BnsInitialData::FirstOrderSystem Struct Reference

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 = implementation defined
using	primal_fluxes = implementation defined
using	background_fields = implementation defined
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 >
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 $\mathrm{\Phi }$. For a background matter distribution (given by the specific enthalpy h) and a background metric ${\gamma }_{ij}$. The velocity potential is defined by $D_i\mathrm{\Phi }=h{u}_i$ with $u_i$ (the spatial part of) the four velocity and where ${\mathrm{\Gamma }}_{jk}^i=\frac{1}{2}{\gamma }^{il}({\partial }_j{\gamma }_{kl}+{\partial }_k{\gamma }_{jl}-{\partial }_l{\gamma }_{jk})$ 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 $\mathrm{\Phi }$ 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):

$$\begin{array}{r}-{\partial }_i{F}^i+S=f\end{array}$$

$$\begin{array}{rl}{F}^i& =D^i\varphi -\frac{B^j{D}_j\varphi }{{\alpha }^2}{B}^i\\ S& =-F^i{D}_i(\ln \frac{\alpha \rho }{h})-{\mathrm{\Gamma }}_{ij}^i{F}^j\\ f& =-D_i\left(\frac{C{B}^i}{{\alpha }^2}\right)-\frac{C}{{\alpha }^2}{B}^i{D}_i(\ln \frac{\alpha \rho }{h})\end{array}$$

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The documentation for this struct was generated from the following file:

• src/Elliptic/Systems/BnsInitialData/FirstOrderSystem.hpp