Explicit discretization of the diffusion equation for a scalar equation defined on cells. More...

Namespaces
module	mod_discretize_cell_explicit_diffusion_term_o2
	Compute cell explicit diffusion term with centered second order scheme.

Functions
subroutine	mod_compute_cell_diffusion_term_explicit_centered::compute_cell_diffusion_term_explicit_centered_o2 (div_grad_phi, phi, flux_coef)
	Compute the diffusive term of an equation defined on cells.

subroutine	mod_compute_cell_diffusion_term_explicit_centered::compute_cell_diffusion_term_explicit_centered_o4 (div_grad_phi, phi, flux_coef)
	Compute the diffusive term of an equation defined on cells - \( 4^{th} \) order.

subroutine	mod_integrate_cell_diffusion_term_explicit_generic::integrate_cell_diffusion_term_explicit_euler (phi_output, time_step, temporal_coefficient, diffusion_coefficient, face_diffusion_coefficient, phi_n, temporal_stability_factor, boundary_condition, equation_has_immersed_boundaries, equation_diffusion_term_scheme, ibc_variable, equation_isd_target)
	Compute the cell field \( \phi \) from the integration of \( a_t\frac{\partial \phi}{\partial t}=\nabla \cdot (D\nabla\phi) \) with Euler first order temporal scheme.

subroutine	mod_integrate_cell_diffusion_term_explicit_generic::integrate_cell_diffusion_term_explicit_rk2 (phi_output, time_step, temporal_coefficient, diffusion_coefficient, face_diffusion_coefficient, phi_n, temporal_stability_factor, boundary_condition, equation_has_immersed_boundaries, equation_diffusion_term_scheme, ibc_variable, equation_isd_target)
	Compute the cell field \( \phi \) from the integration of \( a_t\frac{\partial \phi}{\partial t}=\nabla \cdot (D\nabla\phi) \) with the canonical RK2 temporal scheme.

Detailed Description

This directory provides a set of routines to discretize a scalar diffusion equation defined, in a conservative way, on cells such as:

\[ \frac{\partial \phi}{\partial t} = \nabla \cdot \left( k(\mathbf{x},t) \nabla \phi \right) \]

with a given diffusion coefficient field \( k(\mathbf{x},t) \) ( \( D \)).

For 2D, the term becomes:

\begin{align} \nabla \cdot ( D \nabla \phi)_{0,0} =& \frac{1}{\delta x_0} F_{+\frac12} \phi_{+\frac12,0} -\frac{1}{\delta x_0} \left(F_{+\frac12} + F_{-\frac12}\right) \phi_{0,0} +\frac{1}{\delta x_0} F_{-\frac12} \phi_{-\frac12,0} \\ +& \frac{1}{\delta y_0} G_{+\frac12} \phi_{0,+\frac12} -\frac{1}{\delta y_0} \left(G_{+\frac12} + G_{-\frac12}\right) \phi_{0,0} +\frac{1}{\delta y_0} G_{-\frac12} \phi_{0,-\frac12} \,. \end{align}

where \( \mathbf{F} = (F, G) \) (flux_coef), \( F_{+1/2,0} = \frac{D_{+1/2,0}}{\delta x_{+1/2}} \), is the face diffusive flux coefficients, and \( \delta x_0 \) and \( \delta y_0 \) is the size of the cell \( (0,0) \).

Spatial discretization

For now, only the canonical \( 2^{nd} \) and \( 4^{th} \) order center schemes are implemented.

Function Documentation

◆ compute_cell_diffusion_term_explicit_centered_o2()

subroutine mod_compute_cell_diffusion_term_explicit_centered::compute_cell_diffusion_term_explicit_centered_o2	(	double precision, dimension(:,:,:), intent(inout)	div_grad_phi,
		double precision, dimension(:,:,:), intent(in)	phi,
		type(t_face_field), intent(in)	flux_coef )

The term writes:

\begin{align} \nabla \cdot ( D \nabla \phi)_{0,0} =& \frac{1}{\delta x_0} F_{+\frac12} \phi_{+\frac12,0} -\frac{1}{\delta x_0} \left(F_{+\frac12} + F_{-\frac12}\right) \phi_{0,0} +\frac{1}{\delta x_0} F_{-\frac12} \phi_{-\frac12,0} \\ +& \frac{1}{\delta y_0} G_{+\frac12} \phi_{0,+\frac12} -\frac{1}{\delta y_0} \left(G_{+\frac12} + G_{-\frac12}\right) \phi_{0,0} +\frac{1}{\delta y_0} G_{-\frac12} \phi_{0,-\frac12} \,. \end{align}

where \( \mathbf{F} = (F, G) \) (flux_coef) is the face diffusive flux coefficients, and \( \delta x_0 \) and \( \delta y_0 \) is the size of the cell \( (0,0) \).

