Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale > Struct Template Reference

The Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein constraint equations, formulated as a set of coupled first-order partial differential equations. More...

#include <FirstOrderSystem.hpp>

Public Types
using	primal_fields = implementation defined
using	primal_fluxes = implementation defined
using	background_fields = implementation defined
using	inv_metric_tag = implementation defined
using	fluxes_computer = Fluxes< EnabledEquations, ConformalGeometry >
using	sources_computer = Sources< EnabledEquations, ConformalGeometry, ConformalMatterScale >
using	sources_computer_linearized = LinearizedSources< EnabledEquations, ConformalGeometry, ConformalMatterScale >
using	boundary_conditions_base = elliptic::BoundaryConditions::BoundaryCondition< 3 >
using	modify_boundary_data = void

Static Public Attributes
static constexpr Equations	enabled_equations = EnabledEquations
static constexpr Geometry	conformal_geometry = ConformalGeometry
static constexpr int	conformal_matter_scale = ConformalMatterScale
static constexpr size_t	volume_dim = 3

Detailed Description

template<Equations EnabledEquations, Geometry ConformalGeometry, int ConformalMatterScale>
struct Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale >

The Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein constraint equations, formulated as a set of coupled first-order partial differential equations.

See Xcts for details on the XCTS equations. The system can be formulated in terms of these fluxes and sources (see elliptic::protocols::FirstOrderSystem):

$$\begin{array}{}\text{(1)}& F_{\psi }^i=& {\overline{\gamma }}^{ij}{\partial }_j\psi \text{(2)}& S_{\psi }=& -{\overline{\mathrm{\Gamma }}}_{ij}^i{F}_{\psi }^j+\frac{1}{8}\psi \overline{R}+\frac{1}{12}{\psi }^5{K}^2-\frac{1}{8}{\psi }^{-7}{\overline{A}}^2-2\pi {\psi }^5\rho \end{array}$$

for the Hamiltonian constraint,

$$\begin{array}{}\text{(3)}& F_{\alpha \psi }^i=& {\overline{\gamma }}^{ij}{\partial }_j\alpha \psi \text{(4)}& S_{\alpha \psi }=& -{\overline{\mathrm{\Gamma }}}_{ij}^i{F}_{\alpha \psi }^j+\alpha \psi (\frac{7}{8}{\psi }^{-8}{\overline{A}}^2+\frac{5}{12}{\psi }^4{K}^2+\frac{1}{8}\overline{R}+2\pi {\psi }^4(\rho +2S))\text{(5)}& & -{\psi }^5{\partial }_{t}K+{\psi }^5({\beta }^i{\overline{D}}_{i}K+{\beta }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}^i{\overline{D}}_{i}K)\end{array}$$

for the lapse equation, and

$$\begin{array}{}\text{(6)}& F_{\beta }^{ij}=& {\left(\overline{L}\beta \right)}^{ij}={\overline{\mathrm{\nabla }}}^i{\beta }^j+{\overline{\mathrm{\nabla }}}^j{\beta }^i-\frac{2}{3}{\overline{\gamma }}^{ij}{\overline{\mathrm{\nabla }}}_k{\beta }^k\text{(7)}& S_{\beta }^i=& -{\overline{\mathrm{\Gamma }}}_{jk}^j{F}_{\beta }^{ik}-{\overline{\mathrm{\Gamma }}}_{jk}^i{F}_{\beta }^{jk}+(F_{\beta }^{ij}+{\left(\overline{L}{\beta }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}\right)}^{ij}-{\overline{u}}^{ij}){\overline{\gamma }}_{jk}(\frac{F_{\alpha \psi }^k}{\alpha \psi }-7\frac{F_{\psi }^k}{\psi })\text{(8)}& & -{\overline{D}}_j({\left(\overline{L}{\beta }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}\right)}^{ij}-{\overline{u}}^{ij})+\frac{4}{3}\frac{\alpha \psi }{\psi }{\overline{D}}^{i}K+16\pi \left(\alpha \psi \right){\psi }^3{S}^i\end{array}$$

for the momentum constraint, with

$$\begin{array}{}\text{(9)}& {\overline{A}}^{ij}=& \frac{{\psi }^7}{2\alpha \psi }({\left(\overline{L}\beta \right)}^{ij}+{\left(\overline{L}{\beta }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}\right)}^{ij}-{\overline{u}}^{ij})\end{array}$$

and all $f_{\alpha }=0$.

Note that the symbol $\beta$ in the equations above means ${\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}$. The full shift is ${\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}+{\beta }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}$. See Xcts::Tags::ShiftBackground and Xcts::Tags::ShiftExcess for details on this split. Also note that the background shift is degenerate with $\overline{u}$ so we treat the quantity ${\left(\overline{L}{\beta }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}\right)}^{ij}-{\overline{u}}^{ij}$ as a single background field (see Xcts::Tags::LongitudinalShiftBackgroundMinusDtConformalMetric). The covariant divergence of this quantity w.r.t. the conformal metric is also a background field.

Solving a subset of equations:

This system allows you to select a subset of Xcts::Equations so you don't have to solve for all variables if some are analytically known. Specify the set of enabled equations as the first template parameter. The set of required background fields depends on your choice of equations.

Conformal background geometry:

The equations simplify significantly if the conformal metric is flat ("conformal flatness") and in Cartesian coordinates. In this case you can specify Xcts::Geometry::FlatCartesian as the second template parameter so computations are optimized for a flat background geometry and you don't have to supply geometric background fields. Else, specify Xcts::Geometry::Curved.

Conformal matter scale:

The matter source terms in the XCTS equations have the known defect that they can spoil uniqueness of the solutions. See e.g. [16] for a detailed study. To cure this defect one can conformally re-scale the matter source terms as $\overline{\rho }={\psi }^n\rho$, $\overline{S}={\psi }^{n}S$ and $\overline{S^i}={\psi }^n{S}^i$ and treat the re-scaled fields as freely-specifyable background data for the XCTS equations. You can select the ConformalMatterScale $n$ as the third template parameter. Common choices are $n=0$ for vacuum systems where the matter sources are irrelevant, $n=6$ as suggested in [74] or $n=8$ as suggested in [16].

---

The documentation for this struct was generated from the following file:

• src/Elliptic/Systems/Xcts/FirstOrderSystem.hpp