Add spatial covariant derivative of extrinsic curvature

src/PointwiseFunctions/GeneralRelativity/ExtrinsicCurvature.cpp

@@ -46,6 +46,37 @@ tnsr::ii<DataType, SpatialDim, Frame> extrinsic_curvature(
                       spatial_metric, dt_spatial_metric, deriv_spatial_metric);
   return ex_curvature;
 }
+
+template <typename DataType, size_t SpatialDim, typename Frame>
+void covariant_derivative_of_extrinsic_curvature(
+    const gsl::not_null<tnsr::ijj<DataType, SpatialDim, Frame>*> grad_ex_curv,
+    const tnsr::ijj<DataType, SpatialDim, Frame>& d_ex_curv,
+    const tnsr::ii<DataType, SpatialDim, Frame>& ex_curv,
+    const tnsr::Ijj<DataType, SpatialDim, Frame>&
+        spatial_christoffel_second_kind) {
+  set_number_of_grid_points(grad_ex_curv, ex_curv);
+  tenex::evaluate<ti::i, ti::j, ti::k>(
+      grad_ex_curv, d_ex_curv(ti::i, ti::j, ti::k) -
+                        spatial_christoffel_second_kind(ti::L, ti::i, ti::j) *
+                            ex_curv(ti::l, ti::k) -
+                        spatial_christoffel_second_kind(ti::L, ti::i, ti::k) *
+                            ex_curv(ti::j, ti::l));
+}
+
+template <typename DataType, size_t SpatialDim, typename Frame>
+tnsr::ijj<DataType, SpatialDim, Frame>
+covariant_derivative_of_extrinsic_curvature(
+    const tnsr::ijj<DataType, SpatialDim, Frame>& d_ex_curv,
+    const tnsr::ii<DataType, SpatialDim, Frame>& ex_curv,
+    const tnsr::Ijj<DataType, SpatialDim, Frame>&
+        spatial_christoffel_second_kind) {
+  tnsr::ijj<DataType, SpatialDim, Frame> grad_ex_curv{};
+  covariant_derivative_of_extrinsic_curvature(make_not_null(&grad_ex_curv),
+                                              d_ex_curv, ex_curv,
+                                              spatial_christoffel_second_kind);
+  return grad_ex_curv;
+}
+
 }  // namespace gr
 
 #define DIM(data) BOOST_PP_TUPLE_ELEM(0, data)
@@ -71,7 +102,20 @@ tnsr::ii<DataType, SpatialDim, Frame> extrinsic_curvature(
       const tnsr::ii<DTYPE(data), DIM(data), FRAME(data)>& spatial_metric,    \
       const tnsr::ii<DTYPE(data), DIM(data), FRAME(data)>& dt_spatial_metric, \
       const tnsr::ijj<DTYPE(data), DIM(data), FRAME(data)>&                   \
-          deriv_spatial_metric);
+          deriv_spatial_metric);                                              \
+  template void gr::covariant_derivative_of_extrinsic_curvature(              \
+      const gsl::not_null<tnsr::ijj<DTYPE(data), DIM(data), FRAME(data)>*>    \
+          grad_ex_curv,                                                       \
+      const tnsr::ijj<DTYPE(data), DIM(data), FRAME(data)>& d_ex_curv,        \
+      const tnsr::ii<DTYPE(data), DIM(data), FRAME(data)>& ex_curv,           \
+      const tnsr::Ijj<DTYPE(data), DIM(data), FRAME(data)>&                   \
+          spatial_christoffel_second_kind);                                   \
+  template tnsr::ijj<DTYPE(data), DIM(data), FRAME(data)>                     \
+  gr::covariant_derivative_of_extrinsic_curvature(                            \
+      const tnsr::ijj<DTYPE(data), DIM(data), FRAME(data)>& d_ex_curv,        \
+      const tnsr::ii<DTYPE(data), DIM(data), FRAME(data)>& ex_curv,           \
+      const tnsr::Ijj<DTYPE(data), DIM(data), FRAME(data)>&                   \
+          spatial_christoffel_second_kind);
 
