A Hybrid Moment Method for Multi-scale Kinetic Equations Based on Maximum Entropy Principle

Abstract

We propose a hybrid method for the multi-scale kinetic equations in the framework of the hyperbolic moment method (Cai and Li in SIAM J Sci Comput 32(5):2875–2907, 2010). In this method, the fourth order moment system is chosen as the governing equations in the fluid region, while the hyperbolic moment system with arbitrary order is chosen as the governing equations in the kinetic region. When transiting from the fluid regime to the kinetic regime, the maximum entropy principle is adopted to reconstruct the kinetic distribution function, so that the information in the fluid region can be utilized thoroughly. Moreover, only one uniform set of numerical scheme is needed for both the fluid and kinetic regions. Numerical tests validate this new hybrid method.

Appendices

Appendix A: Numerical Scheme for the Regularized Moment Method

In this section, we will introduce the numerical scheme of the dimensional reduced regularized moment method. We refer readers [7] for more details. Let $\varvec{\omega }= (\varvec{\omega }^{(g)}, \varvec{\omega }^{(h)})$ be the variables of the reduced system, where $\varvec{\omega }^{(f)}, f = g, h$ is the corresponding variable of the distribution function f in (2.28). The reduced moment system is written as

$$\begin{aligned} \dfrac{\partial {\varvec{\omega }}}{\partial {t}} + \sum _{j=1}^2\mathbf{B_j} \dfrac{\partial {\varvec{\omega }}}{\partial {x}} =\mathbf{Q}\varvec{\omega }. \end{aligned}$$

(A.1)

Splitting method is adopted here, where (A.1) is split into the convection part and the collision part

• convection part:

  $$\begin{aligned} \dfrac{\partial {\varvec{\omega }}}{\partial {t}} + \sum _{j=1}^2\mathbf{B_j} \dfrac{\partial {\varvec{\omega }}}{\partial {x}} =0. \end{aligned}$$

  (A.2)

• collision part:

  $$\begin{aligned} \dfrac{\partial {\varvec{\omega }}}{\partial {t}} = \mathbf{Q}\varvec{\omega }. \end{aligned}$$

  (A.3)

The detailed algorithm is as below,

• 1. Let $n = 0$, and give the initial value of $\varvec{\omega }_i^n$.

• 2. Calculate the time step length $\Delta t_n$ by CFL condition.

• 3. Solve the convection part using the finite volume method with the HLL flux, and denote the result as $\varvec{\omega }_{i}^{n, *}$.

• 4. Update the collision step using $\varvec{\omega }_i^{n,*}$ as the initial condition.

• 5. Let $t \leftarrow t + \Delta t_n$ and $n \leftarrow n+1$; then go to Step 2.

The numerical scheme for the convection step and the collision is listed below in detail.

Convection step For the convection part, the moment equation (A.2) is reformulated as

$$\begin{aligned} \dfrac{\partial {\varvec{\omega }}}{\partial {t}} + \dfrac{\partial {\mathbf{F}(\varvec{\omega })}}{\partial {x}} +\mathbf{R}(\varvec{\omega }) \dfrac{\partial {\varvec{\omega }}}{\partial {x}} = 0. \end{aligned}$$

(A.4)

Then, the finite volume method is utilized here, and the detailed scheme is

$$\begin{aligned} \varvec{\omega }_{i}^{n, *} = \varvec{\omega }_i^n - \frac{\Delta t_n}{\Delta x} \left( {\hat{{\varvec{F}}}}_{i+1/2}^n - {\hat{{\varvec{F}}}}_{i - 1/2}^n\right) - \frac{\Delta t_n}{\Delta x}\left( {\hat{{\varvec{R}}}}_{i+1/2}^{n-} - {\hat{{\varvec{R}}}}_{i-1/2}^{n+}\right) , \end{aligned}$$

(A.5)

where ${\hat{{\varvec{F}}}}_{i+1/2}^{n}$ is the HLL numerical flux defined as

$$\begin{aligned} {\hat{{\varvec{F}}}}_{i+1/2}^{n} = \left\{ \begin{array}{ll} {{\varvec{F}}}(\varvec{\omega }_i^n),&{} \lambda _{i+1/2}^{L,n} \geqslant 0, \\[2mm] \frac{\lambda _{i+1/2}^{R,n} {{\varvec{F}}}(\varvec{\omega }_{i}^n) - \lambda _{i+1/2}^{L,n} {{\varvec{F}}}(\varvec{\omega }_{i+1}^n)}{\lambda _{i+1/2}^{R,n} - \lambda _{i+1/2}^{L,n}} + \frac{\lambda _{i+1/2}^{L,n} \lambda _{i+1/2}^{R,n} (\varvec{\omega }_{i+1}^n - \varvec{\omega }_{i}^n)}{\lambda _{i+1/2}^{R,n} - \lambda _{i+1/2}^{L,n}}, &{} \lambda _{i+1/2}^{L,n}< 0 < \lambda _{i+1/2}^{R,n}, \\[2mm] {\varvec{F}}(\varvec{\omega }_{i+1}^n), &{} \lambda _{i+1/2}^{R,n} \leqslant 0. \end{array} \right. \nonumber \\ \end{aligned}$$

