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