Commit 6692a69b by Tianbai Xiao


Former-commit-id: 359e017b
 ... ... @@ -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 +124,63 @@ 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 \end{array} 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}}{c^{3}}\left[n_{g} \log \left(n_{g}\right)-\left(n_{g}+1\right) \log \left(n_{g}+1\right)\right] The solution of :eq:closure is expressed in terms of the Legendre dual .. math:: \eta_{*}(f)=-\frac{2 k \nu^{2}}{c^{3}} \log \left(1-\exp \left(-\frac{h \nu c}{k} f\right)\right) Let .. math:: \mathcal{B}(\boldsymbol{\alpha}):=\eta_{*}^{\prime}\left(\boldsymbol{\alpha}^{T} \mathbf{m}\right)=\frac{2 h \nu^{3}}{c^{2}} \frac{1}{\exp \left(-\frac{h \nu c}{k} \boldsymbol{\alpha}^{T} \mathbf{m}\right)-1} 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 ... ...