Note

As described in Cell diffusion term, a central schema of order 2 is used (regular mesh).
No treatment of boundary condition is done here.

Todo: check order with irregular grids

◆ compute_cell_diffusion_term_explicit_centered_o4()

subroutine mod_compute_cell_diffusion_term_explicit_centered::compute_cell_diffusion_term_explicit_centered_o4	(	double precision, dimension(:,:,:), intent(inout)	div_grad_phi,
		double precision, dimension(:,:,:), intent(in)	phi,
		type(t_face_field), intent(in)	flux_coef )

Note

As described in Cell diffusion term, a central schema of order 2 is used (regular mesh).
No treatment of boundary condition is done here.

Todo: [AJ] Generalize \( 4^{th} \) order scheme for rectilinear grids.

◆ integrate_cell_diffusion_term_explicit_euler()

subroutine mod_integrate_cell_diffusion_term_explicit_generic::integrate_cell_diffusion_term_explicit_euler	(	double precision, dimension(:,:,:), intent(out)	phi_output,
		double precision, intent(in)	time_step,
		double precision, dimension(:,:,:), intent(in)	temporal_coefficient,
		double precision, dimension(:,:,:), intent(in)	diffusion_coefficient,
		type(t_face_field), intent(in)	face_diffusion_coefficient,
		double precision, dimension(:,:,:), intent(in)	phi_n,
		double precision, intent(in)	temporal_stability_factor,
		type(t_boundary_condition), intent(in)	boundary_condition,
		logical, intent(in)	equation_has_immersed_boundaries,
		integer, intent(in)	equation_diffusion_term_scheme,
		type(t_immersed_boundary_condition), dimension(:), intent(in)	ibc_variable,
		integer, dimension(:), intent(in)	equation_isd_target )

Parameters

[out]	phi_output	\( \phi^{n+1} \)
[in]	time_step	the time step
[in]	temporal_coefficient	the physical parameter of temporal term \( a_t \)
[in]	diffusion_coefficient	the physical parameter multiplying the gradient of the field \( D \)
[in]	face_diffusion_coefficient	divided by spatial step: \( F_{+\frac12,0} = \frac{D_{+\frac12,0}}{\delta x_{+\frac12}} \)
[in]	phi_n	\( \phi^{n} \)
[in]	temporal_stability_factor	the cfl coefficient to use (depending on the time and space schemes)
[in]	boundary_condition	the boundary conditions for cell centered \( \phi \)
[in]	equation_has_immersed_boundaries	for immersed boundary cell update
[in]	equation_diffusion_term_scheme	discretization scheme of the diffusion term
[in]	ibc_variable	instance of immersed_boundary_condition associated to \( \phi^{n} \)
[in]	equation_isd_target	immersed subdomain target array of the equation

◆ integrate_cell_diffusion_term_explicit_rk2()

subroutine mod_integrate_cell_diffusion_term_explicit_generic::integrate_cell_diffusion_term_explicit_rk2	(	double precision, dimension(:,:,:), intent(out)	phi_output,
		double precision, intent(in)	time_step,
		double precision, dimension(:,:,:), intent(in)	temporal_coefficient,
		double precision, dimension(:,:,:), intent(in)	diffusion_coefficient,
		type(t_face_field), intent(in)	face_diffusion_coefficient,
		double precision, dimension(:,:,:), intent(in)	phi_n,
		double precision, intent(in)	temporal_stability_factor,
		type(t_boundary_condition), intent(in)	boundary_condition,
		logical, intent(in)	equation_has_immersed_boundaries,
		integer, intent(in)	equation_diffusion_term_scheme,
		type(t_immersed_boundary_condition), dimension(:), intent(in)	ibc_variable,
		integer, dimension(:), intent(in)	equation_isd_target )