time_dependent_coordinate

Module that defines the interface for a class that provides the routines necessary for the use of time dependent coordinates. This is currently tied to supporting the scalar wave equation on a Schwarzschild background provided in scalar_schw.

Starting from Tortoise coordinates $(t,r_*)$ the equations are transformed to time dependent coordinate $(\lambda,\xi)$ as descibed in Field, Hesthaven & Lau, Class. Quant. Grav. 26 (2009) 165010.

The implementation is provided in submodule_time_dependent_coordinate_implementation.f90.

Uses

- kinds
- grid_function
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Used by

- Descendants:
- time_dependent_coordinate_implementation
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Nodes of different colours represent the following:

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Interfaces

tdc_init tdc_set_coefficients tdc_tdc_to_tortoise_cvec tdc_tdc_to_tortoise_cscal tdc_tdc_to_tortoise_rscal tdc_tortoise_to_tdc_cvecb tdc_tortoise_to_tdc_cscalb tdc_tortoise_to_tdc_cscal

Derived Types

tdc

Interfaces

interface

public module subroutine tdc_init(this)

The interface for the initialization routine.

Arguments

Type Intent Optional Attributes Name

class(tdc), intent(inout) :: this
This time dependent coordinate object is being initialized.

Type	Intent	Optional	Attributes		Name
class(tdc),	intent(inout)			::	this	This time dependent coordinate object is being initialized.

interface

public module subroutine tdc_set_coefficients(this, coeffs, lcoeffs, lambda, s, sinv, rho, rstar, rschw, ll)

The interface for the routine that sets the coefficient for the scalar wave equation in Schwarzschild spacetime.

Arguments

Type	Intent	Attributes		Name
class(tdc),	intent(inout)		::	this	The routine is called on this time dependent coordinate object.
type(rgf),	intent(inout),	dimension(:)	::	coeffs	A 1d array of real grid functions containing the wave equation coefficients (should be eq_coeffs). On output it will contain $(c_{\xi\xi}, c_{\lambda\xi}, c_{\xi}, c_{\lambda})$ .
type(rgf),	intent(inout),	dimension(:)	::	lcoeffs	A 1d array of real grid functions containing the $\ell$ dependent coefficients (should be eq_lcoeffs). On output it will contain the potential for all modes.
type(rgfb),	intent(inout),	dimension(:)	::	lambda	A 1d array of real boundary grid functions containing the characteristic speeds at the boundary of the elements (should be eq_lambda). On output it will be updated.
type(rgfb),	intent(inout),	dimension(:,:)	::	s	A 2d array of real boundary grid functions containing the matrix used to convert from characteristic to evolved variables (should be eq_s). On output it will be updated.
type(rgfb),	intent(inout),	dimension(:,:)	::	sinv	A 2d array of real boundary grid functions containing the matrix used to convert from evolved to characteristic variables (should be eq_sinv). On output it will be updated.
type(rgf),	intent(in)		::	rho	A real grid function containing the computational radial coordinate, $\rho$ .
type(rgf),	intent(inout)		::	rstar	A real grid function containing the tortoise radial coordinate, $r_*$ . On output this will be updated.
type(rgf),	intent(inout)		::	rschw	A real grid function containing the Schwarzschild radial coodrinate, $r_{\mathrm{schw}}$ . On output this will be updated.
integer(kind=c_int),	intent(in),	dimension(:)	::	ll	A 1d array of c_int containing the $\ell$ -values of all the modes.

interface

public module subroutine tdc_tdc_to_tortoise_cvec(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

Routine to transform from time dependent $(\lambda,\xi)$ to tortoise coordinates $(t,r_*)$ for complex vector input at element boundaries.