(A.6)

The numerical flux for the non-conservation part ${\hat{{\varvec{R}}}}_{i+1/2}^{n\pm }$ is

$$\begin{aligned} \begin{array}{c} {\hat{{\varvec{R}}}}_{i+1/2}^{n-} = \left\{ \begin{array}{ll} 0, &{} \lambda _{i+1/2}^{L,n} \geqslant 0, \\ -\frac{\lambda _{i+1/2}^{L,n}}{\lambda _{i+1/2}^{R,n} - \lambda _{i+1/2}^{L,n}} {\varvec{g}}_{i+1/2}^n , &{} \lambda _{i+1/2}^{L,n}< 0< \lambda _{i+1/2}^{R,n}, \\ {\varvec{g}}_{i+1/2}^n, &{} \lambda _{i+1/2}^{R,n} \leqslant 0, \end{array} \right. \\ \\[2mm] {\hat{{\varvec{R}}}}_{i+1/2}^{n+} = \left\{ \begin{array}{ll} -{\varvec{g}}_{i+1/2}^n, &{} \lambda _{i+1/2}^{L,n} \geqslant 0, \\ -\frac{\lambda _{i+1/2}^{R,n}}{\lambda _{i+1/2}^{R,n} - \lambda _{i+1/2}^{L,n}} {\varvec{g}}_{i+1/2}^n , &{} \lambda _{i+1/2}^{L,n}< 0 < \lambda _{i+1/2}^{R,n}, \\ 0, &{} \lambda _{i+1/2}^{R,n} \leqslant 0, \end{array} \right. \end{array} \end{aligned}$$

(A.7)

with

$$\begin{aligned} {\varvec{g}}_{i+1/2}^n = \int _0^1 {\varvec{R}}(\Phi (s; {\varvec{q}}_i^n; {\varvec{q}}_{i+1}^n)) \dfrac{\partial {\Phi }}{\partial {s}}(s; {\varvec{q}}_i^n; {\varvec{q}}_{i+1}^n) \,\mathrm {d}s, \end{aligned}$$

(A.8)

where $\Phi (s; \cdot , \cdot )$ is a path to connect the two states. We refer [11] for more details. The characteristic speeds $\lambda _{i+1/2}^R$ and $\lambda _{i+1/2}^L$ are

$$\begin{aligned} \begin{aligned} \lambda _{i+1/2}^{R,n}&= \max \left( u_{1,i}^n + C_{M+1}\sqrt{\theta _i^n}, u_{1,j}^n + C_{M+1} \sqrt{\theta _{i+1}^n}\right) , \\ \lambda _{i+1/2}^{L,n}&= \max \left( u_{1,i}^n - C_{M+1}\sqrt{\theta _i^n}, u_{1,j}^n - C_{M+1} \sqrt{\theta _{i+1}^n}\right) , \end{aligned} \end{aligned}$$

(A.9)

where $C_{M+1}$ is the maximal roots of Hermite polynomial of degree $M+1$. The time step length is also decided by the characteristic velocity as

$$\begin{aligned} \frac{ \Delta t^n \max \limits _i \left\{ \left| \lambda _{i+1/2}^{R,n}\right| , \left| \lambda _{i+1/2}^{L,n}\right| \right\} }{\Delta x} \leqslant \mathrm{CFL}. \end{aligned}$$

(A.10)

Collision step For the BGK and Shakhov model, the collision part can be solved exactly in the reduced regularized moment method. For neatness, the superscript $n, n+1$ and the subscripts i are omitted. The solutions are as below:

• For the BGK model:

  $$\begin{aligned} \begin{aligned} g_{\alpha }&= g_{\alpha }^{*} \exp \left( -\frac{\Delta t}{\tau }\right) , \quad 2 \leqslant |\alpha | \leqslant M, \\ h_{0}&= g_0^{*}\theta ^{*} \left( 1 - \exp \left( \frac{\Delta t}{\tau }\right) \right) + h_{0}^{*} \exp \left( -\frac{\Delta t}{\tau }\right) , \\ h_{\alpha }&= h_{\alpha }^{*} \exp \left( -\frac{\Delta t}{\tau }\right) , \quad 1 \leqslant |\alpha | \leqslant M-2. \end{aligned} \end{aligned}$$

  (A.11)

