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 gradient of the gradient of the lapse.
      15             :  *
      16             :  * \details Computes the gradient of the gradient as:
      17             :  * \f{align}
      18             :  *     \nabla_i \nabla_j \alpha &= \alpha A_i A_j -
      19             :  *                 \alpha \Gamma^k{}_{ij} A_k + \alpha \partial_{(i} A_{j)}
      20             :  * \f}
      21             :  * where \f$\alpha\f$, \f$\Gamma^k{}_{ij}\f$, \f$A_i\f$, and
      22             :  * \f$\partial_j A_i\f$ are the lapse, spatial christoffel symbols of the second
      23             :  * kind, the CCZ4 auxiliary variable defined by `Ccz4::Tags::FieldA`, and its
      24             :  * spatial derivative, respectively.
      25             :  *
      26             :  * \note In second-order Ccz4, we impose symmetry of index i and j
      27             :  * in \f$ \partial_i A_j=\partial_i \partial_j \ln \alpha \f$,
      28             :  * because partial derivatives commute and to use
      29             :  * `second_partial_derivatives()`. \f$ A_i \f$ is evolved in
      30             :  * the first-order system so no such symmetry is imposed.
      31             :  */
      32             : template <typename DataType, size_t Dim, typename Frame, typename TensorType>
      33           1 : void grad_grad_lapse(
      34             :     const gsl::not_null<tnsr::ij<DataType, Dim, Frame>*> result,
      35             :     const Scalar<DataType>& lapse,
      36             :     const tnsr::Ijj<DataType, Dim, Frame>& christoffel_second_kind,
      37             :     const tnsr::i<DataType, Dim, Frame>& field_a, const TensorType& d_field_a);
      38             : 
      39             : template <typename DataType, size_t Dim, typename Frame, typename TensorType>
      40           1 : tnsr::ij<DataType, Dim, Frame> grad_grad_lapse(
      41             :     const Scalar<DataType>& lapse,
      42             :     const tnsr::Ijj<DataType, Dim, Frame>& christoffel_second_kind,
      43             :     const tnsr::i<DataType, Dim, Frame>& field_a, const TensorType& d_field_a);
      44             : 
      45             : /// @}
      46             : 
      47             : /// @{
      48             : /*!
      49             :  * \ingroup GeneralRelativityGroup
      50             :  * \brief Computes the divergence of the lapse.
      51             :  *
      52             :  * \details Computes the divergence as:
      53             :  * \f{align}
      54             :  *     \nabla^i \nabla_i \alpha &= \phi^2 \tilde{\gamma}^{ij}
      55             :  *         (\nabla_i \nabla_j \alpha)
      56             :  * \f}
      57             :  * where \f$\phi\f$, \f$\tilde{\gamma}^{ij}\f$, and
      58             :  * \f$\nabla_i \nabla_j \alpha\f$ are the conformal factor, inverse conformal
      59             :  * spatial metric, and the gradient of the gradient of the lapse defined by
      60             :  * `Ccz4::Tags::ConformalFactor`, `Ccz4::Tags::InverseConformalMetric`, and
      61             :  * `Ccz4::Tags::GradGradLapse`, respectively.
      62             :  */
      63             : template <typename DataType, size_t Dim, typename Frame>
      64           1 : void divergence_lapse(
      65             :     const gsl::not_null<Scalar<DataType>*> result,
      66             :     const Scalar<DataType>& conformal_factor_squared,
      67             :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_metric,
      68             :     const tnsr::ij<DataType, Dim, Frame>& grad_grad_lapse);
      69             : 
      70             : template <typename DataType, size_t Dim, typename Frame>
      71           1 : Scalar<DataType> divergence_lapse(
      72             :     const Scalar<DataType>& conformal_factor_squared,
      73             :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_metric,
      74             :     const tnsr::ij<DataType, Dim, Frame>& grad_grad_lapse);
      75             : /// @}
      76             : }  // namespace Ccz4