Commit 3930d6e9 authored by Tianbai Xiao's avatar Tianbai Xiao
Browse files

Add PN docs


Former-commit-id: 433274cd
parent 0882c593
......@@ -54,12 +54,78 @@ For convenience, it is often reformulated with polar coordinates :math:`\{r, \ph
.. math::
&\left[\frac{1}{v(E)} \partial_{t} +\Omega \cdot \nabla+\Sigma_t (r, E, t)\right] \psi(r, \Omega, E, t) \\
&=\int_{0}^{\infty} d E^{\prime} \int_{\mathcal R^2} d \Omega^{\prime} \Sigma_{s}\left(r, \Omega^{\prime} \bullet \Omega, E^{\prime} \rightarrow E\right) \psi\left(r, \Omega^{\prime}, E^{\prime}, t\right) + Q(r, \Omega, E, t).
&\left[\frac{1}{v(E)} \partial_{t} +\Omega \cdot \nabla+\Sigma_t (t, r, E)\right] \psi(t, r, \Omega, E) \\
&=\int_{0}^{\infty} d E^{\prime} \int_{\mathcal R^2} d \Omega^{\prime} \Sigma_{s}\left(r, \Omega^{\prime} \bullet \Omega, E^{\prime} \rightarrow E\right) \psi\left(t, r, \Omega^{\prime}, E^{\prime}\right) + Q(t, r, \Omega, E).
The particle distribution :math:`\psi(r, \Omega, E, t)` here is often named as angular flux, :math:`\{\Sigma_s, \Sigma_t \}` are the scattering and total cross sections correspondingly, and :math:`Q` denotes a source term.
The spherical harmonics moment equations
---------
The spherical harmonics (:math:`P_N`) method (cf. Brunner and Holloway [2005]) is one of several ways to solve the equation of radiative transfer.
It serves as an approximate method, i.e. the method of moments, to reduce the high dimensionality when the original kinetic equation of radiative transfer, which is formulated on a seven-dimensional domain.
Let us consider the radiative transfer equation in one-dimensional physical space with only one energy group, i.e.
.. math::
\partial_{t} \psi(t, z, \mu) &+\mu \nabla_{z} \psi(t, z, \mu)+\Sigma_{t}(t, z) \psi(t, z, \mu) \\
&=\int_{\mathcal S^{2}} \Sigma_{s}\left(t, z, \mu \cdot \Omega^{\prime}\right) \psi\left(t, z, \mu^{\prime}\right) d \mu^{\prime}+q(t, z, \mu),
where :math:`\mu` is the projected angular variable on :math:`z` axis.
To obtain the :math:`P_N` equations, we express the angular dependence of the distribution function in terms of a Fourier series,
.. math::
\psi(t, z, \mu)=\sum_{\ell=0}^{\infty} \psi_{\ell}(t, z) \frac{2 \ell+1}{2} P_{\ell}(\mu),
where :math:`P_{\ell}` are the Legendre polynomials.
These form an orthogonal basis of the space
of polynomials with respect to the standard scalar product on :math:`[−1, 1]`.
We can then obtain
.. math::
\partial_{t} \psi_{\ell}+\partial_{z} \int_{=1}^{1} \mu P_{\ell} \psi \mathrm{d} \mu+\Sigma_{t \ell} \psi_{\ell}=q_{\ell},
where
.. math::
\Sigma_{t \ell}=\Sigma_{t}-\Sigma_{s \ell}=\Sigma_{a}+\Sigma_{s 0}-\Sigma_{s \ell}, \quad \Sigma_{s \ell}=2 \pi \int_{-1}^{1} P_{\ell}(\mu) \Sigma_{s}(\mu) \mathrm{d} \mu.
Two properties of the spherical harmonics are crucial for our method. These appear here as properties of the Legendre polynomials. First, we observe that, by this
procedure, we have diagonalized the scattering operator on the right-hand side (the
Legendre polynomials are the eigenfunctions of scattering operator).
Second, a general property of orthogonal polynomials is that they satisfy a recursion relation. In
particular, the Legendre polynomials satisfy
.. math::
\mu P_{\ell}(\mu)=\frac{\ell}{2 \ell+1} P_{\ell-1}(\mu)+\frac{\ell+1}{2 \ell+1} P_{\ell+1}(\mu).
Using this fact and truncating the expansion at :math:`\ell = N`, we arrive at the slab-geometry
:math:`P_N` equations,
.. math::
\partial_{t} \psi_{\ell}+\partial_{z}\left(\frac{\ell+1}{2 \ell+1} \psi_{\ell+1}+\frac{\ell}{2 \ell+1} \psi_{\ell-1}\right)+\Sigma_{t \ell} \psi_{\ell}=q_{\ell}.
The above method can be extended to multi-dimensional case with the help of spherical harmonics, which are defined as
.. math::
Y_{\ell}^{m}(\mu, \phi)=(-1)^{m} \sqrt{\frac{2 \ell+1}{4 \pi} \frac{(\ell-m) !}{(\ell+m) !}} e^{i m \phi} P_{\ell}^{m}(\mu),
where :math:`\ell \leq 0` and :math:`\ell \leq m \leq -\ell`.
The entropy closure moment equations
---------
The continuous slowing down approximation
---------
......
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