Implicit cell advection term

Implicit advection schemes of a scalar equation defined on cells. More...

Functions
subroutine	mod_add_cell_advection_div_u_term::add_cell_advection_div_u_term (matrix, coefficient, divergence, stencil_size, equation_ls_map)
	Add to the matrix (diagonal element) the compressive part of the cell advection term.
subroutine	mod_apply_cell_advection_scheme::apply_flux_scheme_to_system (coefficient, velocity, matrix, equation_stencil, equation_ls_map, apply_cell_advection_scheme)
	Modify the matrix by adding each cell's stencil matrix.
subroutine	mod_apply_cell_advection_scheme::apply_cell_advection_o2_centered_scheme (coefficient, velocity, matrix_line, ns, i, j, k)
	Compute the implicit second order advection scheme stencil matrix.
subroutine	mod_apply_cell_advection_scheme::apply_cell_advection_o1_upwind_scheme (coefficient, velocity, matrix_line, ns, i, j, k)
	Compute the implicit first order upwind advection scheme stencil matrix.
subroutine	mod_apply_cell_advection_scheme::apply_cell_advection_o2_upwind_scheme (coefficient, velocity, matrix_line, ns, i, j, k)
	Compute the implicit second order upwind advection scheme stencil matrix.
subroutine	mod_apply_cell_advection_scheme::apply_cell_advection_quick_scheme (coefficient, velocity, matrix_line, ns, i, j, k)
	Compute the implicit second/third order QUICK advection scheme stencil matrix.
subroutine	mod_apply_cell_advection_scheme::apply_cell_advection_houc3_scheme (coefficient, velocity, matrix_line, ns, i, j, k)
	Compute the implicit third order HOUC3 advection scheme stencil matrix.
subroutine	mod_apply_cell_advection_scheme::apply_cell_advection_generic_upwind_reconstruction_scheme (coefficient, velocity, matrix_line, rec_fast_scheme_backward, rec_fast_scheme_forward, ns, i, j, k)
	Compute the advection upwind scheme stencil matrix using a given reconstruction scheme.

Detailed Description

Implicit advection schemes of a scalar equation defined on cells.

This module provides several numerical schemes to discretize the continuity/advection term in a conservative manner:

\[ \varrho \underbrace{ \nabla \cdot \left( \mathbf{u^{n+1}} \varphi^{n+1} \right) }_{\text{conservative advection term}} \]

as well as the supplementary compressive term when solving pure advection:

\[ \varrho \underbrace{ \mathbf{u^{n+1}} \nabla \cdot \varphi^{n+1} }_{\text{advection term}} = \varrho \underbrace{ \nabla \cdot \left( \mathbf{u^{n+1}} \varphi^{n+1} \right) }_{\text{conservative advection term}} - \varrho \underbrace{ \varphi^{n+1} \nabla \cdot \mathbf{u^{n+1}} }_{\text{compression term}} \]

Numerical schemes

The module provides several schemes for the discretization of the conservative advection term:

• First order upwind: apply_cell_advection_o1_upwind_scheme;
• Second order centered: apply_cell_advection_o2_centered_scheme;
• Second order upwind: apply_cell_advection_o2_upwind_scheme;
• Second/third order upwind QUICK: apply_cell_advection_quick_scheme;
• Third order upwind HOUC3: apply_cell_advection_houc3_scheme. as well as a generic routine apply_cell_advection_generic_upwind_reconstruction_scheme that can be used in conjonction with reconstruction schemes (see Interpolation at grid elements).

The routine add_cell_advection_div_u_term handles the discretization of the compression term.

The routine apply_flux_scheme_to_system can be called to fill the linear system matrix with a given scheme routine.

Function Documentation

add_cell_advection_div_u_term()

subroutine mod_add_cell_advection_div_u_term::add_cell_advection_div_u_term	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:,:,:), intent(in)	coefficient,
		double precision, dimension(:,:,:), intent(in)	divergence,
		integer, intent(in)	stencil_size,
		type(t_ls_map), intent(in)	equation_ls_map )

Add to the matrix (diagonal element) the compressive part of the cell advection term.

Parameters

[in]	coefficient	the coefficient field
[in]	divergence	the \( \nabla \cdot \mathbf{u^{n+1} \) term
[in,out]	matrix	the linear system matrix
[in]	equation_ls_map	See type_ls_map::t_ls_map
[in]	stencil_size	the stencil size

The term discretizes \( \varrho \phi^{n+1} \nabla \cdot \mathbf{u^{n+1}} \). It is usually neglected when the velocity field is solenoidal. However, it has to be added in the general case, when solving a pure transport equation.

apply_cell_advection_generic_upwind_reconstruction_scheme()

subroutine mod_apply_cell_advection_scheme::apply_cell_advection_generic_upwind_reconstruction_scheme	(	double precision, dimension(:,:,:), intent(in)	coefficient,
		type(t_face_field), intent(in)	velocity,
		double precision, dimension(-ns:ns,-ns:ns,-ns:ns), intent(inout)	matrix_line,
		class(t_rec_fast_scheme), intent(inout)	rec_fast_scheme_backward,
		class(t_rec_fast_scheme), intent(inout)	rec_fast_scheme_forward,
		integer, intent(in)	ns,
		integer, intent(in)	i,
		integer, intent(in)	j,
		integer, intent(in)	k )

Compute the advection upwind scheme stencil matrix using a given reconstruction scheme.