 GENERATE_INSTANTIATIONS(INSTANTIATE, (1, 2, 3), (double, DataVector),
                         (Frame::Grid, Frame::Distorted, Frame::Inertial))

src/PointwiseFunctions/GeneralRelativity/ExtrinsicCurvature.hpp

@@ -6,6 +6,7 @@
 #include <cstddef>
 
 #include "DataStructures/Tensor/TypeAliases.hpp"
+#include "PointwiseFunctions/GeneralRelativity/Tags.hpp"
 #include "Utilities/Gsl.hpp"
 
 /// \ingroup GeneralRelativityGroup
@@ -47,4 +48,60 @@ void extrinsic_curvature(
     const tnsr::ii<DataType, SpatialDim, Frame>& dt_spatial_metric,
     const tnsr::ijj<DataType, SpatialDim, Frame>& deriv_spatial_metric);
 /// @}
+
+/// @{
+/*!
+ * \ingroup GeneralRelativityGroup
+ * \brief  Computes the spatial covariant derivative of the extrinsic curvature.
+ *  \details The spatial covariant derivative is computed as
+ * \f[ D_k K_{ij} = \partial_k K_{ij} - {^{(3)}\Gamma^{l}_{ki}} K_{lj}
+ * - {^{(3)}\Gamma^{l}_{kj}}K_{il} \f]
+ * where \f$ {^{(3)}\Gamma^{k}_{ij}} \f$ is the spatial Christoffel symbol of
+ * the second kind.
+ */
+template <typename DataType, size_t SpatialDim, typename Frame>
+tnsr::ijj<DataType, SpatialDim, Frame>
+covariant_derivative_of_extrinsic_curvature(
+    const tnsr::ijj<DataType, SpatialDim, Frame>& d_ex_curv,
+    const tnsr::ii<DataType, SpatialDim, Frame>& ex_curv,
+    const tnsr::Ijj<DataType, SpatialDim, Frame>&
+        spatial_christoffel_second_kind);
+template <typename DataType, size_t SpatialDim, typename Frame>
+void covariant_derivative_of_extrinsic_curvature(
+    gsl::not_null<tnsr::ijj<DataType, SpatialDim, Frame>*> grad_ex_curv,
+    const tnsr::ijj<DataType, SpatialDim, Frame>& d_ex_curv,
+    const tnsr::ii<DataType, SpatialDim, Frame>& ex_curv,
+    const tnsr::Ijj<DataType, SpatialDim, Frame>&
+        spatial_christoffel_second_kind);
+/// @}
+
+namespace Tags {
+/// \copydoc covariant_derivative_of_extrinsic_curvature
+template <size_t SpatialDim, typename Frame>
+struct CovariantDerivativeOfExtrinsicCurvatureCompute
+    : gr::Tags::CovariantDerivativeOfExtrinsicCurvature<DataVector, SpatialDim,
+                                                        Frame>,
+      db::ComputeTag {
+  using argument_tags = tmpl::list<
+      ::Tags::deriv<gr::Tags::ExtrinsicCurvature<DataVector, SpatialDim, Frame>,
+                    tmpl::size_t<SpatialDim>, Frame>,
+      gr::Tags::ExtrinsicCurvature<DataVector, SpatialDim, Frame>,
+      gr::Tags::SpatialChristoffelSecondKind<DataVector, SpatialDim, Frame>>;
+
+  using return_type = tnsr::ijj<DataVector, SpatialDim, Frame>;
+
+  static constexpr auto function = static_cast<void (*)(
+      const gsl::not_null<tnsr::ijj<DataVector, SpatialDim, Frame>*>,
+      const tnsr::ijj<DataVector, SpatialDim, Frame>&,
+      const tnsr::ii<DataVector, SpatialDim, Frame>&,
+      const tnsr::Ijj<DataVector, SpatialDim, Frame>&
+          spatial_christoffel_second_kind)>(
+      &covariant_derivative_of_extrinsic_curvature<DataVector, SpatialDim,
+                                                   Frame>);
+
+  using base =
+      gr::Tags::CovariantDerivativeOfExtrinsicCurvature<DataVector, SpatialDim,
+                                                        Frame>;
+};
+}  // namespace Tags
 }  // namespace gr

