Abstract
Hyperbolic moment equations based on Burnett’s expansion of the distribution function are derived for the Boltzmann equation with linearized collision operator. Boundary conditions are equipped for these models, and it is proven that the number of boundary conditions is correct for a large class of moment models. A new second-order numerical scheme is proposed for solving these moment equations, and the new method is suitable for both ordered- and full-moment theories. Numerical experiments are carried out for both one- and two-dimensional problems to show the performance of the moment methods.
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Acknowledgements
We thank Dr. Xiaojun Gu and Dr. Anirudh Rana for providing DSMC results for the two-dimensional problems. This research was supported by Humboldt Research Fellowship for Postdoctoral Researchers provided by Alexander von Humboldt Foundation.
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Appendices
Appendix 1: Proof of Theorem 1
Lemma 3
For \(\varvec{\xi }= (\xi _1, \xi _2, \xi _3)^T\) and \(\varvec{\xi }^* = (\xi _1, \xi _2, -\xi _3)^T\), it holds that
Proof
Obviously \(|\varvec{\xi }| = |\varvec{\xi }^*|\). Thus we only need to prove \(Y_l^m(\varvec{\xi }/ |\varvec{\xi }|) = (-1)^{l+m} Y_l^m(\varvec{\xi }^* / |\varvec{\xi }|)\). If \(\varvec{\xi }/ |\varvec{\xi }| = (\sin \theta \cos \varphi , \sin \theta \sin \varphi , \cos \theta )^T\), then \(\varvec{\xi }^* / |\varvec{\xi }| = (\sin \theta ^* \cos \varphi , \sin \theta ^* \sin \varphi , \cos \theta ^*)^T\) with \(\theta ^* = \pi - \theta \). According to (3.4), it is sufficient to prove \(P_l^m(\cos \theta ) = (-1)^{l+m} P_l^m(\cos \theta ^*) = (-1)^{l+m} P_l^m(-\cos \theta )\). Since \((x^2-1)^l\) is an even function, its \((l+m)\)th order derivative is even/odd if \(l+m\) is even/odd. Thus according to (3.5), the Legendre function \(P_l^m(x)\) is even/odd if \(l+m\) is even/odd, which completes the proof. \(\square \)
Proof of Theorem 1
When \(\varvec{n}= \varvec{n}_z\), the conclusion is exactly the same as Lemma 3. If \(\varvec{n}\ne \varvec{n}_z\), then
Using
we immediately have \(\varvec{\eta }_{\varvec{n}} \big ( \varvec{\xi }- 2(\varvec{\xi }\cdot \varvec{n}) \varvec{n}\big ) = \varvec{\eta }_{\varvec{n}}(\varvec{\xi }) - 2(\varvec{\xi }\cdot \varvec{n}) \varvec{n}_z\). Therefore
and for any vector \(\varvec{n}_z^{\perp }\) perpendicular to \(\varvec{n}_z\),
Similar as (6.3), we have \(2\varvec{n}_z \cdot (\varvec{n}_z - \varvec{n}) = |\varvec{n}_z - \varvec{n}|^2\), and therefore
Inserting the above equation into (6.4), we get
Thus (3.37) is a direct result of (6.5), (6.7) and Lemma 3. \(\square \)
Appendix 2: Derivation of the Boundary Condition for the Velocity
In this section, we derive the boundary condition (3.43a). For simplicity, the parameters t and \(\varvec{x}\) are omitted. In (3.38), we let \((l,m,n) = (1,0,0)\), and get a linear polynomial \([(\varvec{c}- \varvec{v}^W) \cdot \varvec{n}] / \sqrt{RT}\). Let \(p(\varvec{c})\) be this polynomial in (3.34), we have
By the definition of \(\rho ^W\) in (3.33), the right-hand side of the above equation is zero. The left hand side equals \(\rho (\varvec{v}- \varvec{v}^W) \cdot \varvec{n}\). Thus we have \(\varvec{v}\cdot \varvec{n}= \varvec{v}^W \cdot \varvec{n}\).
Appendix 3: Proof of Theorem 2
The proof of Theorem 2 is given in this section. We first assume \(\varvec{n}= (0,0,1)^T\) and study the eigenvalues of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \Pi _3\). Define the operator \(\tilde{\Pi }_k^{[\varvec{\alpha }]}\), \(k = 1,2,3\) as
and function spaces
Obviously \(\mathbb {H}_{L,\varvec{N}}^{[\varvec{\alpha }]} = \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} \oplus \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\), and we have the following lemma:
Lemma 4
If \(f \in \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\), then \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]} f \in \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\). If \(f \in \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\), then \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]} f \in \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\).
