Merge pull request #5918 from nilsvu/xcts_christoffel

XCTS: output spatial Christoffels
sxs-collaboration · Apr 23, 2024 · 9b01d30 · 9b01d30
2 parents 7bf9ea3 + 665ec43
commit 9b01d30
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diff --git a/src/Elliptic/Executables/Xcts/SolveXcts.hpp b/src/Elliptic/Executables/Xcts/SolveXcts.hpp
@@ -72,6 +72,8 @@ struct Metavariables {
                  gr::Tags::HamiltonianConstraint<DataVector>,
                  gr::Tags::MomentumConstraint<DataVector, 3>,
                  gr::Tags::SpatialMetric<DataVector, 3>,
+                 gr::Tags::InverseSpatialMetric<DataVector, 3>,
+                 gr::Tags::SpatialChristoffelSecondKind<DataVector, 3>,
                  gr::Tags::Lapse<DataVector>, gr::Tags::Shift<DataVector, 3>,
                  gr::Tags::ExtrinsicCurvature<DataVector, 3>>>;
   using hydro_quantities_compute = Xcts::Tags::HydroQuantitiesCompute<

diff --git a/src/PointwiseFunctions/Xcts/SpacetimeQuantities.cpp b/src/PointwiseFunctions/Xcts/SpacetimeQuantities.cpp
@@ -72,6 +72,39 @@ void SpacetimeQuantitiesComputer::operator()(
                      inv_jacobian);
 }
 
+void SpacetimeQuantitiesComputer::operator()(
+    const gsl::not_null<tnsr::Ijj<DataVector, 3>*>
+        spatial_christoffel_second_kind,
+    const gsl::not_null<Cache*> cache,
+    gr::Tags::SpatialChristoffelSecondKind<DataVector, 3> /*meta*/) const {
+  // Eq. (3.7) in Baumgarte/Shapiro
+  const auto& conformal_factor =
+      cache->get_var(*this, Tags::ConformalFactor<DataVector>{});
+  const auto& deriv_conformal_factor =
+      cache->get_var(*this, ::Tags::deriv<Tags::ConformalFactor<DataVector>,
+                                          tmpl::size_t<3>, Frame::Inertial>{});
+  *spatial_christoffel_second_kind = conformal_christoffel_second_kind;
+  for (size_t i = 0; i < 3; ++i) {
+    for (size_t j = 0; j < 3; ++j) {
+      for (size_t k = 0; k <= j; ++k) {
+        if (i == j) {
+          spatial_christoffel_second_kind->get(i, j, k) +=
+              2. * deriv_conformal_factor.get(k) / get(conformal_factor);
+        }
+        if (i == k) {
+          spatial_christoffel_second_kind->get(i, j, k) +=
+              2. * deriv_conformal_factor.get(j) / get(conformal_factor);
+        }
+        for (size_t l = 0; l < 3; ++l) {
+          spatial_christoffel_second_kind->get(i, j, k) -=
+              2. * conformal_metric.get(j, k) * inv_conformal_metric.get(i, l) *
+              deriv_conformal_factor.get(l) / get(conformal_factor);
+        }
+      }
+    }
+  }
+}
+
 void SpacetimeQuantitiesComputer::operator()(
     const gsl::not_null<Scalar<DataVector>*>
         conformal_laplacian_of_conformal_factor,