src/PointwiseFunctions/GeneralRelativity/Tags.hpp

@@ -143,6 +143,10 @@ template <typename DataType>
 struct TraceExtrinsicCurvature : db::SimpleTag {
   using type = Scalar<DataType>;
 };
+template <typename DataType, size_t Dim, typename Frame>
+struct CovariantDerivativeOfExtrinsicCurvature : db::SimpleTag {
+  using type = tnsr::ijj<DataType, Dim, Frame>;
+};
 
 /*!
  * \brief Holds a quantity that's similar to the shift, but isn't the shift.

src/PointwiseFunctions/GeneralRelativity/TagsDeclarations.hpp

@@ -61,6 +61,8 @@ template <typename DataType, size_t Dim, typename Frame = Frame::Inertial>
 struct ExtrinsicCurvature;
 template <typename DataType>
 struct TraceExtrinsicCurvature;
+template <typename DataType, size_t Dim, typename Frame>
+struct CovariantDerivativeOfExtrinsicCurvature;
 template <typename DataType, size_t Dim, typename Frame = Frame::Inertial>
 struct SpatialRicci;
 template <typename DataType>

src/PointwiseFunctions/GeneralRelativity/WeylMagnetic.hpp

@@ -84,8 +84,7 @@ template <typename DataType, size_t Dim, typename Frame>
 struct WeylMagneticCompute : WeylMagnetic<DataType, Dim, Frame>,
                              db::ComputeTag {
   using argument_tags = tmpl::list<
-      ::Tags::deriv<gr::Tags::ExtrinsicCurvature<DataType, Dim, Frame>,
-                    tmpl::size_t<Dim>, Frame>,
+      gr::Tags::CovariantDerivativeOfExtrinsicCurvature<DataType, Dim, Frame>,
       gr::Tags::SpatialMetric<DataType, Dim, Frame>,
       gr::Tags::SqrtDetSpatialMetric<DataType>>;
 

tests/Unit/PointwiseFunctions/GeneralRelativity/ComputeSpacetimeQuantities.py

@@ -153,6 +153,22 @@ def extrinsic_curvature(
     return ext_curve
 
 
+def covariant_deriv_extrinsic_curvture_adm(
+    d_ex_curv,
+    ex_curv,
+    spatial_christoffel_second_kind,
+):
+    grad_ex_curv = d_ex_curv
+    grad_ex_curv += -np.einsum(
+        "lki,lj->kij", spatial_christoffel_second_kind, ex_curv
+    )
+    grad_ex_curv += -np.einsum(
+        "lkj,il->kij", spatial_christoffel_second_kind, ex_curv
+    )
+
+    return grad_ex_curv
+
+
 def deriv_inverse_spatial_metric(inverse_spatial_metric, d_spatial_metric):
     return -np.einsum(
         "in,mj,knm->kij",

tests/Unit/PointwiseFunctions/GeneralRelativity/Test_ComputeSpacetimeQuantities.cpp

