Poisson::FirstOrderSystem< Dim, BackgroundGeometry, DataType > Struct Template Reference

The Poisson equation formulated as a set of coupled first-order PDEs. More...

#include <FirstOrderSystem.hpp>

Public Types
using	primal_fields = tmpl::list<Tags::Field<DataType>>
using	primal_fluxes
using	background_fields
using	inv_metric_tag
using	fluxes_computer = Fluxes<Dim, BackgroundGeometry, DataType>
using	sources_computer
using	boundary_conditions_base
using	modify_boundary_data = void

Static Public Attributes
static constexpr size_t	volume_dim = Dim

Detailed Description

template<size_t Dim, Geometry BackgroundGeometry, typename DataType = DataVector>
struct Poisson::FirstOrderSystem< Dim, BackgroundGeometry, DataType >

The Poisson equation formulated as a set of coupled first-order PDEs.

Details

This system formulates the Poisson equation \(-\Delta_\gamma u(x) = f(x)\) on a background metric \(\gamma_{ij}\) as the set of coupled first-order PDEs

\begin{align*}-\partial_i v^i - \Gamma^i_{ij} v^j = f(x) \\ v^i = \gamma^{ij} \partial_j u(x) \end{align*}

where \(\Gamma^i_{jk}=\frac{1}{2}\gamma^{il}\left(\partial_j\gamma_{kl} +\partial_k\gamma_{jl}-\partial_l\gamma_{jk}\right)\) are the Christoffel symbols of the second kind of the background metric \(\gamma_{ij}\). The background metric \(\gamma_{ij}\) and the Christoffel symbols derived from it are assumed to be independent of the variables \(u\), i.e. constant throughout an iterative elliptic solve.

The system can be formulated in terms of these fluxes and sources (see elliptic::protocols::FirstOrderSystem):

\begin{align*}F^i &= v^i = \gamma^{ij} \partial_j u \\ S &= -\Gamma^i_{ij} v^j \\ f &= f(x) \text{.} \end{align*}

The fluxes and sources simplify significantly when the background metric is flat and we employ Cartesian coordinates so \(\gamma_{ij} = \delta_{ij}\) and \(\Gamma^i_{jk} = 0\). Set the template parameter BackgroundGeometry to Poisson::Geometry::FlatCartesian to specialize the system for this case. Set it to Poisson::Geometry::Curved for the general case.

Complex Poisson equation

This system can also be used to solve the complex Poisson equation where \(u(x)\) and \(f(x)\) are complex-valued, by setting the DataType template parameter to ComplexDataVector. Note that the real and imaginary sectors of the equations decouple, so they are essentially two independent Poisson equations. This is useful for testing the elliptic solver with complex-valued fields, and also as building blocks for other Poisson-like systems of equations that have additional complex-valued source terms.

Member Typedef Documentation

background_fields

template<size_t Dim, Geometry BackgroundGeometry, typename DataType = DataVector>

using Poisson::FirstOrderSystem< Dim, BackgroundGeometry, DataType >::background_fields

Initial value:

 tmpl::conditional_t<
      BackgroundGeometry == Geometry::FlatCartesian, tmpl::list<>,
      tmpl::list<
          gr::Tags::InverseSpatialMetric<DataVector, Dim>,
          gr::Tags::SpatialChristoffelSecondKindContracted<DataVector, Dim>>>

boundary_conditions_base

template<size_t Dim, Geometry BackgroundGeometry, typename DataType = DataVector>

using Poisson::FirstOrderSystem< Dim, BackgroundGeometry, DataType >::boundary_conditions_base

Initial value:

 
      elliptic::BoundaryConditions::BoundaryCondition<Dim>

inv_metric_tag

template<size_t Dim, Geometry BackgroundGeometry, typename DataType = DataVector>

using Poisson::FirstOrderSystem< Dim, BackgroundGeometry, DataType >::inv_metric_tag

Initial value:

 
      tmpl::conditional_t<BackgroundGeometry == Geometry::FlatCartesian, void,
                          gr::Tags::InverseSpatialMetric<DataVector, Dim>>

primal_fluxes

template<size_t Dim, Geometry BackgroundGeometry, typename DataType = DataVector>

using Poisson::FirstOrderSystem< Dim, BackgroundGeometry, DataType >::primal_fluxes

Initial value:

 tmpl::list<
      ::Tags::Flux<Tags::Field<DataType>, tmpl::size_t<Dim>, Frame::Inertial>>

sources_computer

template<size_t Dim, Geometry BackgroundGeometry, typename DataType = DataVector>

using Poisson::FirstOrderSystem< Dim, BackgroundGeometry, DataType >::sources_computer

Initial value:

 
      tmpl::conditional_t<BackgroundGeometry == Geometry::FlatCartesian, void,
                          Sources<Dim, BackgroundGeometry, DataType>>

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

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