Commit b719a2c8 authored by Steffen Schotthöfer's avatar Steffen Schotthöfer
Browse files

Merge branch 'develop' of into develop

Former-commit-id: f9798581
parents 61f75dff f5a65b82
......@@ -10,8 +10,8 @@ The framework provides rich deterministic solver types for different domain-spec
A list of current supported models and equations is as follows.
- linear Boltzmann (:math:`S_N`) equation
- spherical harmonics (:math:`P_N`) equations
- entropy-closure moment (:math:`M_N`) equations
- spherical harmonics (:math:`P_N`) moment equations
- entropy-closure (:math:`M_N`) moment equations
- continuous slowing down equation
......@@ -75,6 +75,7 @@ Let us consider the radiative transfer equation in one-dimensional physical spac
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),
......@@ -83,6 +84,7 @@ 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},
......@@ -122,8 +124,68 @@ where :math:`\ell \leq 0` and :math:`\ell \leq m \leq -\ell`.
The entropy closure moment equations
Another method of moments comes from the minimal principle of a convex entropy to close the moment system.
Derivation of such moment system begins with the choice of a vector-valued function
:math:`m: \mathbb{S}^{2} \rightarrow \mathbb{R}^{n}, \Omega \mapsto\left[m_{0}(\Omega), \ldots, m_{n-1}(\Omega)\right]^{T}`,
whose n components are linearly independent functions of :math:`\Omega`.
Evolution equations for the moments u(x, t) :=
hmψ(x, ·, t)i are found by multiplying the transport equation by m and integrating
over all angles to give
.. math::
\frac{1}{v} \partial_{t} u+\nabla_{x} \cdot\langle\Omega m \psi\rangle=\langle m \mathcal{C}(\psi)\rangle.
The system above is not closed; a recipe, or closure, must be prescribed to express
unknown quantities in terms of the given moments. Often this is done via an
approximation for :math:`\psi` that depends on :math:`u`,
.. math::
\psi(x, \Omega, t) \simeq \mathcal{E}(u(x, t))(\Omega).
A general strategy for prescribing a closure is to
use the solution of a constrained optimization problem
.. math::
:label: closure
\min_{g \in \operatorname{Dom}(\mathcal{H})} & \mathcal{H}(g) \\
\quad \text { s.t. } & \langle\mathbf{m} g\rangle=\langle\mathbf{m} \psi\rangle=u,
where :math:`\mathcal H(g)=\langle \eta(g) \rangle` and $\eta: \mathbb R \rightarrow \mathbb R$
is a convex function that is related to
the entropy of the system. For photons, the physically relevant entropy comes from
Bose-Einstein statistics
.. math::
\eta(g)=\frac{2 k \nu^{2}}{v^{3}}\left[n_{g} \log \left(n_{g}\right)-\left(n_{g}+1\right) \log \left(n_{g}+1\right)\right],
where :math:`n_g` is the occupation number associated with g,
.. math::
n_{g}:=\frac{v^{2}}{2 h \nu^{3}} g.
The solution of :eq:`closure` is expressed in terms of the Legendre dual
.. math::
\eta_{*}(f)=-\frac{2 k \nu^{2}}{v^{3}} \log \left(1-\exp \left(-\frac{h \nu c}{k} f\right)\right).
.. math::
\mathcal{B}(\boldsymbol{\alpha}):=\eta_{*}^{\prime}\left(\boldsymbol{\alpha}^{T} \mathbf{m}\right)=\frac{2 h \nu^{3}}{v^{2}} \frac{1}{\exp \left(-\frac{h \nu c}{k} \boldsymbol{\alpha}^{T} \mathbf{m}\right)-1},
then the solution of :eq:`closure` is given by :math:`\mathcal B(\hat \alpha)`, where :math:`\hat \alpha= \hat \alpha(u)` solves the
dual problem
.. math::
\min _{\boldsymbol{\alpha} \in \mathbb{R}^{n}}\left\{\left\langle\eta_{*}\left(\boldsymbol{\alpha}^{T} \mathbf{m}\right)\right\rangle-\boldsymbol{\alpha}^{T} \mathbf{u}\right\}.
The continuous slowing down approximation
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