• For the Shakhov model:

  $$\begin{aligned} \begin{aligned} g_{3e_1}&= g_{3e_1}^{*}\exp \left( -\frac{\Delta t}{\tau }\right) + \frac{1}{5}q_i^{*}\left( \exp \left( -\frac{ \Pr \Delta t}{\tau }\right) -\exp \left( -\frac{\Delta t}{\tau }\right) \right) , \\ g_{\alpha }&= g_{\alpha }^{*} \exp \left( -\frac{\Delta t}{\tau }\right) , \quad |\alpha | = 2 ~\mathrm{or}~ 4 \leqslant |\alpha | \leqslant M, \\ h_{0}&= g_0^{*}\theta ^{*} \left( 1 - \exp \left( \frac{\Delta t}{\tau }\right) \right) + h_{0}^{*} \exp \left( -\frac{\Delta t}{\tau }\right) , \\ h_{e_1}&= h_{e_1}^{*}\exp \left( -\frac{\Delta t}{\tau }\right) + \frac{1}{5}q_i^{*}\left( \exp \left( -\frac{ \Pr \Delta t}{\tau }\right) -\exp \left( -\frac{\Delta t}{\tau }\right) \right) , \\ h_{3e_1}&= h_{3e_1}^{*}\exp \left( -\frac{\Delta t}{\tau }\right) + \frac{\theta ^{*}}{5}q_i^{*}\left( \exp \left( -\frac{ \Pr \Delta t}{\tau }\right) -\exp \left( -\frac{\Delta t}{\tau }\right) \right) , \\ h_{\alpha }&= h_{\alpha }^{*} \exp \left( -\frac{\Delta t}{\tau }\right) , \quad |\alpha | = 2 ~\mathrm{or}~ 4 \leqslant |\alpha | \leqslant M-2. \end{aligned} \end{aligned}$$

  (A.12)

Appendix B: Reduced Moment System

When the distribution function has some symmetric properties in the direction $v_2$ and $v_3$. The Chu reduction is utilized here and we refer [7] for more details. With this dimension reduction method, we will introduce two distribution functions in 1D microscopic velocity space (3.17), which is approximated as (3.18). The reduced Boltzmann with Shakhov collision model for $g(t,x, v_1)$ and $h(t, x, v_1)$ is

$$\begin{aligned} \begin{aligned} \dfrac{\partial {g}}{\partial {t}} + v_1 \dfrac{\partial {g}}{\partial {x}}&= \frac{1}{\epsilon } \left( g^N - g\right) , \\ \dfrac{\partial {h}}{\partial {t}} + v_1 \dfrac{\partial {h}}{\partial {x}}&= \frac{1}{\epsilon } \left( h^N - h\right) , \end{aligned} \end{aligned}$$

(B.1)

where

$$\begin{aligned} \begin{aligned} g^N&= \left[ 1 + \dfrac{(1 - \mathrm{Pr})(v_1 - u_1(t, x)) q(t, x)}{5\rho (t, x) [\theta (t, x)]^2}\left( \frac{|v_1 - u_1(t, x)|^2}{\theta (t, x)} - 5\right) \right] g_{\mathrm{eq}}, \\ h^N&= \left[ 1 + \dfrac{(1 - \mathrm{Pr})(v_1 - u_1(t, x)) q(t, x)}{5\rho (t, x) [\theta (t, x)]^2}\left( \frac{|v_1 - u_1(t, x)|^2}{\theta (t, x)} - 3\right) \right] h_{\mathrm{eq}}, \end{aligned} \end{aligned}$$

(B.2)

with

$$\begin{aligned} g_{\mathrm{eq}} = \frac{\rho }{\sqrt{2 \pi \theta }} \exp \left( -\frac{(v_1 - u_1(t,x))^2}{2 \theta }\right) , \qquad h_\mathrm{eq} = \theta g_{\mathrm{eq}}. \end{aligned}$$

(B.3)

The relationship of the macroscopic variables and the distribution functions are

$$\begin{aligned} \begin{aligned} \rho&= \int _{{\mathbb {R}}} g(t, x, v_1) \,\mathrm {d}v_1, \qquad \rho u_1 = \int _{{\mathbb {R}}} v_1 g(t, x ,v_1) \,\mathrm {d}v_1, \\ \frac{3}{2}\rho \theta&= \int _{{\mathbb {R}}} \left( \frac{1}{2} |v_1 -u_1|^2 g + h\right) \,\mathrm {d}v_1, \qquad q_1 = \int _{{\mathbb {R}}} (v_1 - u_1) \left( \frac{1}{2} |v_1 - u_1|^2 g + h\right) \,\mathrm {d}v_1. \end{aligned} \end{aligned}$$

(B.4)

