The Irrotational Bns equations From Baumgarte and Shapiro Chapter 15 formulated as a set of coupled first-order PDEs. More...

#include <FirstOrderSystem.hpp>

Public Types
using	primal_fields = tmpl::list<velocity_potential>
using	primal_fluxes
using	background_fields
using	inv_metric_tag = gr::Tags::InverseSpatialMetric<DataVector, 3>
using	fluxes_computer = Fluxes
using	sources_computer = Sources
using	boundary_conditions_base
using	modify_boundary_data = void

Static Public Attributes
static constexpr size_t	volume_dim = 3

Detailed Description

The Irrotational Bns equations From Baumgarte and Shapiro Chapter 15 formulated as a set of coupled first-order PDEs.

Details

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*}

Member Typedef Documentation

◆ background_fields

using BnsInitialData::FirstOrderSystem::background_fields

Initial value:

 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>>