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Commit: 817e13c5144619b701c7cd870655d8dbf94ab8ce Lines: 5 6 83.3 %
Date: 2024-07-19 22:17:05
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          Line data    Source code
       1           0 : // Distributed under the MIT License.
       2             : // See LICENSE.txt for details.
       3             : 
       4             : #pragma once
       5             : 
       6             : #include <array>
       7             : #include <cstddef>
       8             : 
       9             : #include "DataStructures/Tensor/TypeAliases.hpp"
      10             : #include "Utilities/ContainerHelpers.hpp"
      11             : 
      12             : /// \cond
      13             : class DataVector;
      14             : namespace gsl {
      15             : template <class T>
      16             : class not_null;
      17             : }  // namespace gsl
      18             : /// \endcond
      19             : 
      20             : namespace gh::BoundaryConditions {
      21             : /// \brief Detailed implementation of Bjorhus-type boundary corrections
      22           1 : namespace Bjorhus {
      23             : /*!
      24             :  * \brief Computes the expression needed to set boundary conditions on the time
      25             :  * derivative of the characteristic field \f$v^{\psi}_{ab}\f$
      26             :  *
      27             :  * \details In the Bjorhus scheme, the time derivatives of evolved variables are
      28             :  * characteristic projected. A constraint-preserving correction term is added
      29             :  * here to the resulting characteristic (time-derivative) field:
      30             :  *
      31             :  * \f{align}
      32             :  * \Delta \partial_t v^{\psi}_{ab} = \lambda_{\psi} n^i C_{iab}
      33             :  * \f}
      34             :  *
      35             :  * where \f$n^i\f$ is the local unit normal to the external boundary,
      36             :  * \f$C_{iab} = \partial_i \psi_{ab} - \Phi_{iab}\f$ is the three-index
      37             :  * constraint, and \f$\lambda_{\psi}\f$ is the characteristic speed of the field
      38             :  * \f$v^{\psi}_{ab}\f$.
      39             :  */
      40             : template <size_t VolumeDim, typename DataType>
      41           1 : void constraint_preserving_bjorhus_corrections_dt_v_psi(
      42             :     gsl::not_null<tnsr::aa<DataType, VolumeDim, Frame::Inertial>*> bc_dt_v_psi,
      43             :     const tnsr::I<DataType, VolumeDim, Frame::Inertial>&
      44             :         unit_interface_normal_vector,
      45             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>&
      46             :         three_index_constraint,
      47             :     const std::array<DataType, 4>& char_speeds);
      48             : 
      49             : /*!
      50             :  * \brief Computes the expression needed to set boundary conditions on the time
      51             :  * derivative of the characteristic field \f$v^{0}_{iab}\f$
      52             :  *
      53             :  * \details In the Bjorhus scheme, the time derivatives of evolved variables are
      54             :  * characteristic projected. A constraint-preserving correction term is added
      55             :  * here to the resulting characteristic (time-derivative) field \f$v^0_{iab}\f$:
      56             :  *
      57             :  * \f{align}
      58             :  * \Delta \partial_t v^{0}_{iab} = \lambda_{0} n^j C_{jiab}
      59             :  * \f}
      60             :  *
      61             :  * where \f$n^i\f$ is the local unit normal to the external boundary,
      62             :  * \f$C_{ijab} = \partial_i\Phi_{jab} - \partial_j\Phi_{iab}\f$ is the
      63             :  * four-index constraint, and \f$\lambda_{0}\f$ is the characteristic speed of
      64             :  * the field \f$v^0_{iab}\f$.
      65             :  *
      66             :  * \note In 3D, only the non-zero and unique components of the four-index
      67             :  * constraint are stored, as \f$\hat{C}_{iab} = \epsilon_i^{jk} C_{jkab}\f$. In
      68             :  * 2D the input expected here is \f$\hat{C}_{0ab} = C_{01ab}, \hat{C}_{1ab} =
      69             :  * C_{10ab}\f$, and in 1D \f$\hat{C}_{0ab} = 0\f$.
      70             :  */
      71             : template <size_t VolumeDim, typename DataType>
      72           1 : void constraint_preserving_bjorhus_corrections_dt_v_zero(
      73             :     gsl::not_null<tnsr::iaa<DataType, VolumeDim, Frame::Inertial>*>
      74             :         bc_dt_v_zero,
      75             :     const tnsr::I<DataType, VolumeDim, Frame::Inertial>&
      76             :         unit_interface_normal_vector,
      77             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>&
      78             :         four_index_constraint,
      79             :     const std::array<DataType, 4>& char_speeds);
      80             : 
      81             : /// @{
      82             : /*!