diff --git a/src/PointwiseFunctions/Xcts/SpacetimeQuantities.hpp b/src/PointwiseFunctions/Xcts/SpacetimeQuantities.hpp
@@ -53,6 +53,7 @@ using SpacetimeQuantities = CachedTempBuffer<
     gr::Tags::SpatialMetric<DataVector, 3>,
     gr::Tags::InverseSpatialMetric<DataVector, 3>, gr::Tags::Lapse<DataVector>,
     gr::Tags::Shift<DataVector, 3>, gr::Tags::ExtrinsicCurvature<DataVector, 3>,
+    gr::Tags::SpatialChristoffelSecondKind<DataVector, 3>,
     // Constraints
     gr::Tags::HamiltonianConstraint<DataVector>,
     gr::Tags::MomentumConstraint<DataVector, 3>>;
@@ -80,6 +81,10 @@ struct SpacetimeQuantitiesComputer {
       gsl::not_null<Cache*> cache,
       ::Tags::deriv<Tags::ConformalFactor<DataVector>, tmpl::size_t<3>,
                     Frame::Inertial> /*meta*/) const;
+  void operator()(
+      gsl::not_null<tnsr::Ijj<DataVector, 3>*> spatial_christoffel_second_kind,
+      gsl::not_null<Cache*> cache,
+      gr::Tags::SpatialChristoffelSecondKind<DataVector, 3> /*meta*/) const;
   void operator()(
       gsl::not_null<Scalar<DataVector>*>
           conformal_laplacian_of_conformal_factor,

diff --git a/support/Pipelines/Bbh/InitialData.yaml b/support/Pipelines/Bbh/InitialData.yaml
@@ -218,6 +218,8 @@ EventsAndTriggers:
             - Shift
             - ShiftExcess
             - SpatialMetric
+            - InverseSpatialMetric
+            - SpatialChristoffelSecondKind
             - ExtrinsicCurvature
             - RadiallyCompressedCoordinates
           InterpolateToMesh: None

diff --git a/tests/Unit/PointwiseFunctions/Xcts/SpacetimeQuantities.py b/tests/Unit/PointwiseFunctions/Xcts/SpacetimeQuantities.py
@@ -13,6 +13,29 @@ def inv_spatial_metric(conformal_factor, inv_conformal_metric):
     return conformal_factor ** (-4) * inv_conformal_metric
 
 
+def spatial_christoffel_second_kind(
+    conformal_factor,
+    deriv_conformal_factor,
+    conformal_metric,
+    inv_conformal_metric,
+    conformal_christoffel_second_kind,
+):
+    # Eq. (3.7) in Baumgarte/Shapiro
+    krond = np.eye(3)
+    return conformal_christoffel_second_kind + 2.0 * (
+        np.einsum("ij,k->ijk", krond, deriv_conformal_factor / conformal_factor)
+        + np.einsum(
+            "ik,j->ijk", krond, deriv_conformal_factor / conformal_factor
+        )
+        - np.einsum(
+            "jk,il,l->ijk",
+            conformal_metric,
+            inv_conformal_metric,
+            deriv_conformal_factor / conformal_factor,
+        )
+    )
+
+
 def lapse(conformal_factor, lapse_times_conformal_factor):
     return lapse_times_conformal_factor / conformal_factor
 

diff --git a/tests/Unit/PointwiseFunctions/Xcts/Test_SpacetimeQuantities.cpp b/tests/Unit/PointwiseFunctions/Xcts/Test_SpacetimeQuantities.cpp
@@ -126,6 +126,13 @@ SPECTRE_TEST_CASE("Unit.PointwiseFunctions.Xcts.SpacetimeQuantities",
   check_with_python(get<gr::Tags::InverseSpatialMetric<DataVector, 3>>(vars),
                     "inv_spatial_metric", conformal_factor,
                     inv_conformal_metric);
+  check_with_python(
+      get<gr::Tags::SpatialChristoffelSecondKind<DataVector, 3>>(vars),
+      "spatial_christoffel_second_kind", conformal_factor,
+      get<::Tags::deriv<Tags::ConformalFactor<DataVector>, tmpl::size_t<3>,
+                        Frame::Inertial>>(vars),
+      conformal_metric, inv_conformal_metric,
+      conformal_christoffel_second_kind);
   check_with_python(get<gr::Tags::Lapse<DataVector>>(vars), "lapse",
                     conformal_factor, lapse_times_conformal_factor);
   check_with_python(get<gr::Tags::Shift<DataVector, 3>>(vars), "shift",