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/DataBox/Prefixes.hpp"
9 : #include "DataStructures/Tensor/EagerMath/Magnitude.hpp"
10 : #include "Elliptic/BoundaryConditions/BoundaryCondition.hpp"
11 : #include "Elliptic/Protocols/FirstOrderSystem.hpp"
12 : #include "Elliptic/Systems/Xcts/FluxesAndSources.hpp"
13 : #include "Elliptic/Systems/Xcts/Geometry.hpp"
14 : #include "Elliptic/Systems/Xcts/Tags.hpp"
15 : #include "NumericalAlgorithms/LinearOperators/PartialDerivatives.hpp"
16 : #include "PointwiseFunctions/GeneralRelativity/Tags.hpp"
17 : #include "PointwiseFunctions/GeneralRelativity/Tags/Conformal.hpp"
18 : #include "Utilities/ProtocolHelpers.hpp"
19 : #include "Utilities/TMPL.hpp"
20 :
21 : namespace Xcts {
22 :
23 : /*!
24 : * \brief The Extended Conformal Thin Sandwich (XCTS) decomposition of the
25 : * Einstein constraint equations, formulated as a set of coupled first-order
26 : * partial differential equations
27 : *
28 : * See \ref Xcts for details on the XCTS equations.
29 : * The system can be formulated in terms of these fluxes and sources (see
30 : * `elliptic::protocols::FirstOrderSystem`):
31 : *
32 : * \f{align}
33 : * F^i_\psi ={} &\bar{\gamma}^{ij} \partial_j \psi \\
34 : * S_\psi ={} &-\bar{\Gamma}^i_{ij} F^j_\psi
35 : * + \frac{1}{8}\psi\bar{R} + \frac{1}{12}\psi^5 K^2
36 : * - \frac{1}{8}\psi^{-7}\bar{A}^2 - 2\pi\psi^5\rho
37 : * \f}
38 : *
39 : * for the Hamiltonian constraint,
40 : *
41 : * \f{align}
42 : * F^i_{\alpha\psi} ={} &\bar{\gamma}^{ij} \partial_j \alpha\psi \\
43 : * S_{\alpha\psi} ={} &-\bar{\Gamma}^i_{ij} F^j_{\alpha\psi}
44 : * + \alpha\psi \left(\frac{7}{8}\psi^{-8} \bar{A}^2
45 : * + \frac{5}{12} \psi^4 K^2 + \frac{1}{8}\bar{R}
46 : * + 2\pi\psi^4\left(\rho + 2S\right) \right) \\
47 : * &- \psi^5\partial_t K + \psi^5\left(\beta^i\bar{D}_i K
48 : * + \beta_\mathrm{background}^i\bar{D}_i K\right)
49 : * \f}
50 : *
51 : * for the lapse equation, and
52 : *
53 : * \f{align}
54 : * F^{ij}_\beta ={} &\left(\bar{L}\beta\right)^{ij} = \bar{\nabla}^i \beta^j
55 : * + \bar{\nabla}^j \beta^i
56 : * - \frac{2}{3} \bar{\gamma}^{ij} \bar{\nabla}_k \beta^k \\
57 : * S^i_\beta ={} &-\bar{\Gamma}^j_{jk} F^{ik}_\beta
58 : * - \bar{\Gamma}^i_{jk} F^{jk}_\beta
59 : * + \left(F^{ij}_\beta
60 : * + \left(\bar{L}\beta_\mathrm{background}\right)^{ij} - \bar{u}^{ij}\right)
61 : * \bar{\gamma}_{jk} \left(\frac{F^k_{\alpha\psi}}{\alpha\psi}
62 : * - 7 \frac{F^k_\psi}{\psi}\right) \\
63 : * &- \bar{D}_j\left(\left(\bar{L}\beta_\mathrm{background}\right)^{ij}
64 : * - \bar{u}^{ij}\right)
65 : * + \frac{4}{3}\frac{\alpha\psi}{\psi}\bar{D}^i K
66 : * + 16\pi\left(\alpha\psi\right)\psi^3 S^i
67 : * \f}
68 : *
69 : * for the momentum constraint, with
70 : *
71 : * \f{align}
72 : * \bar{A}^{ij} ={} &\frac{\psi^7}{2\alpha\psi}\left(
73 : * \left(\bar{L}\beta\right)^{ij} +
74 : * \left(\bar{L}\beta_\mathrm{background}\right)^{ij} - \bar{u}^{ij} \right)
75 : * \f}
76 : *
77 : * and all \f$f_\alpha=0\f$.