      83             :  * \brief Computes the expression needed to set boundary conditions on the time
      84             :  * derivative of the characteristic field \f$v^{-}_{ab}\f$
      85             :  *
      86             :  * \details In the Bjorhus scheme, the time derivatives of evolved variables are
      87             :  * characteristic projected. Constraint-preserving correction terms
      88             :  * \f$T^{\mathrm C}_{ab}\f$, Sommerfeld condition terms for gauge degrees of
      89             :  * freedom \f$T^{\mathrm G}_{ab}\f$, and terms constraining physical degrees of
      90             :  * freedom \f$T^{\mathrm P}_{ab}\f$ are added here to the resulting
      91             :  * characteristic (time-derivative) field:
      92             :  *
      93             :  * \f{align}
      94             :  * \Delta \partial_t v^{-}_{ab} = T^{\mathrm C}_{ab} + T^{\mathrm G}_{ab}
      95             :  *                              + T^{\mathrm P}_{ab}.
      96             :  * \f}
      97             :  *
      98             :  * These terms are given by Eq. (64) of \cite Lindblom2005qh :
      99             :  *
     100             :  * \f{eqnarray}
     101             :  * T^{\mathrm C}_{ab} &=& \frac{1}{2}
     102             :  *      \left(2 k^c k^d l_a l_b - k^c l_b P^d_a - k^c l_a P^d_b - k^d l_b P^c_a
     103             :  *            - k^d l_a P^c_b + P^{cd} P_{ab}\right) \partial_t v^{-}_{cd}\\
     104             :  *     &&+ \frac{1}{\sqrt{2}} \lambda_{-}c^{\hat{0}-}_c \left(l_a l_b k^c +
     105             :  *          P_{ab} l^c - P^c_b l_a - P^c_a l_b \right) \nonumber
     106             :  * \f}
     107             :  *
     108             :  * where \f$l^a (l_a)\f$ is the outgoing null vector (one-form), \f$k^a (k_a)\f$
     109             :  * is the incoming null vector (one-form), \f$P_{ab}, P^{ab}, P^a_b\f$ are
     110             :  * spacetime projection operators defined in `transverse_projection_operator()`,
     111             :  * \f$\partial_t v^{-}_{ab}\f$ is the characteristic projected time derivative
     112             :  * of evolved variables (corresponding to the \f$v^{-}\f$ field), and
     113             :  * \f$c^{\hat{0}\pm}_a\f$ are characteristic modes of the constraint evolution
     114             :  * system:
     115             :  *
     116             :  * \f{align}\nonumber
     117             :  * c^{\hat{0}\pm}_a = F_a \mp n^k C_{ka},
     118             :  * \f}
     119             :  *
     120             :  * where \f$F_a\f$ is the generalized-harmonic (GH) F constraint [Eq. (43) of
     121             :  * \cite Lindblom2005qh], \f$C_{ka}\f$ is the GH 2-index constraint [Eq. (44)
     122             :  * of \cite Lindblom2005qh], and \f$n^k\f$ is the unit spatial normal to the
     123             :  * outer boundary. Boundary correction terms that prevent strong reflections of
     124             :  * gauge perturbations are given by Eq. (25) of \cite Rinne2007ui :
     125             :  *
     126             :  * \f{align}
     127             :  * T^{\mathrm G}_{ab} =
     128             :  *     \left(k_a P^c_b l^d + k_b P^c_a l^d -
     129             :  *          \left(k_a l_b k^c l^d + k_b l_a k_c l^d + k_a k_b l^c l^d
     130             :  *          \right) \right) \left(\gamma_2 - \frac{1}{r}
     131             :  *                          \right) \partial_t v^{\psi}_{cd}
     132             :  * \f}
     133             :  *
     134             :  * where \f$r\f$ is the radial coordinate at the outer boundary, which is
     135             :  * assumed to be spherical, \f$\gamma_2\f$ is a GH constraint damping parameter,
     136             :  * and \f$\partial_t v^{\psi}_{ab}\f$ is the characteristic projected time
     137             :  * derivative of evolved variables (corresponding to the \f$v^{\psi}\f$ field).