@@ -153,7 +153,18 @@ void test_compute_extrinsic_curvature(const DataType& used_for_size) {
       "ComputeSpacetimeQuantities", "extrinsic_curvature", {{{-10., 10.}}},
       used_for_size);
 }
-
+template <size_t SpatialDim, typename DataType>
+void test_cov_deriv_extrinsic_curvature_adm(const DataType& used_for_size) {
+  pypp::check_with_random_values<1>(
+      static_cast<tnsr::ijj<DataType, SpatialDim, Frame::Inertial> (*)(
+          const tnsr::ijj<DataType, SpatialDim, Frame::Inertial>&,
+          const tnsr::ii<DataType, SpatialDim, Frame::Inertial>&,
+          const tnsr::Ijj<DataType, SpatialDim, Frame::Inertial>&)>(
+          &::gr::covariant_derivative_of_extrinsic_curvature<
+              DataType, SpatialDim, Frame::Inertial>),
+      "ComputeSpacetimeQuantities", "covariant_deriv_extrinsic_curvture_adm",
+      {{{-1., 1.}}}, used_for_size);
+}
 template <size_t Dim, typename T>
 void test_compute_spatial_metric_lapse_shift(const T& used_for_size) {
   // Set up random values for lapse, shift, and spatial_metric.
@@ -232,6 +243,8 @@ SPECTRE_TEST_CASE("Unit.PointwiseFunctions.GeneralRelativity.SpacetimeDecomp",
                                     (1, 2, 3));
   CHECK_FOR_DOUBLES_AND_DATAVECTORS(test_compute_deriv_inverse_spatial_metric,
                                     (1, 2, 3));
+  CHECK_FOR_DOUBLES_AND_DATAVECTORS(test_cov_deriv_extrinsic_curvature_adm,
+                                    (1, 2, 3));
 
   // Check that compute items work correctly in the DataBox
   // First, check that the names are correct
@@ -263,6 +276,9 @@ SPECTRE_TEST_CASE("Unit.PointwiseFunctions.GeneralRelativity.SpacetimeDecomp",
   TestHelpers::db::test_compute_tag<
       gr::Tags::DerivativesOfSpacetimeMetricCompute<3, Frame::Inertial>>(
       "DerivativesOfSpacetimeMetric");
+  TestHelpers::db::test_compute_tag<
+      gr::Tags::CovariantDerivativeOfExtrinsicCurvatureCompute<
+          3, Frame::Inertial>>("CovariantDerivativeOfExtrinsicCurvature");
 
   // Second, put the compute items into a data box and check that they
   // put the correct results

tests/Unit/PointwiseFunctions/GeneralRelativity/Test_Tags.cpp

@@ -81,6 +81,9 @@ void test_simple_tags() {
       gr::Tags::ExtrinsicCurvature<Type, Dim, Frame>>("ExtrinsicCurvature");
   TestHelpers::db::test_simple_tag<gr::Tags::TraceExtrinsicCurvature<Type>>(
       "TraceExtrinsicCurvature");
+  TestHelpers::db::test_simple_tag<
+      gr::Tags::CovariantDerivativeOfExtrinsicCurvature<Type, Dim, Frame>>(
+      "CovariantDerivativeOfExtrinsicCurvature");
   TestHelpers::db::test_simple_tag<gr::Tags::SpatialRicci<Type, Dim, Frame>>(
       "SpatialRicci");
   TestHelpers::db::test_simple_tag<gr::Tags::EnergyDensity<Type>>(

tests/Unit/PointwiseFunctions/GeneralRelativity/Test_WeylMagnetic.cpp

@@ -74,8 +74,8 @@ void test_compute_item_in_databox(const DataType& used_for_size) {
                       make_not_null(&inverse_spatial_metric), used_for_size);
 
   const auto box = db::create<
-      db::AddSimpleTags<::Tags::deriv<gr::Tags::ExtrinsicCurvature<DataType, 3>,
-                                      tmpl::size_t<3>, Frame::Inertial>,
+      db::AddSimpleTags<gr::Tags::CovariantDerivativeOfExtrinsicCurvature<
+                            DataType, 3, Frame::Inertial>,
                         gr::Tags::SpatialMetric<DataType, 3>,
                         gr::Tags::SqrtDetSpatialMetric<DataType>,
                         gr::Tags::InverseSpatialMetric<DataType, 3>>,