Substituting (3.18) into (B.1), and matching order of the basis functions, we can derive the moment equations for the reduced distribution function as

$$\begin{aligned} \begin{aligned}&\dfrac{\partial {\psi _{i}}}{\partial {t}} + \left( \theta \dfrac{\partial {\psi _{i -1}}}{\partial {x}} +u_1\dfrac{\partial {\psi _{i}}}{\partial {x_1}} +(1-\delta _{N,i})(i + 1) \dfrac{\partial {\psi _{i+e_j}}}{\partial {x}} \right) + \dfrac{\partial {u_1}}{\partial {t}} \psi _{i-1} \\&\quad + \dfrac{\partial {u_1}}{\partial {x}} \left[ \theta \psi _{i-2} +u_1\psi _{i-1} + (1-\delta _{N,i})(i + 1) \psi _{i} \right] + \frac{1}{2} \dfrac{\partial {\theta }}{\partial {t}} \psi _{i-2} \\&\quad + \frac{1}{2}\theta \dfrac{\partial {\theta }}{\partial {x_1}} \left[ \theta \psi _{i-3} +u_1 \psi _{i-2} + (1-\delta _{N,i})(i + 1) \psi _{i-1} \right] = \frac{1}{\epsilon } (\Delta _{i}(\psi ) - \psi _{i}), \qquad i \leqslant N, \end{aligned} \end{aligned}$$

(B.5)

where $\psi = g, h$. When $\psi = g, N = M$, while $N = M - 2$, if $\psi = h$. Here $\Delta _i(\psi )$ is the deduced by the collision term as

$$\begin{aligned} \begin{array}{cc} \Delta _i(g) = \left\{ \begin{array}{ll} g_0, &{} i = 0, \\ (1 - \mathrm Pr) q_i / 5, &{} i = 3, \\ 0, &{} \mathrm{otherwise,} \end{array} \right. &{} \Delta _i(h) = \left\{ \begin{array}{ll} \theta g_0, &{} i = 0, \\ (1 - \mathrm Pr) q_i / 5, &{} i = 1, 3, \\ 0, &{} \mathrm{otherwise,} \end{array} \right. \end{array} \end{aligned}$$

(B.6)

Appendix C: Integration Algorithm in Computing the Maximum Entropy Distribution Function

For computing the gradient and Hessian matrix when applying the Newton iteration to solve the maximum entropy distribution ansatz, we need to compute the high order moments of the distribution ansatz. We employ an adaptive integration technique based on the location of the peaks of the distribution function. Since the maximum entropy distribution function takes the form of $\exp (P(v))$, where P(v) is a quartic function, its peak values could be computed by taking the roots of $P'(v)$. In [29], the detailed algorithm has been provided, and we briefly review it here. Since the distribution function decays exponentially, we truncate the integration domain from ${\mathbb {R}}$ to one or two bounded intervals, by taking the intercept of P(v) at a certain value, such that P(v) is cut somewhere sufficiently low below its peak values. The truncation procedure is done by considering two possible scenarios separately. Let $P(v) = \sum \limits _{i=0}^4 \beta _i v^i$,

• 1. If P(v) has only one peak, which we denote as $v_1$, the integration interval is decided as $[c_1, c_2]$, where $c_1, c_2$ are the two roots of

  $$\begin{aligned} P(v) - (P(v_1)-c) = 0, \end{aligned}$$

  (C.1)

  with c a sufficiently large constant.

• 2. If P(v) has two peaks $v_1$ and $v_2$, we solve

  $$\begin{aligned} P(v) - (P(v_1)-c) = 0,\quad P(v) -(P(v_2)-c) = 0, \end{aligned}$$

  (C.2)

  to obtain the two intervals as $[c_1,c_2]$ and $[c_3,c_4]$. If $c_2>c_3$, then the interval $[c_1,c_4]$ is used for computation.

The value of c is taken according to experience, such that it is sufficiently large. In our computation, it is usually enough to take $c = 50$. There are two additional special cases which we treat separately:

• 1. If $|\beta _4| < 10^{-16}$ and $\beta _2 < 0$, the distribution function is almost Maxwellian. In this case, we could directly specify the integration interval, as the position of the peak is simply $-\frac{\beta _1}{2 \beta _2}$. Considering that the smaller the value of $|\beta _2|$, the larger the integration interval needs to be, we take the length of the truncated integration interval to be $4\sqrt{-\frac{50}{\beta _2}}$.

• 2. If $\beta _4 < -50$, the moments are close to the boundary of the realizable region. In this case, c needs to be larger as the peaks are very sharp. Considering that the larger $|\beta _4|$ is, the sharper the peaks are, for moments near the boundary we modify the value of c to be $c = 50\sqrt{-\frac{\beta _4}{10}}$.

Once the integration interval is determined, we use Gauss–Chebyshev quadrature to calculate the integral. In our numerical examples, 400 quadrature points are used to evaluate each integration. If there are two intervals, the quadrature points are divided between the two intervals proportional to the interval lengths.