Arguments

Type	Intent	Attributes		Name
class(tdc),	intent(inout)		::	this	The routine is called on this time dependent coordinate object.
integer(kind=ip),	intent(in)		::	ielem	The index of the element that contains the coordinate transformation information.
integer(kind=ip),	intent(in)		::	dir	The index of the boundary within the element that contains the coordinate transformation. Left boundary: 1, right boundary: 2.
complex(kind=wp),	intent(in),	dimension(:)	::	dudlambda	The vector of $\frac{\partial\Psi}{\partial\lambda}$ values to transform.
complex(kind=wp),	intent(in),	dimension(:)	::	dudxi	The vector of $\frac{\partial\Psi}{\partial\xi}$ values to transform.
complex(kind=wp),	intent(out),	dimension(:)	::	dudt	On output contains the $\frac{\partial\Psi}{\partial t}$ vector.
complex(kind=wp),	intent(out),	dimension(:)	::	dudrstar	On output contains the $\frac{\partial\Psi}{\partial r_*}$ vector.

interface

public module subroutine tdc_tdc_to_tortoise_cscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

Routine to transform from time dependent $(\lambda,\xi)$ to tortoise coordinates $(t,r_*)$ for complex scalar input at element boundaries.

Arguments

Type	Intent		Name
class(tdc),	intent(inout)	::	this	The routine is called on this time dependent coordinate object.
integer(kind=ip),	intent(in)	::	ielem	The index of the element that contains the coordinate transformation information.
integer(kind=ip),	intent(in)	::	dir	The index of the boundary within the element that contains the coordinate transformation. Left boundary: 1, right boundary: 2.
complex(kind=wp),	intent(in)	::	dudlambda	The value of $\frac{\partial\Psi}{\partial\lambda}$ to transform.
complex(kind=wp),	intent(in)	::	dudxi	The value of $\frac{\partial\Psi}{\partial\xi}$ to transform.
complex(kind=wp),	intent(out)	::	dudt	On output contains the $\frac{\partial\Psi}{\partial t}$ value.
complex(kind=wp),	intent(out)	::	dudrstar	On output contains the $\frac{\partial\Psi}{\partial r_*}$ value.

interface

public module subroutine tdc_tdc_to_tortoise_rscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

Routine to transform from time dependent $(\lambda,\xi)$ to tortoise coordinates $(t,r_*)$ for real scalar input at element boundaries.

Arguments

Type	Intent		Name
class(tdc),	intent(inout)	::	this	The routine is called on this time dependent coordinate object.
integer(kind=ip),	intent(in)	::	ielem	The index of the element that contains the coordinate transformation information.
integer(kind=ip),	intent(in)	::	dir	The index of the boundary within the element that contains the coordinate transformation. Left boundary: 1, right boundary: 2.
real(kind=wp),	intent(in)	::	dudlambda	The value of $\frac{\partial\Psi}{\partial\lambda}$ to transform.
real(kind=wp),	intent(in)	::	dudxi	The value of $\frac{\partial\Psi}{\partial\xi}$ to transform.
real(kind=wp),	intent(out)	::	dudt	On output contains the $\frac{\partial\Psi}{\partial t}$ value.
real(kind=wp),	intent(out)	::	dudrstar	On output contains the $\frac{\partial\Psi}{\partial r_*}$ value.

interface

public module subroutine tdc_tortoise_to_tdc_cvecb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi)

Routine to transform from tortoise coordinates $(t,r_*)$ to time dependent $(\lambda,\xi)$ for complex vector input at element boundaries.

Arguments

Type	Intent	Attributes		Name
class(tdc),	intent(inout)		::	this	The routine is called on this time dependent coordinate object.
integer(kind=ip),	intent(in)		::	ielem	The index of the element that contains the coordinate transformation information.
integer(kind=ip),	intent(in)		::	dir	The index of the boundary within the element that contains the coordinate transformation. Left boundary: 1, right boundary: 2.
complex(kind=wp),	intent(in),	dimension(:)	::	dudt	The vector of $\frac{\partial\Psi}{\partial t}$ to transform.
complex(kind=wp),	intent(in),	dimension(:)	::	dudrstar	The vector of $\frac{\partial\Psi}{\partial r_*}$ to transform.
complex(kind=wp),	intent(out),	dimension(:)	::	dudlambda	On output contains the $\frac{\partial\Psi}{\partial\lambda}$ vector.
complex(kind=wp),	intent(out),	dimension(:)	::	dudxi	On output contains the $\frac{\partial\Psi}{\partial\xi}$ vector.

interface

public module subroutine tdc_tortoise_to_tdc_cscalb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi)

Routine to transform from tortoise coordinates $(t,r_*)$ to time dependent $(\lambda,\xi)$ for complex scalar input at element boundaries.

