Stability and Conservation Properties of Hermite-Based Approximations of the Vlasov-Poisson System

Abstract

Spectral approximation based on Hermite-Fourier expansion of the Vlasov-Poisson model for a collisionless plasma in the electrostatic limit is provided by adding high-order artificial collision operators of Lenard-Bernstein type. These differential operators are suitably designed in order to preserve the physically-meaningful invariants (number of particles, momentum, energy). In view of time-discretization, stability results in appropriate norms are presented. In this study, necessary conditions link the magnitude of the artificial collision term, the number of spectral modes of the discretization, as well as the time-step. The analysis, carried out in full for the Hermite discretization of a simple linear problem in one-dimension, is then partly extended to cover the complete nonlinear Vlasov-Poisson model.

Appendix A

We prove here some inequalities concerning Hermite expansions. In what follows, $\varphi$ is supposed to be a function such that $\varphi =\sum _{n=0}^\infty C_{n}H_{n}$, where the coefficients are computed with the help of  (13). We first present a particular version of the Poincaré inequality.

Theorem A.1

If $\varphi =\sum _{n=0}^\infty C_{n}H_{n}$, then

$$\begin{aligned} \int _{\mathbb {R}}\varphi ^2e^{-v^2}\,dv\le \frac{1}{2}\int _{\mathbb {R}}\big (\varphi ^{\prime }\big )^2e^{-v^2}\,dv+\sqrt{\pi }C_{0}^2. \end{aligned}$$

(A.1)

The inequality can be generalized to derivatives of order $m>1$

$$\begin{aligned} \int _{\mathbb {R}}\varphi ^2e^{-v^2}\,dv\le \frac{1}{2^{m}\,m!}\int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv+\sqrt{\pi }\sum _{\ell =0}^{m-1}\,2^{\ell }\,\ell !\,C_{\ell }^2. \end{aligned}$$

(A.2)

Proof

The orthogonality of the first derivatives of the Hermite polynomials, Eq. (12), and the fact that $2n\ge 2$ for $n\ge 1$, imply

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{\prime }\big )^2e^{-v^2}\,dv&=\int _{\mathbb {R}}\big ( \sum _{n=1}^\infty C_{n}H^{\prime }_{n} \big )^2e^{-v^2}\,dv= \sum _{n=1}^\infty C_{n}^2\int _{\mathbb {R}}\big (H^{\prime }_{n}\big )^2e^{-v^2}\,dv\nonumber \\&= \sum _{n=1}^\infty C_{n}^2\,2n\,\int _{\mathbb {R}}H^2_{n}e^{-v^2}\,dv\ge 2\sum _{n=1}^\infty C_{n}^2\int _{\mathbb {R}}H_{n}^2e^{-v^2}\,dv, \end{aligned}$$

(A.3)

where all summations start from $n=1$ since $H_{0}=1$ and $H^{\prime }_{0}=0$.

Successively, we add and subtract the weighted integral of the zeroth-order mode, i.e, $C^2_{0}H^2_{0}$, to the last member of inequality (A.3) and use the expansion of $\varphi$, so to have

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{\prime }\big )^2e^{-v^2}\,dv&\ge 2\sum _{n=0}^\infty C_{n}^2\int _{\mathbb {R}}H_{n}^2e^{-v^2}\,dv-2C_{0}^2\int _{\mathbb {R}}H_{0}^2e^{-v^2}\,dv\nonumber \\&= 2\int _{\mathbb {R}}\varphi ^2e^{-v^2}\,dv- 2 \sqrt{\pi }C_{0}^2. \end{aligned}$$

(A.4)

By reversing this inequality we find that

$$\begin{aligned} \int _{\mathbb {R}}\varphi ^2e^{-v^2}\,dv\le \frac{1}{2}\int _{\mathbb {R}}\big (\varphi ^{\prime }\big )^2e^{-v^2}\,dv+\sqrt{\pi }C_{0}^2, \end{aligned}$$

which is the first inequality  (A.1). We generalize the result to derivatives of order $m>1$ as follows. Since $H^{(m)}_{n}=0$ for $n<m$, using formulas (9) and (11), we find