     138             :  * Finally, we constrain physical degrees of freedom using corrections from
     139             :  * Eq. (68) of \cite Lindblom2005qh :
     140             :  *
     141             :  * \f{align}
     142             :  * T^{\mathrm P}_{ab} = \left( P^c_a P^d_b - \frac{1}{2} P_{ab} P^{cd} \right)
     143             :  *     \left(\partial_t v^{-}_{cd} + \lambda_{-}
     144             :  *           \left(U^{3-}_{cd} - \gamma_2 n^i C_{icd}\right)\right),
     145             :  * \f}
     146             :  *
     147             :  * where \f$C_{icd}\f$ is the GH 3-index constraint [c.f. Eq. (26) of
     148             :  * \cite Lindblom2005qh, see also `three_index_constraint()`], and
     149             :  *
     150             :  * \f{align}\nonumber
     151             :  * U^{3-}_{ab} = 2 P^i_a P^j_b U^{8-}_{ij}
     152             :  * \f}
     153             :  *
     154             :  * is the inward propagating characteristic mode of the Weyl tensor evolution
     155             :  * \f$U^{8-}_{ab}\f$ [c.f. Eq. 75 of \cite Kidder2004rw ] projected onto the
     156             :  * outer boundary using the spatial-spacetime projection operators \f$P^i_a\f$.
     157             :  * Note that (A) the covariant derivative of extrinsic curvature needed to
     158             :  * get the Weyl propagating modes is calculated substituting the evolved
     159             :  * variable \f$\Phi_{iab}\f$ for spatial derivatives of the spacetime metric;
     160             :  * and (B) the spatial Ricci tensor used in the same calculation is also
     161             :  * calculated using the same substituion, and also includes corrections
     162             :  * proportional to the GH 4-index constraint \f$C_{ijab}\f$ [c.f. Eq. (45) of
     163             :  * \cite Lindblom2005qh , see also `four_index_constraint()`]:
     164             :  *
     165             :  * \f{align}\nonumber
     166             :  * R_{ij} \rightarrow R_{ij} + C_{(iklj)} + \frac{1}{2} n^k q^a C_{(ikj)a},
     167             :  * \f}
     168             :  *
     169             :  * where \f$q^a\f$ is the future-directed spacetime normal vector.
     170             :  */
     171             : template <size_t VolumeDim, typename DataType>
     172           1 : void constraint_preserving_bjorhus_corrections_dt_v_minus(
     173             :     gsl::not_null<tnsr::aa<DataType, VolumeDim, Frame::Inertial>*>
     174             :         bc_dt_v_minus,
     175             :     const Scalar<DataType>& gamma2,
     176             :     const tnsr::I<DataType, VolumeDim, Frame::Inertial>& inertial_coords,
     177             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>& incoming_null_one_form,
     178             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>& outgoing_null_one_form,
     179             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& incoming_null_vector,
     180             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& outgoing_null_vector,
     181             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>& projection_ab,
     182             :     const tnsr::Ab<DataType, VolumeDim, Frame::Inertial>& projection_Ab,
     183             :     const tnsr::AA<DataType, VolumeDim, Frame::Inertial>& projection_AB,
     184             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>&
     185             :         char_projected_rhs_dt_v_psi,
     186             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>&
     187             :         char_projected_rhs_dt_v_minus,
     188             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>&
     189             :         constraint_char_zero_plus,
     190             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>&
     191             :         constraint_char_zero_minus,
     192             :     const std::array<DataType, 4>& char_speeds);
     193             : 
     194             : template <size_t VolumeDim, typename DataType>
     195           1 : void constraint_preserving_physical_bjorhus_corrections_dt_v_minus(
     196             :     gsl::not_null<tnsr::aa<DataType, VolumeDim, Frame::Inertial>*>
     197             :         bc_dt_v_minus,
     198             :     const Scalar<DataType>& gamma2,
     199             :     const tnsr::I<DataType, VolumeDim, Frame::Inertial>& inertial_coords,
     200             :     const tnsr::i<DataType, VolumeDim, Frame::Inertial>&
     201             :         unit_interface_normal_one_form,
     202             :     const tnsr::I<DataType, VolumeDim, Frame::Inertial>&
     203             :         unit_interface_normal_vector,
     204             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>&
     205             :         spacetime_unit_normal_vector,
     206             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>& incoming_null_one_form,
     207             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>& outgoing_null_one_form,
     208             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& incoming_null_vector,
     209             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& outgoing_null_vector,
     210             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>& projection_ab,
     211             :     const tnsr::Ab<DataType, VolumeDim, Frame::Inertial>& projection_Ab,
     212             :     const tnsr::AA<DataType, VolumeDim, Frame::Inertial>& projection_AB,
     213             :     const tnsr::II<DataType, VolumeDim, Frame::Inertial>&
     214             :         inverse_spatial_metric,
     215             :     const tnsr::ii<DataType, VolumeDim, Frame::Inertial>& extrinsic_curvature,
     216             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>& spacetime_metric,
     217             :     const tnsr::AA<DataType, VolumeDim, Frame::Inertial>&