Arguments

Type	Intent		Name
class(tdc),	intent(inout)	::	this	The routine is called on this time dependent coordinate object.
integer(kind=ip),	intent(in)	::	ielem	The index of the element that contains the coordinate transformation information.
integer(kind=ip),	intent(in)	::	dir	The index of the boundary within the element that contains the coordinate transformation. Left boundary: 1, right boundary: 2.
complex(kind=wp),	intent(in)	::	dudt	The value of $\frac{\partial\Psi}{\partial t}$ to transform.
complex(kind=wp),	intent(in)	::	dudrstar	The value of $\frac{\partial\Psi}{\partial r_*}$ to transform.
complex(kind=wp),	intent(out)	::	dudlambda	On output contains the $\frac{\partial\Psi}{\partial\lambda}$ value.
complex(kind=wp),	intent(out)	::	dudxi	On output contains the $\frac{\partial\Psi}{\partial\xi}$ value.

interface

public module subroutine tdc_tortoise_to_tdc_cscal(this, elem, node, dpsidt, dpsidr)

Routine to transform from tortoise coordinates $(t,r_*)$ to time dependent $(\lambda,\xi)$ for complex scalar input.

Arguments

Type	Intent		Name
class(tdc),	intent(inout)	::	this	The routine is called on this time dependent coordinate object.
integer(kind=ip),	intent(in)	::	elem	The element index of the point to transform.
integer(kind=ip),	intent(in)	::	node	The node index of the point to transform.
complex(kind=wp),	intent(inout)	::	dpsidt	On input contains: $\frac{\partial\Psi}{\partial t}$ . On output contains $\frac{\partial\Psi}{\partial\lambda}$ .
complex(kind=wp),	intent(inout)	::	dpsidr	On input contains: $\frac{\partial\Psi}{\partial r_*}$ . On output contains $\frac{\partial\Psi}{\partial\xi}$ .

Derived Types

type, public :: tdc

A class that defines a time dependent coordinate transformation for the scalar wave equation on s Schwarzschild spacetime.

Components

Type	Visibility		Name
type(rgfb),	public	::	dxdlambda_b	A real boundary grid function that stores $\frac{\partial r_*}{\partial \lambda}$ on the element boundaries.
type(rgfb),	public	::	dxdxi_b	A real boundary grid function that stores $\frac{\partial r_*}{\partial \xi}$ on the element boundaries.
real(kind=wp),	public	::	maxspeed	The maximum coordinate speed (needed for CFL timestep condition).
type(rgf),	public	::	dxdlambda	A real grid function that stores $\frac{\partial r_*}{\partial \lambda}$ everywhere.
type(rgf),	public	::	dxdxi	A real grid function that stores $\frac{\partial r_*}{\partial \xi}$ everywhere.
type(rgf),	public	::	d2xdlambda2	A real grid function that stores $\frac{\partial^2 r_*}{\partial \lambda^2}$ everywhere.
type(rgf),	public	::	d2xdxi2	A real grid function that stores $\frac{\partial^2 r_*}{\partial \xi^2}$ everywhere.
type(rgf),	public	::	d2xdlambdadxi	A real grid function that stores $\frac{\partial^2 r_*}{\partial \lambda \partial \xi}$ everywhere.
type(rgf),	public	::	rm2m	A real grid function that stores $r-2M$ everywhere.