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv&= \int _{\mathbb {R}}\big ( \sum _{n=m}^\infty C_{n}H^{(m)}_{n} \big )^2e^{-v^2}\,dv= \sum _{n=m}^\infty C_{n}^2\int _{\mathbb {R}}\big (H^{(m)}_{n}\big )^2e^{-v^2}\,dv\nonumber \\&= \sum _{n=m}^\infty C_{n}^2\,2^{m}\frac{n!}{(n-m)!}\,\int _{\mathbb {R}}H^2_{n}e^{-v^2}\,dv\ge 2^{m}\,m!\sum _{n=m}^\infty C_{n}^2\int _{\mathbb {R}}H_{n}^2e^{-v^2}\,dv, \end{aligned}$$

(A.5)

as $n!/{(n-m)!}\ge m!$, when $n\ge m$. Now, we add and subtract the weighted integral of the first m modes, i.e., $\big (C_{\ell }H_{\ell }\big )^2$, $\ell =0,\ldots ,m-1$, to the last member of (A.5), and use the normalization of the Hermite polynomials to find out that

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv&\ge 2^{m}\,m!\left( \sum _{n=0}^\infty C_{n}^2\int _{\mathbb {R}}H_{n}^2e^{-v^2}\,dv-\sum _{\ell =0}^{m-1}C_{\ell }^2\int _{\mathbb {R}}H_{\ell }^2e^{-v^2}\,dv\right) \nonumber \\&= 2^{m}\,m!\left( \int _{\mathbb {R}}\varphi ^2e^{-v^2}\,dv-\sqrt{\pi }\sum _{\ell =0}^{m-1}\,2^{\ell }\,\ell !\,C_{\ell }^2 \right) . \end{aligned}$$

(A.6)

By reversing this inequality we easily arrive at  (A.2). $\square$

A further generalization of the previous result is provided by the following statement.

Theorem A.2

(Generalized Poincaré-type inequality) If $\varphi =\sum _{n=0}^\infty C_{n}H_{n}$, then for $m>p$ one has

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{(p)}\big )^2e^{-v^2}\,dv\le \frac{1}{2^{m-p}\,(m-p)!}\int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv+ 2^{p}\,\sqrt{\pi }\sum _{\ell =p}^{m-1}\,2^{\ell }\, \frac{(\ell !)^2}{(\ell -p)!}\,C_{\ell }^2. \end{aligned}$$

(A.7)

Proof

By noting that $H^{m}=H^{(p+(m-p))}=\big (H^{(p)}\big )^{(m-p)}$, a straightforward calculation exploiting the orthogonality of the derivatives of the Hermite polynomials yields

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv&= \int _{\mathbb {R}}\big ( \sum _{n=m}^\infty C_{n}H^{(m)}_{n} \big )^2e^{-v^2}\,dv= \int _{\mathbb {R}}\big ( \sum _{n=m}^\infty C_{n}H^{(p+(m-p))}_{n} \big )^2e^{-v^2}\,dv\nonumber \\&= \sum _{n=m}^\infty C_{n}^2\,2^{m-p}\frac{n!}{(n-(m-p))!}\,\int _{\mathbb {R}}\big (H^{(p)}_{n}\big )^2e^{-v^2}\,dv\nonumber \\&\ge 2^{m-p}\,(m-p)!\sum _{n=m}^\infty C_{n}^2\int _{\mathbb {R}}\big (H_{n}^{(p)}\big )^2e^{-v^2}\,dv, \end{aligned}$$

(A.8)

where we also used the fact that $n!/{(n-(m-p))!}>(m-p)!$, for $n>1$.

We add and subtract the weighted integrals of $C_{\ell }^2\big (H_{\ell }^{(p)}\big )^2$ for $\ell =p,\ldots ,p+(m-p)-1$, to the last member of (A.8) and we repeat the same argument as above until we obtain

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{(p)}\big )^2e^{-v^2}\,dv&\le \frac{1}{2^{m-p}\,(m-p)!}\int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv+ \sum _{\ell =p}^{m-1}C_{\ell }^2\int _{\mathbb {R}}\big (H_{\ell }^{(p)}\big )^2e^{-v^2}\,dv\\&= \frac{1}{2^{m-p}\,(m-p)!}\int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv+ \sum _{\ell =p}^{m-1}C_{\ell }^2\,2^{p}\,\frac{\ell !}{(\ell -p)!}\int _{\mathbb {R}}H_{\ell }^{2}e^{-v^2}\,dv\\&= \frac{1}{2^{m-p}\,(m-p)!}\int _{\mathbb {R}}\big (\varphi ^{(m)}\big )^2e^{-v^2}\,dv+ 2^{p}\,\sqrt{\pi }\sum _{\ell =p}^{m-1}\,2^{\ell }\,\frac{(\ell !)^2}{(\ell -p)!}\,C_{\ell }^2, \end{aligned}$$

