SpECTRE
v2024.06.18

The Irrotational Bns equations From Baumgarte and Shapiro Chapter 15 formulated as a set of coupled firstorder PDEs. More...
#include <FirstOrderSystem.hpp>
Public Types  
using  primal_fields = tmpl::list< velocity_potential > 
using  primal_fluxes = tmpl::list< ::Tags::Flux< velocity_potential, tmpl::size_t< 3 >, Frame::Inertial > > 
using  background_fields = tmpl::list< gr::Tags::InverseSpatialMetric< DataVector, 3 >, gr::Tags::SpatialChristoffelSecondKindContracted< DataVector, 3 >, gr::Tags::Lapse< DataVector >, ::Tags::deriv< gr::Tags::Lapse< DataVector >, tmpl::integral_constant< size_t, 3 >, Frame::Inertial >, gr::Tags::Shift< DataVector, 3 >, ::Tags::deriv< gr::Tags::Shift< DataVector, 3 >, tmpl::integral_constant< size_t, 3 >, Frame::Inertial >, Tags::RotationalShift< DataVector >, Tags::DerivLogLapseTimesDensityOverSpecificEnthalpy< DataVector >, Tags::RotationalShiftStress< DataVector > > 
using  inv_metric_tag = gr::Tags::InverseSpatialMetric< DataVector, 3 > 
using  fluxes_computer = Fluxes 
using  sources_computer = Sources 
using  boundary_conditions_base = elliptic::BoundaryConditions::BoundaryCondition< 3 > 
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 firstorder 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*}