Type-Bound Procedures

procedure, public :: init => tdc_init	The initialization routine.
procedure, public :: set_coefficients => tdc_set_coefficients	The routine that sets the coefficients for the wave equation.
procedure, public :: tdc_to_tortoise_cvec => tdc_tdc_to_tortoise_cvec	Routine to transform from $\lambda,\xi)$ to $(t,r_*)$ for complex vector arguments at the boundary of an element.
procedure, public :: tdc_to_tortoise_cscal => tdc_tdc_to_tortoise_cscal	Routine to transform from $\lambda,\xi)$ to $(t,r_*)$ for complex scalar arguments at the boundary of an element.
procedure, public :: tdc_to_tortoise_rscal => tdc_tdc_to_tortoise_rscal	Routine to transform from $\lambda,\xi)$ to $(t,r_*)$ for real scalar arguments at the boundary of an element.
generic, public :: tdc_to_tortoise => tdc_to_tortoise_cvec, tdc_to_tortoise_cscal, tdc_to_tortoise_rscal	Generic name for routines that transform from $\lambda,\xi)$ to $(t,r_*)$ at the boundary of an element.
procedure, public :: tortoise_to_tdc_cvecb => tdc_tortoise_to_tdc_cvecb	Routine to transform from $(t,r_*)$ to $\lambda,\xi)$ for complex vector arguments at the boundary of an element.
procedure, public :: tortoise_to_tdc_cscalb => tdc_tortoise_to_tdc_cscalb	Routine to transform from $(t,r_*)$ to $\lambda,\xi)$ for complex scalar arguments at the boundary of an element.
procedure, public :: tortoise_to_tdc_cscal => tdc_tortoise_to_tdc_cscal	Routine to transform from $(t,r_*)$ to $\lambda,\xi)$ for complex scalar arguments at any node in any element.
generic, public :: tortoise_to_tdc => tortoise_to_tdc_cvecb, tortoise_to_tdc_cscalb, tortoise_to_tdc_cscal	Generic name for routines that transform from $(t,r_*)$ to $\lambda,\xi)$ .

time_dependent_coordinate Module

Contents

Interfaces

Derived Types

Uses

Used by

Descendants:

Contents

Interfaces

Derived Types

Interfaces

interface

public module subroutine tdc_init(this)

Arguments

interface

public module subroutine tdc_set_coefficients(this, coeffs, lcoeffs, lambda, s, sinv, rho, rstar, rschw, ll)

Arguments

interface

public module subroutine tdc_tdc_to_tortoise_cvec(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

Arguments

interface

public module subroutine tdc_tdc_to_tortoise_cscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

Arguments

interface

public module subroutine tdc_tdc_to_tortoise_rscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

Arguments

interface

public module subroutine tdc_tortoise_to_tdc_cvecb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi)

Arguments

interface

public module subroutine tdc_tortoise_to_tdc_cscalb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi)

Arguments

interface

public module subroutine tdc_tortoise_to_tdc_cscal(this, elem, node, dpsidt, dpsidr)

Arguments

Derived Types

type, public :: tdc

Components

Type-Bound Procedures

time_dependent_coordinate Module

Contents

Interfaces

Derived Types

Uses

Used by

Descendants:

Contents

Interfaces

Derived Types

Interfaces

interface

public module subroutine tdc_init(this) Implementation →

Arguments

interface

public module subroutine tdc_set_coefficients(this, coeffs, lcoeffs, lambda, s, sinv, rho, rstar, rschw, ll) Implementation →

Arguments

interface

public module subroutine tdc_tdc_to_tortoise_cvec(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar) Implementation →

Arguments

interface

public module subroutine tdc_tdc_to_tortoise_cscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar) Implementation →

Arguments

interface

public module subroutine tdc_tdc_to_tortoise_rscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar) Implementation →

Arguments

interface

public module subroutine tdc_tortoise_to_tdc_cvecb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi) Implementation →

Arguments

interface

public module subroutine tdc_tortoise_to_tdc_cscalb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi) Implementation →

Arguments

interface

public module subroutine tdc_tortoise_to_tdc_cscal(this, elem, node, dpsidt, dpsidr) Implementation →

Arguments

Derived Types

type, public :: tdc

Components

Type-Bound Procedures

public module subroutine tdc_init(this)

public module subroutine tdc_set_coefficients(this, coeffs, lcoeffs, lambda, s, sinv, rho, rstar, rschw, ll)

public module subroutine tdc_tdc_to_tortoise_cvec(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

public module subroutine tdc_tdc_to_tortoise_cscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

public module subroutine tdc_tdc_to_tortoise_rscal(this, ielem, dir, dudlambda, dudxi, dudt, dudrstar)

public module subroutine tdc_tortoise_to_tdc_cvecb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi)

public module subroutine tdc_tortoise_to_tdc_cscalb(this, ielem, dir, dudt, dudrstar, dudlambda, dudxi)

public module subroutine tdc_tortoise_to_tdc_cscal(this, elem, node, dpsidt, dpsidr)