which is our assertion. $\square$

In particular, if $\varphi$ belongs to the space of polynomials of degree at most N, we have $2n\le 2N$, so that the relations in (A.3) can be adjusted to obtain the so called inverse inequality

$$\begin{aligned} \int _{\mathbb {R}}\big (\varphi ^{\prime }\big )^2e^{-v^2}\,dv\le 2N \int _{\mathbb {R}}\varphi ^2e^{-v^2}\,dv. \end{aligned}$$

(A.9)

Another useful inequality is

$$\begin{aligned} \int _{\mathbb {R}}v^2H_{n}^2e^{-v^2}\,dv\le \frac{3}{4}\int _{\mathbb {R}}\big (H^{\prime }_{n}\big )^2e^{-v^2}\,dv, \qquad \forall n\ge 1. \end{aligned}$$

(A.10)

In order to prove it, we combine (7) and (8) to get

$$\begin{aligned} 2vH_{n} = H_{n}^{\prime } + H_{n+1} = 2nH_{n-1} + H_{n+1} \quad \forall n\ge 1. \end{aligned}$$

(A.11)

Finally, we deduce (A.10) from the sequence of relations

$$\begin{aligned} \int _{\mathbb {R}}v^2H_{n}^2e^{-v^2}\,dv&= \int _{\mathbb {R}}n^2H_{n-1}^2e^{-v^2}\,dv+\frac{1}{4}\int _{\mathbb {R}}H_{n+1}^2e^{-v^2}\,dv\nonumber \\&= \sqrt{\pi } \left[ n^2\,2^{n-1}\,(n-1)! + \frac{1}{4}\,2^{n+1}\,(n+1)! \right] = \sqrt{\pi }\left[ 2^{n-1}\,n\,n! + 2^{n-1}(n+1)n! \right] \nonumber \\&= \sqrt{\pi } 2^{n-1}(2n+1)n! \ \le \ \sqrt{\pi }\,\frac{3}{4}\,2^{n+1}\,n\,n! = \frac{3}{4}\int _{\mathbb {R}}\big (H^{\prime }_{n}\big )^2e^{-v^2}\,dv, \qquad \forall n\ge 1, \end{aligned}$$

(A.12)

where we noted that $2n+1\le 2n+n=3n$. The last equality follows from (12).

The inequality (A.10) is the starting point to show that

$$\begin{aligned} \int _{\mathbb {R}}v^2\varphi ^2e^{-v^2}\,dv\le \frac{3}{4}N\int _{\mathbb {R}}\big (\varphi '\big )^2e^{-v^2}\,dv, \end{aligned}$$

(A.13)

which holds for every polynomial $\varphi =\sum _{n=1}^NC_{n}H_{n}$ with degree less or equal to N and $C_{0}=0$. We argue as follows. For a given set of values $\alpha _n$, the relation here below is a consequence of the Schwartz inequality

$$\begin{aligned} \left( \sum _{n=1}^{N}\alpha _n \right) ^2 = \left( \sum _{n=1}^{N} 1\cdot \alpha _n \right) ^2 \le \sum _{n=1}^{N}1^2 \ \sum _{n=1}^{N}\alpha _n^2 = N \sum _{n=1}^{N}\alpha _n^2. \end{aligned}$$

(A.14)

With the help of the above inequality, the orthogonality of the Hermite polynomials implies

$$\begin{aligned} \int _{\mathbb {R}}v^2\varphi ^2e^{-v^2}\,dv&= \int _{\mathbb {R}}v^2\Big (\sum _{n=1}^{N}C_nH_{n}\Big )^2e^{-v^2}\,dv\le N\sum _{n=1}^{N}C_{n}^2\int _{\mathbb {R}}v^2H^2_{n}e^{-v^2}\,dv\\&\le \frac{3}{4}N\sum _{n=1}^{N}C_{n}^2\int _{\mathbb {R}}\big (H^{\prime }_{n}\big )^2e^{-v^2}\,dv= \frac{3}{4}N\int _{\mathbb {R}}\big (\varphi '\big )^2e^{-v^2}\,dv, \end{aligned}$$

that actually corresponds to (A.13).