Proof
This lemma is a direct result of the recursion relation of the basis functions:
If \(l+m\) is odd, the right-hand side shows that the above function is the linear combination of \(\psi _{l'm'n'}\) with \(l'+m'\) being even. Similarly, when \(l+m\) is even, the above function is the linear combination of \(\psi _{l'm'n'}\) with \(l'+m'\) being odd. Since the operator \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} g\) becomes a truncation of g when the function g is expanded into series with basis function \(\psi _{lmn}\), the result of the lemma is obtained immediately. \(\square \)
This lemma inspires us to define \(\mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}: \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} \rightarrow \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) as the restriction of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) on \(\mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\), and define \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}: \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \rightarrow \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) as the restriction of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) on \(\mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\). Below we consider \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) as a linear transformation on \(\mathbb {H}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) and study the distribution of its eigenvalues. We start from proving that \(\mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) is injective:
Lemma 5
If \(N_0 \geqslant N_1 \geqslant \cdots \geqslant N_L\) and \(f \in \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\), then \(\mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f \ne 0\) if and only if \(f \ne 0\).
Proof
Obviously \(\mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f \ne 0\) implies \(f \ne 0\). Now we assume \(f \ne 0\) and has the following expansion:
Let \(l_0\) be the minimum l such that \(f_{lmn} \ne 0\) holds for some m and n, and let \(n_0\) be the minimum n such that \(f_{l_0 m n} \ne 0\) holds for some m. Thus for all \(l' < l_0\) and \(n' < n_0\), \(f_{l'mn} = 0\) and \(f_{l_0mn'} = 0\) for every available m and n, and there exists an \(m_0\) such that \(f_{l_0 m_0 n_0} \ne 0\). Since \(l_0 + m_0\) is odd, we have \(l_0 \geqslant 1\) and \(|m_0| < l_0\). Let \(\mu = 0\) in (3.27), and we obtain
From \(N_{l_0-1} \geqslant N_{l_0}\) and \(|m_0| < l_0\), we know that \((l_0-1, m_0, n_0) \in \mathcal {I}_{L,\varvec{N}}\). According to the definition of \((l_0, m_0, n_0)\), one easily finds that in the above sum, the coefficient of \(\psi _{l_0-1,m_0,n_0}\) is \(\sqrt{2(n_0 + l_0) + 1} \gamma _{-l_0,m_0}^0 f_{l_0 m_0 n_0}\). When \(l_0 + m_0\) is odd, \(\gamma _{-l_0,m_0}^0 \ne 0\), and therefore \(\mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f \ne 0\). \(\square \)
Lemma 6
If \(N_0 \geqslant N_1 \geqslant \cdots \geqslant N_L\), then \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) has only positive eigenvalues.
Proof
For any \(f, g \in \mathbb {H}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) and
define
According to the definition of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) (3.15), we have
Using the orthogonality (3.6), one can easily find that the second integral in the above equation equals \(\mathfrak {m}^{-1} \overline{g_{lmn}}\). By replacing f with its series form, we get
Using the same method, one can show that \(\langle f, \mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3 g \rangle ^{[\varvec{\alpha }]}\) also equals the right-hand side of the above equation. Thus
If \(f \in \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) and \(g \in \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\), Eq. (6.17) becomes \(\langle \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f, g \rangle ^{[\varvec{\alpha }]} = \langle f, \mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} g \rangle ^{[\varvec{\alpha }]}\). If \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f = \lambda f\) and \(f \ne 0\), then \(\langle f, \lambda f \rangle ^{[\varvec{\alpha }]} = \langle f, \mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f \rangle ^{[\varvec{\alpha }]} = \langle \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f, \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f \rangle ^{[\varvec{\alpha }]}\). By Lemma 5, when \(f \ne 0\), \(\mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f \ne 0\). Hence \(\lambda = \langle \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f, \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f \rangle ^{[\varvec{\alpha }]} / \langle f, f \rangle ^{[\varvec{\alpha }]} > 0\). \(\square \)
Theorem 7
\(\lambda \) is a nonzero eigenvalue of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) with multiplicity M if and only if \(\lambda ^2\) is an eigenvalue of \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) with multiplicity M.
