Line data    Source code

       1            0 : // Distributed under the MIT License.
       2              : // See LICENSE.txt for details.
       3              : 
       4              : #pragma once
       5              : 
       6              : #include <cstddef>
       7              : 
       8              : #include "DataStructures/Tensor/TypeAliases.hpp"
       9              : #include "Utilities/Gsl.hpp"
      10              : 
      11              : namespace Ccz4 {
      12              : /// @{
      13              : /*!
      14              :  * \brief Computes the spatial derivative of the conformal spatial christoffel
      15              :  * symbols of the second kind
      16              :  *
      17              :  * \details Computes the derivative as:
      18              :  * \f{align}
      19              :  *     \partial_k \tilde{\Gamma}^m{}_{ij} &=
      20              :  *       -2 D_k{}^{ml} (D_{ijl} + D_{jil} - D_{lij}) +
      21              :  *       \tilde{\gamma}^{ml}(\partial_{(k} D_{i)jl} + \partial_{(k} D_{j)il} -
      22              :  *       \partial_{(k} D_{l)ij})
      23              :  * \f}
      24              :  * where \f$\tilde{\gamma}^{ij}\f$, \f$D_{ijk}\f$, \f$\partial_l D_{ijk}\f$, and
      25              :  * \f$D_k{}^{ij}\f$ are the inverse conformal spatial metric defined by
      26              :  * `Ccz4::Tags::InverseConformalMetric`, the CCZ4 auxiliary variable defined by
      27              :  * `Ccz4::Tags::FieldD`, its spatial derivative, and the CCZ4 identity defined
      28              :  * by `Ccz4::Tags::FieldDUp`.
      29              :  * \note In second-order Ccz4, we impose symmetry of index k and l
      30              :  * in \f$ \partial_l D_{kij}=\frac{1}{2}\partial_l \partial_k
      31              :  * \tilde{\gamma}_{ij} \f$, because partial derivatives commute and to use
      32              :  * `second_partial_derivatives()`. \f$ D_{kij} \f$ is evolved in
      33              :  * the first-order system so no such symmetry is imposed.
      34              :  */
      35              : template <typename DataType, size_t Dim, typename Frame, typename TensorType>
      36            1 : void deriv_conformal_christoffel_second_kind(
      37              :     const gsl::not_null<tnsr::iJkk<DataType, Dim, Frame>*> result,
      38              :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_spatial_metric,
      39              :     const tnsr::ijj<DataType, Dim, Frame>& field_d, const TensorType& d_field_d,
      40              :     const tnsr::iJJ<DataType, Dim, Frame>& field_d_up);
      41              : 
      42              : template <typename DataType, size_t Dim, typename Frame, typename TensorType>
      43            1 : tnsr::iJkk<DataType, Dim, Frame> deriv_conformal_christoffel_second_kind(
      44              :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_spatial_metric,
      45              :     const tnsr::ijj<DataType, Dim, Frame>& field_d, const TensorType& d_field_d,
      46              :     const tnsr::iJJ<DataType, Dim, Frame>& field_d_up);
      47              : 
      48              : /// @}
      49              : 
      50              : /// @{
      51              : /*!
      52              :  * \brief Computes the spatial derivative of the contraction of the conformal
      53              :  * spatial Christoffel symbols of the second kind
      54              :  *
      55              :  * \details Computes the derivative as:
      56              :  *
      57              :  * \f{align}
      58              :  *     \partial_k \tilde{\Gamma}^i &= -2 D_k{}^{jl} \tilde{\Gamma}^i_{jl} +
      59              :  *       \tilde{\gamma}^{jl} \partial_k \tilde{\Gamma}^i_{jl}
      60              :  * \f}
      61              :  *
      62              :  * where \f$\tilde{\gamma}^{ij}\f$ is the inverse conformal spatial metric
      63              :  * defined by `Ccz4::Tags::InverseConformalMetric`, \f$D_k{}^{ij}\f$ is the CCZ4
      64              :  * identity defined by `Ccz4::Tags::FieldDUp`, \f$\tilde{\Gamma}^k_{ij}\f$ is
      65              :  * the conformal spatial Christoffel symbols of the second kind defined by
      66              :  * `Ccz4::Tags::ConformalChristoffelSecondKind`, and
      67              :  * \f$\partial_k \tilde{\Gamma}^k_{ij}\f$ is its spatial derivative defined by
      68              :  * `Ccz4::Tags::DerivConformalChristoffelSecondKind`.
      69              :  */
      70              : template <typename DataType, size_t Dim, typename Frame>
      71            1 : void deriv_contracted_conformal_christoffel_second_kind(
      72              :     const gsl::not_null<tnsr::iJ<DataType, Dim, Frame>*> result,
      73              :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_spatial_metric,
      74              :     const tnsr::iJJ<DataType, Dim, Frame>& field_d_up,
      75              :     const tnsr::Ijj<DataType, Dim, Frame>& conformal_christoffel_second_kind,
      76              :     const tnsr::iJkk<DataType, Dim, Frame>&
      77              :         d_conformal_christoffel_second_kind);
      78              : 
      79              : template <typename DataType, size_t Dim, typename Frame>
      80              : tnsr::iJ<DataType, Dim, Frame>
      81            1 : deriv_contracted_conformal_christoffel_second_kind(
      82              :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_spatial_metric,
      83              :     const tnsr::iJJ<DataType, Dim, Frame>& field_d_up,
      84              :     const tnsr::Ijj<DataType, Dim, Frame>& conformal_christoffel_second_kind,
      85              :     const tnsr::iJkk<DataType, Dim, Frame>&
      86              :         d_conformal_christoffel_second_kind);
      87              : /// @}
      88              : }  // namespace Ccz4