MomentGauge.Statistic.PolyStatistics#

Module Contents#

Classes#

PolyStatistics

The base class for store pre-defined polynomial statistics.

Maxwellian_1D_stats

The polynomial statistics for 1D Maxwellian distribution.

ESBGK_1D_stats

The polynomial statistics for 1D ESBGK distribution.

M35_1D_stats

The 1D version of polynomial statistics for 35 moments.

M35_P2_1D_stats

The 1D version of polynomial statistics for 35 moments with .
class MomentGauge.Statistic.PolyStatistics.PolyStatistics#

The base class for store pre-defined polynomial statistics.

suff_stats#

a list of statistics [\(\phi_i,i=0,\cdots,M\)], in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u , \(*\) gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

\(*\) gauge_paras : - Arbitrary many extra parameters. The \(*\) refers to the unpacking operator in python.

Returns:

float – the moment value

The lengh of the list may vary. Its first element must satisfy \(\phi_0\) ( u , \(*\) gauge_paras ) = 1

Type:

None

class MomentGauge.Statistic.PolyStatistics.Maxwellian_1D_stats#

Bases: PolyStatistics

The polynomial statistics for 1D Maxwellian distribution.

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,2\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

Returns:

float – the moment value

Specifically,

\(\phi_0\) (u ) = 1.

\(\phi_1\) (u ) = \(u_x\)

\(\phi_2\) (u ) = \(u_x^2 + u_y^2 + u_z^2\).

Type:

list of length (3)

class MomentGauge.Statistic.PolyStatistics.ESBGK_1D_stats#

Bases: PolyStatistics

The polynomial statistics for 1D ESBGK distribution.

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,3\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

Returns:

float – the moment value

Specifically,

\(\phi_0\) (u ) = 1.

\(\phi_1\) (u ) = \(u_x\)

\(\phi_2\) (u ) = \(u_x^2\)

\(\phi_2\) (u ) = \(u_y^2 + u_z^2\).

Type:

list of length (4)

class MomentGauge.Statistic.PolyStatistics.M35_1D_stats#

Bases: PolyStatistics

The 1D version of polynomial statistics for 35 moments.

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,8\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

Returns:

float – the moment value

Specifically,

\begin{equation} \{\phi_i,i=0,\cdots,8\} = \left\{1, {u}_x,\frac{ {u}_x^2-1}{\sqrt{2}},\frac{ {u}_r^2}{2} -1,\frac{ {u}_x^3-3 {u}_x}{\sqrt{6}},\frac{ {u}_x^4-6 {u}_x^2+3}{2 \sqrt{6}},\frac{1}{8} {u}_r^4- {u}_r^2+1,\frac{1}{2} {u}_x ( {u}_r^2-1),\frac{( {u}_x^2 -1)( {u}_r^2-2)}{2 \sqrt{2}}\right\} \end{equation}

in which \(u_r = \sqrt{u_y^2+u_z^2}\)

Type:

list of length (9)

class MomentGauge.Statistic.PolyStatistics.M35_P2_1D_stats#

Bases: PolyStatistics

The 1D version of polynomial statistics for 35 moments with .

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,10\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

Returns:

float – the moment value

Specifically,

\begin{equation} \{\phi_i,i=0,\cdots,10\} = \left\{1, {u}_x,\frac{ {u}_x^2-1}{\sqrt{2}},\frac{ {u}_r^2}{2} -1,\frac{ {u}_x^3-3 {u}_x}{\sqrt{6}},\frac{ {u}_x^4-6 {u}_x^2+3}{2 \sqrt{6}},\frac{1}{8} {u}_r^4- {u}_r^2+1,\frac{1}{2} {u}_x ( {u}_r^2-1),\frac{( {u}_x^2 -1)( {u}_r^2-2)}{2 \sqrt{2}} , \frac{u_x^5}{2 \sqrt{30}}-\sqrt{\frac{5}{6}} u_x^3+\frac{1}{2} \sqrt{\frac{15}{2}} u_x, \frac{u_x^6}{12 \sqrt{5}}-\frac{\sqrt{5} u_x^4}{4}+\frac{3 \sqrt{5} u_x^2}{4}-\frac{\sqrt{5}}{4} \right\} \end{equation}

in which \(u_r = \sqrt{u_y^2+u_z^2}\)

Type:

list of length (11)