Line data    Source code

       1           0 : 
       2             : // Distributed under the MIT License.
       3             : // See LICENSE.txt for details.
       4             : 
       5             : #pragma once
       6             : 
       7             : #include <cstddef>
       8             : 
       9             : #include "DataStructures/DataBox/PrefixHelpers.hpp"
      10             : #include "DataStructures/DataBox/Prefixes.hpp"
      11             : #include "Elliptic/BoundaryConditions/BoundaryCondition.hpp"
      12             : #include "Elliptic/Protocols/FirstOrderSystem.hpp"
      13             : #include "Elliptic/Systems/BnsInitialData/Equations.hpp"
      14             : #include "Elliptic/Systems/BnsInitialData/Tags.hpp"
      15             : #include "NumericalAlgorithms/LinearOperators/PartialDerivatives.hpp"
      16             : #include "PointwiseFunctions/GeneralRelativity/TagsDeclarations.hpp"
      17             : #include "Utilities/ProtocolHelpers.hpp"
      18             : #include "Utilities/TMPL.hpp"
      19             : 
      20             : namespace BnsInitialData {
      21             : 
      22             : /*!
      23             :  * \brief The Irrotational Bns equations From Baumgarte and Shapiro Chapter 15
      24             :  *  formulated as a set of coupled first-order PDEs.
      25             :  *
      26             :  * \details This system formulates the Irrotational Bns Hydrostatic Equilibrium
      27             :  * equations for the velocity potential \f$\Phi\f$. For a background matter
      28             :  * distribution (given by the specific enthalpy h) and a background metric
      29             :  * \f$\gamma_{ij}\f$. The velocity potential is defined by \f$D_i \Phi = h
      30             :  * u_i\f$ with \f$u_i\f$ (the spatial part of) the four velocity and where
      31             :  * \f$\Gamma^i_{jk}=\frac{1}{2}\gamma^{il}\left(\partial_j\gamma_{kl}
      32             :  * +\partial_k\gamma_{jl}-\partial_l\gamma_{jk}\right)\f$ are the Christoffel
      33             :  * symbols of the second kind of the background (spatial) metric
      34             :  * \f$\gamma_{ij}\f$. The
      35             :  * background metric \f$\gamma_{ij}\f$ and the Christoffel symbols derived from
      36             :  * it are assumed to be independent of the variables \f$\Phi\f$ and \f$u_i\f$,
      37             :  * i.e.
      38             :  * constant throughout an iterative elliptic solve.  Additionally a background
      39             :  * lapse (\f$\alpha\f$) and
      40             :  * shift (\f$\beta\f$) must be provided.  Finally, a ``rotational killing
      41             :  * vector" \f$k^i\f$ (with magnitude
      42             :  * proportional to the angular velocity of the orbital motion) is provided.  The
      43             :  * rotational shift is defined as \f$B^i = \beta^i + k^i\f$ which is
      44             :  * heuristically the background
      45             :  * motion of the spacetime.
      46             :  *
      47             :  * The system can be formulated in terms of these fluxes and sources (see
      48             :  * `elliptic::protocols::FirstOrderSystem`):
      49             :  *
      50             :  *
      51             :  * \f{align*}
      52             :  * -\partial_i F^i + S = f
      53             :  * \f}
      54             :  *
      55             :  * \f{align*}
      56             :  * F^i &=  D_i \phi - \frac{B^j D_j \phi}{\alpha^2}B^i  \\
      57             :  * S &= -F^iD_i \left( \ln \frac{\alpha \rho}{h}\right) -\Gamma^i_{ij}F^j \\
      58             :  * f &= -D_i \left(\frac{C B^i}{\alpha^2}\right) -
      59             :  * \frac{C}{\alpha^2}B^iD_i\left(
      60             :  * \ln \frac{\alpha \rho}{h}\right)\\
      61             :  * \f}
      62             :  */
      63           1 : struct FirstOrderSystem
      64             :     : tt::ConformsTo<elliptic::protocols::FirstOrderSystem> {
      65             :  private:
      66           0 :   using velocity_potential = Tags::VelocityPotential<DataVector>;
      67             : 
      68             :  public:
      69           0 :   static constexpr size_t volume_dim = 3;
      70             : 
      71           0 :   using primal_fields = tmpl::list<velocity_potential>;
      72             : 
      73             :   // We just use the standard `Flux` prefix because the fluxes don't have
      74             :   // symmetries and we don't need to give them a particular meaning.
      75           0 :   using primal_fluxes = tmpl::list<
      76             :       ::Tags::Flux<velocity_potential, tmpl::size_t<3>, Frame::Inertial>>;
      77             : 
      78           0 :   using background_fields = tmpl::list<
      79             :       gr::Tags::InverseSpatialMetric<DataVector, 3>,
      80             :       gr::Tags::SpatialChristoffelSecondKindContracted<DataVector, 3>,
      81             :       gr::Tags::Lapse<DataVector>,
      82             :       ::Tags::deriv<gr::Tags::Lapse<DataVector>,
      83             :                     tmpl::integral_constant<size_t, 3>, Frame::Inertial>,
      84             :       gr::Tags::Shift<DataVector, 3>,
      85             :       ::Tags::deriv<gr::Tags::Shift<DataVector, 3>,
      86             :                     tmpl::integral_constant<size_t, 3>, Frame::Inertial>,
      87             :       Tags::RotationalShift<DataVector>,
      88             :       Tags::DerivLogLapseTimesDensityOverSpecificEnthalpy<DataVector>,
      89             :       Tags::RotationalShiftStress<DataVector>>;
      90           0 :   using inv_metric_tag = gr::Tags::InverseSpatialMetric<DataVector, 3>;
      91             : 
      92           0 :   using fluxes_computer = Fluxes;
      93           0 :   using sources_computer = Sources;
      94             : 
      95           0 :   using boundary_conditions_base =
      96             :       elliptic::BoundaryConditions::BoundaryCondition<3>;
      97             : };
      98             : }  // namespace BnsInitialData