefore, also \(\mathbf {v}_{k}\) and \(\mathbf {v}_{l}\)) are diagonal corners, such that Eq. (23) and

*k*and

*l*interchanged in Eqs. (23) and (24) otherwise, i.e., if \(\alpha (i)=\alpha (l).\)

### 4.1 Normal DVMs for Mixtures with Polyatomic Molecules

In correspondence with the definitions in Sect. 2.1 we can introduce the following definitions.

### Definition 4

### Definition 5

A DVM (27) is called semi-supernormal if it is normal and the restriction to each velocity set \(\mathrm {V}_{i}^{j},\) \(1\le i\le s,\,1\le j\le r_{i},\) is a normal DVM.

### Definition 6

A DVM (27) is called supernormal if the restriction to each non-empty subset of \(\mathscr {V}\) constitutes a normal DVM.

In a natural way we can extend Theorem 2 [7] (see also corresponding theorem for mixtures in [10]).

### Theorem 4

A DVM (27) is supernormal if and only if the restriction to each pair of sets in \(\mathscr {V}\) constitutes a supernormal DVM.

Moreover, by combining the arguments in the proofs of the corresponding result for the particular cases of single species with polyatomic molecules (with \(s=1\)) in Theorem 3 and mixtures of monatomic molecules (with \(r_{1}=\cdots =r_{s}=1\)) in [10]

### Theorem 5

For any given number *s* of species with given rational masses \( m_{\alpha _{1}},\ldots ,m_{\alpha _{s}},\) and with \(r_{i}\) given internal energies \(\{ r_{i1}E,\ldots ,r_{ir_{i}}E\} \) for a fixed \(E\in \mathbb {R}_{+}\) and rational numbers \(r_{i1},\ldots ,r_{ir_{i}},\) \(i=1,\ldots ,s,\) there is a supernormal DVM for the mixture of polyatomic molecules.

### 4.2 Algorithm for Construction of Supernormal DVMs

This section concerns an algorithm for for construction of supernormal DVMs for mixtures with polyatomic molecules.

*Algorithm for construction of supernormal DVMs*

- (1)
- (a)
Choose a set of velocities \(\mathrm {V}_{1}^{1}\) such that it corresponds to a normal DVM for a monatomic single species. Here, and in all the steps below, the set should be chosen in such a way, that we can obtain normal models for any mass ratio and/or energy levels we intend to consider; otherwise we might also be able to extend the set(s) later, as we realize that it is needed.

- (b)
For \(j=2,\ldots ,r_{1}\): choose a set of velocities \(\mathrm {V} _{1}^{j} \) corresponding to a normal DVM such that \(\{ \{ \mathrm {V }_{1}^{k},\,E_{1}^{k}\} ,\,\{ \mathrm {V}_{1}^{j},\,E_{1}^{j}\}\} \) is a normal DVM for each \(1\le k<j.\)

- (a)
- (2)For \(i=2,\ldots ,s\):
- (a)
Choose a normal set of velocities \(\mathrm {V}_{i}^{1}\) such that it, together with each of \(\mathrm {V}_{1}^{1},\ldots ,\mathrm {V}_{1}^{r_{1}},\ldots , \mathrm {V}_{i-1}^{1},\ldots ,\mathrm {V}_{i-1}^{r_{i-1}},\) corresponds to a supernormal DVM for binary mixtures.

- (b)For \(j=2,\ldots ,r_{i}\): choose a set of velocities \(\mathrm {V} _{i}^{j} \) such that
- (i)
\(\mathrm {V}_{i}^{1}\) together with each of \(\mathrm {V}_{1}^{1},\ldots , \mathrm {V}_{1}^{r_{1}},\ldots ,\mathrm {V}_{i-1}^{1},\ldots ,\mathrm {V} _{i-1}^{r_{i-1}}\) corresponds to a supernormal DVM for binary mixtures;

- (ii)
\(\{ \{ \mathrm {V}_{i}^{k},\,E_{i}^{k}\} ,\,\{ \mathrm { V}_{i}^{j},\,E_{i}^{j}\} \} \) is a normal DVM for each \(1\le k<j.\)

- (i)

- (a)

### Remark 3

In each case, if we do not allow any collisions between the two species/levels of internal energies, we will have \(2d+4\) linearly independent collision invariants, but we would like to have \(d+3/d+2\) linearly independent collision invariants for mixtures/polyatomic molecules respectively. Hence, cf. [16], we need to have \(d+1/d+2\) linearly independent (also with respect to the collisions inside the two species/energy levels) collisions between the two species/energy levels, cf. Remark 1.

In the case of explicit constructions we could, as in Sect. 3, use an odd-integer grid as our basic universe, instead of the usual integer grid (by the same arguments as in Sect. 3) and use the same base models as in Sect. 3. Here we are concerned by finding \(d+1/d+2\) linearly independent (also with respect to the collisions inside the two species/energy levels) collisions between each two species/energy levels. Again, we stress that those collisions are not the only ones between each two species/energy levels, but all collisions between each species/energy levels can be obtained by combining (one or more of) those linearly independent collisions (including corresponding reverse collisions) with the collisions inside the species/energy levels.

### Example 3

*m*and 3

*m*, and the internal energies

*E*and 2

*E*. We denote

*E*and 2

*E*(heavy species), and

*E*and 4

*E*/ 3 (light species), if the (brown/chained) “basal” inelastic collision is changed, see Fig. 7 [where also the velocity set \(\mathrm {V}\) for the heavy species at the lower internal energy level is reduced to (28)].

### 4.3 Bimolecular Chemical Reactions

We can also add bimolecular reactive collisions [11] (by changing corresponding collision coefficients to be nonzero) to DVMs for mixtures of polyatomic molecules and by that extend to DVMs for bimolecular chemical reactions. For each linearly independent (also with respect to all other collisions) reactive collision, we obtain one new relation on the masses. Note that the maximal number of linearly independent bimolecular reactive collisions are \(d-1,\) since the total number of particles will still be conserved. In [8] an example (cf. [29, 33]) is considered. However, our method is not limited to that case, but can be used also in more general cases.

## Notes

### Acknowledgements

The first ideas from which this paper originate were obtained during a visit at Parma University. The author wants to thank M. Groppi, G. Spiga, and M. Bisi at Parma University for valuable discussions.

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