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SpECTRE
v2025.04.21
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The Irrotational Bns equations From Baumgarte and Shapiro Chapter 15 formulated as a set of coupled first-order PDEs. More...
#include <FirstOrderSystem.hpp>
Static Public Attributes | |
| static constexpr size_t | volume_dim = 3 |
The Irrotational Bns equations From Baumgarte and Shapiro Chapter 15 formulated as a set of coupled first-order PDEs.
This system formulates the Irrotational Bns Hydrostatic Equilibrium equations for the velocity potential \(\Phi\). For a background matter distribution (given by the specific enthalpy h) and a background metric \(\gamma_{ij}\). The velocity potential is defined by \(D_i \Phi = h u_i\) with \(u_i\) (the spatial part of) the four velocity and where \(\Gamma^i_{jk}=\frac{1}{2}\gamma^{il}\left(\partial_j\gamma_{kl} +\partial_k\gamma_{jl}-\partial_l\gamma_{jk}\right)\) are the Christoffel symbols of the second kind of the background (spatial) metric \(\gamma_{ij}\). The background metric \(\gamma_{ij}\) and the Christoffel symbols derived from it are assumed to be independent of the variables \(\Phi\) and \(u_i\), i.e. constant throughout an iterative elliptic solve. Additionally a background lapse ( \(\alpha\)) and shift ( \(\beta\)) must be provided. Finally, a "rotational killing vector" \(k^i\) (with magnitude proportional to the angular velocity of the orbital motion) is provided. The rotational shift is defined as \(B^i = \beta^i + k^i\) which is heuristically the background motion of the spacetime.
The system can be formulated in terms of these fluxes and sources (see elliptic::protocols::FirstOrderSystem):
\begin{align*} -\partial_i F^i + S = f \end{align*}
\begin{align*} F^i &= D^i \phi - \frac{B^j D_j \phi}{\alpha^2}B^i \\ S &= -F^iD_i \left( \ln \frac{\alpha \rho}{h}\right) -\Gamma^i_{ij}F^j \\ f &= -D_i \left(\frac{C B^i}{\alpha^2}\right) - \frac{C}{\alpha^2}B^iD_i\left( \ln \frac{\alpha \rho}{h}\right)\\ \end{align*}