Proof
Suppose \(\lambda \ne 0\) and \(\{ f_1, \ldots , f_M \} \subset \mathbb {H}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) are linearly independent and satisfy \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]} f_k = \lambda f_k\). For any \(k = 1, \ldots , M\), there exist unique \(f_k^o \in \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) and \(f_k^e \in \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) such that \(f_k = f_k^o + f_k^e\). Thus
Therefore \(f_k^o = \lambda ^{-1} \mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} f_k^e\), \(f_k^e = \lambda ^{-1} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f_k^o\), which yields that
If there exists \((\alpha _1, \ldots , \alpha _M)^T \in \mathbb {C}^M\) such that \(\alpha _1 f_1^o + \cdots + \alpha _M f_M^o = 0\), then
Since \(f_1, \ldots , f_M\) are linearly independent, \(\alpha _1 = \cdots = \alpha _M = 0\). Thus \(f_1^o, \ldots , f_M^o\) are also linearly independent, and therefore \(\lambda ^2\) is an eigenvalue of \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) with multiplicity at least M. If there exists \(f_{M+1}^o \in \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) such that \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f_{M+1}^o = \lambda ^2 f_{M+1}^o\), then
Therefore \(f_{M+1}^o + \lambda ^{-1} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f_{M+1}^o\) is a linear combination of \(f_1, \ldots , f_M\), and then \(f_{M+1}^o\) is a linear combination of \(f_1^o, \ldots , f_M^o\). This shows that the multiplicity of the eigenvalue \(\lambda ^2\) is M.
Now we assume that \(\{f_1^o, \ldots , f_M^o\} \subset \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) are linearly independent and fulfill
Equation (6.21) shows that \(f_k^o + \lambda ^{-1} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f_k^o\), \(k=1,\ldots ,M\) are eigenfunctions of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) with eigenvalue \(\lambda \). Obviously these functions are also linearly independent, and therefore \(\lambda \) is an eigenvalue of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) with multiplicity at least M. If \(f_{M+1} = f_{M+1}^o + f_{M+1}^e\) with \(f_{M+1}^o \in \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) and \(f_{M+1}^e \in \mathbb {E}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) satisfies \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]} f_{M+1} = \lambda f_{M+1}\), similar as (6.19), we have \(f_{M+1}^e = \lambda ^{-1} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f_{M+1}^o\) and \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]} f_{M+1}^o = \lambda ^2 f_{M+1}^o\). Thus there exist \(\alpha _1, \ldots , \alpha _M\) such that \(f_{M+1}^o = \alpha _1 f_1^o + \cdots + \alpha _M f_M^o\), which gives
This shows that the multiplicity of the eigenvalue \(\lambda \) is M. \(\square \)
A direct corollary of the above theorem is
Corollary 8
If \(\lambda \) is a nonzero eigenvalue of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) with multiplicity M, then \(-\lambda \) is also a nonzero eigenvalue of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) with multiplicity M.
Theorem 9
If \(N_0 \geqslant N_1 \geqslant \cdots \geqslant N_L\), then the multiplicity of zero eigenvalue of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) is \(\dim \mathbb {H}_{L,\varvec{N}}^{[\varvec{\alpha }]} - 2 \dim \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\).
Proof
By Lemma 6, we can assume that all eigenvalues of \(\mathcal {E}_{L,\varvec{N}}^{[\varvec{\alpha }]} \mathcal {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) are \(\lambda _1^2, \lambda _2^2, \ldots , \lambda _{M_o}^2\), where \(M_o = \dim \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\). Using Theorem 7, we know that \(\pm \lambda _1, \ldots , \pm \lambda _M\) are all the nonzero eigenvalues of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\). Thus the conclusion of the theorem follows naturally. \(\square \)
Corollary 8 and Theorem 9 show that the eigenvalues of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]}\) are symmetrically distributed with respect to the origin, and the number of positive or negative eigenvalues are the same as the number of odd basis functions. The definition of \(\tilde{\Pi }_k^{[\varvec{\alpha }]}\) (6.9) shows that \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3^{[\varvec{\alpha }]} = \mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \Pi _3 - v_3 \mathrm {Id}\). Hence the distribution of eigenvalues of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \Pi _3\) can be described as follows:
Corollary 10
Let \(M = \dim \mathbb {H}_{L,\varvec{N}}^{[\varvec{\alpha }]}\) and \(M_o = \dim \mathbb {O}_{L,\varvec{N}}^{[\varvec{\alpha }]}\), and suppose \(\lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _M\) are all the eigenvalues of \(\mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \Pi _3\). If \(N_0 \geqslant N_1 \geqslant \cdots \geqslant N_L\), then it holds that
From (6.11), it is clear that the eigenvalues of \((RT)^{-1/2} \mathcal {P}_{L,\varvec{N}}^{[\varvec{\alpha }]} \tilde{\Pi }_3\) is independent of \(\varvec{\alpha }\). Thus, through a rotation in the velocity space, Theorem 2 is naturally obtained.
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Cai, Z., Torrilhon, M. Numerical Simulation of Microflows Using Moment Methods with Linearized Collision Operator. J Sci Comput 74, 336–374 (2018). https://doi.org/10.1007/s10915-017-0442-7
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DOI: https://doi.org/10.1007/s10915-017-0442-7