78 : *
79 : * Note that the symbol \f$\beta\f$ in the equations above means
80 : * \f$\beta_\mathrm{excess}\f$. The full shift is \f$\beta_\mathrm{excess} +
81 : * \beta_\mathrm{background}\f$. See `Xcts::Tags::ShiftBackground` and
82 : * `Xcts::Tags::ShiftExcess` for details on this split. Also note that the
83 : * background shift is degenerate with \f$\bar{u}\f$ so we treat the quantity
84 : * \f$\left(\bar{L}\beta_\mathrm{background}\right)^{ij} - \bar{u}^{ij}\f$ as a
85 : * single background field (see
86 : * `Xcts::Tags::LongitudinalShiftBackgroundMinusDtConformalMetric`). The
87 : * covariant divergence of this quantity w.r.t. the conformal metric is also a
88 : * background field.
89 : *
90 : * \par Solving a subset of equations:
91 : * This system allows you to select a subset of `Xcts::Equations` so you don't
92 : * have to solve for all variables if some are analytically known. Specify the
93 : * set of enabled equations as the first template parameter. The set of required
94 : * background fields depends on your choice of equations.
95 : *
96 : * \par Conformal background geometry:
97 : * The equations simplify significantly if the conformal metric is flat
98 : * ("conformal flatness") and in Cartesian coordinates. In this case you can
99 : * specify `Xcts::Geometry::FlatCartesian` as the second template parameter so
100 : * computations are optimized for a flat background geometry and you don't have
101 : * to supply geometric background fields. Else, specify
102 : * `Xcts::Geometry::Curved`.
103 : *
104 : * \par Conformal matter scale:
105 : * The matter source terms in the XCTS equations have the known defect that they
106 : * can spoil uniqueness of the solutions. See e.g. \cite Baumgarte2006ug for a
107 : * detailed study. To cure this defect one can conformally re-scale the matter
108 : * source terms as \f$\bar{\rho}=\psi^n\rho\f$, \f$\bar{S}=\psi^n S\f$ and
109 : * \f$\bar{S^i}=\psi^n S^i\f$ and treat the re-scaled fields as
110 : * freely-specifyable background data for the XCTS equations. You can select the
111 : * `ConformalMatterScale` \f$n\f$ as the third template parameter. Common
112 : * choices are \f$n=0\f$ for vacuum systems where the matter sources are
113 : * irrelevant, \f$n=6\f$ as suggested in \cite Foucart2008qt or \f$n=8\f$ as
114 : * suggested in \cite Baumgarte2006ug.
115 : */
116 : template <Equations EnabledEquations, Geometry ConformalGeometry,
117 : int ConformalMatterScale>
118 1 : struct FirstOrderSystem
119 : : tt::ConformsTo<elliptic::protocols::FirstOrderSystem> {
120 0 : static constexpr Equations enabled_equations = EnabledEquations;
121 0 : static constexpr Geometry conformal_geometry = ConformalGeometry;
122 0 : static constexpr int conformal_matter_scale = ConformalMatterScale;
123 0 : static constexpr size_t volume_dim = 3;
124 :
125 0 : using primal_fields = tmpl::flatten<tmpl::list<
126 : Tags::ConformalFactorMinusOne<DataVector>,
127 : tmpl::conditional_t<
128 : EnabledEquations == Equations::HamiltonianAndLapse or
129 : EnabledEquations == Equations::HamiltonianLapseAndShift,
130 : Tags::LapseTimesConformalFactorMinusOne<DataVector>, tmpl::list<>>,
131 : tmpl::conditional_t<
132 : EnabledEquations == Equations::HamiltonianLapseAndShift,
133 : Tags::ShiftExcess<DataVector, 3, Frame::Inertial>, tmpl::list<>>>>;
134 :
135 : // As fluxes we use the gradients with raised indices for the Hamiltonian and
136 : // lapse equation, and the longitudinal shift excess for the momentum
137 : // constraint. The gradient fluxes don't have symmetries and no particular
138 : // meaning so we use the standard `Flux` tags, but for the symmetric
139 : // longitudinal shift we use the corresponding symmetric tag.