     218             :         inverse_spacetime_metric,
     219             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>&
     220             :         three_index_constraint,
     221             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>&
     222             :         char_projected_rhs_dt_v_psi,
     223             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>&
     224             :         char_projected_rhs_dt_v_minus,
     225             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>&
     226             :         constraint_char_zero_plus,
     227             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>&
     228             :         constraint_char_zero_minus,
     229             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>& phi,
     230             :     const tnsr::ijaa<DataType, VolumeDim, Frame::Inertial>& d_phi,
     231             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>& d_pi,
     232             :     const std::array<DataType, 4>& char_speeds);
     233             : /// @}
     234             : 
     235             : namespace detail {
     236             : template <size_t VolumeDim, typename DataType>
     237             : void add_gauge_sommerfeld_terms_to_dt_v_minus(
     238             :     const gsl::not_null<tnsr::aa<DataType, VolumeDim, Frame::Inertial>*>
     239             :         bc_dt_v_minus,
     240             :     const Scalar<DataType>& gamma2,
     241             :     const tnsr::I<DataType, VolumeDim, Frame::Inertial>& inertial_coords,
     242             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>& incoming_null_one_form,
     243             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>& outgoing_null_one_form,
     244             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& incoming_null_vector,
     245             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& outgoing_null_vector,
     246             :     const tnsr::Ab<DataType, VolumeDim, Frame::Inertial>& projection_Ab,
     247             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>&
     248             :         char_projected_rhs_dt_v_psi);
     249             : 
     250             : template <size_t VolumeDim, typename DataType>
     251             : void add_constraint_dependent_terms_to_dt_v_minus(
     252             :     const gsl::not_null<tnsr::aa<DataType, VolumeDim, Frame::Inertial>*>
     253             :         bc_dt_v_minus,
     254             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>& outgoing_null_one_form,
     255             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& incoming_null_vector,
     256             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>& outgoing_null_vector,
     257             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>& projection_ab,
     258             :     const tnsr::Ab<DataType, VolumeDim, Frame::Inertial>& projection_Ab,
     259             :     const tnsr::AA<DataType, VolumeDim, Frame::Inertial>& projection_AB,
     260             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>&
     261             :         constraint_char_zero_plus,
     262             :     const tnsr::a<DataType, VolumeDim, Frame::Inertial>&
     263             :         constraint_char_zero_minus,
     264             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>&
     265             :         char_projected_rhs_dt_v_minus,
     266             :     const std::array<DataType, 4>& char_speeds);
     267             : 
     268             : template <size_t VolumeDim, typename DataType>
     269             : void add_physical_terms_to_dt_v_minus(
     270             :     const gsl::not_null<tnsr::aa<DataType, VolumeDim, Frame::Inertial>*>
     271             :         bc_dt_v_minus,
     272             :     const Scalar<DataType>& gamma2,
     273             :     const tnsr::i<DataType, VolumeDim, Frame::Inertial>&
     274             :         unit_interface_normal_one_form,
     275             :     const tnsr::I<DataType, VolumeDim, Frame::Inertial>&
     276             :         unit_interface_normal_vector,
     277             :     const tnsr::A<DataType, VolumeDim, Frame::Inertial>&
     278             :         spacetime_unit_normal_vector,
     279             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>& projection_ab,
     280             :     const tnsr::Ab<DataType, VolumeDim, Frame::Inertial>& projection_Ab,
     281             :     const tnsr::AA<DataType, VolumeDim, Frame::Inertial>& projection_AB,
     282             :     const tnsr::II<DataType, VolumeDim, Frame::Inertial>&
     283             :         inverse_spatial_metric,
     284             :     const tnsr::ii<DataType, VolumeDim, Frame::Inertial>& extrinsic_curvature,
     285             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>& spacetime_metric,
     286             :     const tnsr::AA<DataType, VolumeDim, Frame::Inertial>&
     287             :         inverse_spacetime_metric,
     288             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>&
     289             :         three_index_constraint,
     290             :     const tnsr::aa<DataType, VolumeDim, Frame::Inertial>&
     291             :         char_projected_rhs_dt_v_minus,
     292             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>& phi,
     293             :     const tnsr::ijaa<DataType, VolumeDim, Frame::Inertial>& d_phi,
     294             :     const tnsr::iaa<DataType, VolumeDim, Frame::Inertial>& d_pi,
     295             :     const std::array<DataType, 4>& char_speeds);
     296             : }  // namespace detail
     297             : }  // namespace Bjorhus
     298             : }  // namespace gh::BoundaryConditions

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