140 0 : using primal_fluxes = tmpl::flatten<tmpl::list<
141 : ::Tags::Flux<Tags::ConformalFactorMinusOne<DataVector>, tmpl::size_t<3>,
142 : Frame::Inertial>,
143 : tmpl::conditional_t<
144 : EnabledEquations == Equations::HamiltonianAndLapse or
145 : EnabledEquations == Equations::HamiltonianLapseAndShift,
146 : ::Tags::Flux<Tags::LapseTimesConformalFactorMinusOne<DataVector>,
147 : tmpl::size_t<3>, Frame::Inertial>,
148 : tmpl::list<>>,
149 : tmpl::conditional_t<
150 : EnabledEquations == Equations::HamiltonianLapseAndShift,
151 : Tags::LongitudinalShiftExcess<DataVector, 3, Frame::Inertial>,
152 : tmpl::list<>>>>;
153 :
154 0 : using background_fields = tmpl::flatten<tmpl::list<
155 : // Quantities for Hamiltonian constraint
156 : gr::Tags::Conformal<gr::Tags::EnergyDensity<DataVector>,
157 : ConformalMatterScale>,
158 : gr::Tags::TraceExtrinsicCurvature<DataVector>,
159 : tmpl::conditional_t<
160 : ConformalGeometry == Geometry::Curved,
161 : tmpl::list<
162 : Tags::InverseConformalMetric<DataVector, 3, Frame::Inertial>,
163 : Tags::ConformalRicciTensor<DataVector, 3, Frame::Inertial>,
164 : Tags::ConformalRicciScalar<DataVector>,
165 : Tags::ConformalChristoffelContracted<DataVector, 3,
166 : Frame::Inertial>,
167 : ::Tags::deriv<
168 : Tags::ConformalMetric<DataVector, 3, Frame::Inertial>,
169 : tmpl::size_t<3>, Frame::Inertial>>,
170 : tmpl::list<>>,
171 : tmpl::conditional_t<
172 : EnabledEquations ==
173 : Equations::Hamiltonian,
174 : Tags::LongitudinalShiftMinusDtConformalMetricOverLapseSquare<
175 : DataVector>,
176 : tmpl::list<>>,
177 : // Additional quantities for lapse equation
178 : tmpl::conditional_t<
179 : EnabledEquations == Equations::HamiltonianAndLapse or
180 : EnabledEquations ==
181 : Equations::HamiltonianLapseAndShift,
182 : tmpl::list<gr::Tags::Conformal<gr::Tags::StressTrace<DataVector>,
183 : ConformalMatterScale>,
184 : ::Tags::dt<gr::Tags::TraceExtrinsicCurvature<DataVector>>>,
185 : tmpl::list<>>,
186 : tmpl::conditional_t<
187 : EnabledEquations ==
188 : Equations::HamiltonianAndLapse,
189 : tmpl::list<
190 : Tags::LongitudinalShiftMinusDtConformalMetricSquare<DataVector>,
191 : Tags::ShiftDotDerivExtrinsicCurvatureTrace<DataVector>>,
192 : tmpl::list<>>,
193 : // Additional quantities for momentum constraint
194 : tmpl::conditional_t<
195 : EnabledEquations ==
196 : Equations::HamiltonianLapseAndShift,
197 : tmpl::list<
198 : gr::Tags::Conformal<gr::Tags::MomentumDensity<DataVector, 3>,
199 : ConformalMatterScale>,
200 : ::Tags::deriv<gr::Tags::TraceExtrinsicCurvature<DataVector>,
201 : tmpl::size_t<3>, Frame::Inertial>,
202 : Tags::ShiftBackground<DataVector, 3, Frame::Inertial>,
203 : // The deriv(ShiftBackground) is only needed for some observables,
204 : // not to evaluate the equations
205 : ::Tags::deriv<
206 : Tags::ShiftBackground<DataVector, 3, Frame::Inertial>,
207 : tmpl::size_t<3>, Frame::Inertial>,
208 : Tags::LongitudinalShiftBackgroundMinusDtConformalMetric<
209 : DataVector, 3, Frame::Inertial>,
210 : // Note that this is the plain divergence, i.e. with no
211 : // Christoffel symbol terms added
212 : ::Tags::div<
213 : Tags::LongitudinalShiftBackgroundMinusDtConformalMetric<
214 : DataVector, 3, Frame::Inertial>>,
215 : tmpl::conditional_t<
216 : ConformalGeometry == Geometry::Curved,
217 : tmpl::list<
218 : Tags::ConformalMetric<DataVector, 3, Frame::Inertial>,
219 : Tags::ConformalChristoffelFirstKind<DataVector, 3,
220 : Frame::Inertial>,
221 : Tags::ConformalChristoffelSecondKind<DataVector, 3,
222 : Frame::Inertial>>,
223 : tmpl::list<>>>,
224 : tmpl::list<>>>>;
225 0 : using inv_metric_tag = tmpl::conditional_t<
226 : ConformalGeometry == Geometry::FlatCartesian, void,
227 : Tags::InverseConformalMetric<DataVector, 3, Frame::Inertial>>;
228 :
229 0 : using fluxes_computer = Fluxes<EnabledEquations, ConformalGeometry>;
230 0 : using sources_computer =
231 : Sources<EnabledEquations, ConformalGeometry, ConformalMatterScale>;
232 0 : using sources_computer_linearized =
233 : LinearizedSources<EnabledEquations, ConformalGeometry,
234 : ConformalMatterScale>;
235 :
236 0 : using boundary_conditions_base =
237 : elliptic::BoundaryConditions::BoundaryCondition<3>;
238 0 : using modify_boundary_data = void;
239 : };
240 :
241 : } // namespace Xcts
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