Parameters

[in]	coefficient	the coefficient field
[in]	velocity	the velocity field
[in,out]	matrix_line	the resulting stencil matrix
[in]	rec_fast_scheme_backward	the backward reconstruction scheme
[in]	rec_fast_scheme_forward	the forward reconstruction scheme
[in]	ns	the stencil's bound
[in]	i,j,k	the cell indices

Compute the advection upwind scheme stencil matrix using a given reconstruction scheme with variable steps. See Interpolation at grid elements for more information about interpolation/reconstruction schemes.

Todo

The coefficient argument could be factorized.

apply_cell_advection_houc3_scheme()

subroutine mod_apply_cell_advection_scheme::apply_cell_advection_houc3_scheme	(	double precision, dimension(:,:,:), intent(in)	coefficient,
		type(t_face_field), intent(in)	velocity,
		double precision, dimension(-ns:ns,-ns:ns,-ns:ns), intent(inout)	matrix_line,
		integer, intent(in)	ns,
		integer, intent(in)	i,
		integer, intent(in)	j,
		integer, intent(in)	k )

Compute the implicit third order HOUC3 advection scheme stencil matrix.

Parameters

[in]	coefficient	the coefficient field
[in]	velocity	the velocity field
[in,out]	matrix_line	the resulting stencil matrix
[in]	ns	the stencil's bound
[in]	i,j,k	the cell indices

Compute the third order HOUC3 scheme with variable steps.

Todo

The coefficient argument could be factorized.

apply_cell_advection_o1_upwind_scheme()

subroutine mod_apply_cell_advection_scheme::apply_cell_advection_o1_upwind_scheme	(	double precision, dimension(:,:,:), intent(in)	coefficient,
		type(t_face_field), intent(in)	velocity,
		double precision, dimension(-ns:ns,-ns:ns,-ns:ns), intent(inout)	matrix_line,
		integer, intent(in)	ns,
		integer, intent(in)	i,
		integer, intent(in)	j,
		integer, intent(in)	k )

Compute the implicit first order upwind advection scheme stencil matrix.

Parameters

[in]	coefficient	the coefficient field
[in]	velocity	the velocity field
[in,out]	matrix_line	the resulting stencil matrix
[in]	ns	the stencil's bound
[in]	i,j,k	the cell indices

Compute the first order upwind scheme with variable steps. For uniform steps, in 1D, with a positive velocity the right and left fluxes are computed as:

\[ F_l = u_l \phi_l \simeq u_l \phi_L \\ F_r = u_r \phi_r \simeq u_r \phi_C \]

where \(u_l\) and \(u_r\) are the interpolated velocity at the left/right faces.

Todo

The coefficient argument could be factorized.

apply_cell_advection_o2_centered_scheme()

subroutine mod_apply_cell_advection_scheme::apply_cell_advection_o2_centered_scheme	(	double precision, dimension(:,:,:), intent(in)	coefficient,
		type(t_face_field), intent(in)	velocity,
		double precision, dimension(-ns:ns,-ns:ns,-ns:ns), intent(inout)	matrix_line,
		integer, intent(in)	ns,
		integer, intent(in)	i,
		integer, intent(in)	j,
		integer, intent(in)	k )

Compute the implicit second order advection scheme stencil matrix.

Parameters

[in]	coefficient	the coefficient field
[in]	velocity	the velocity field
[in,out]	matrix_line	the resulting stencil matrix
[in]	ns	the stencil's bound
[in]	i,j,k	the cell indices

Compute the second order centered scheme with variable steps. For uniform steps, in 1D, the right and left fluxes are computed as:

\[ F_l = u_l \phi_l \simeq u_l (\phi_L + \phi_C)/2 \\ F_r = u_r \phi_r \simeq u_r (\phi_C + \phi_R)/2 \]

where \(u_l\) and \(u_r\) are the interpolated velocity at the left/right faces.

Todo

The coefficient argument could be factorized.

apply_cell_advection_o2_upwind_scheme()

subroutine mod_apply_cell_advection_scheme::apply_cell_advection_o2_upwind_scheme	(	double precision, dimension(:,:,:), intent(in)	coefficient,
		type(t_face_field), intent(in)	velocity,
		double precision, dimension(-ns:ns,-ns:ns,-ns:ns), intent(inout)	matrix_line,
		integer, intent(in)	ns,
		integer, intent(in)	i,
		integer, intent(in)	j,
		integer, intent(in)	k )

Compute the implicit second order upwind advection scheme stencil matrix.

Parameters

[in]	coefficient	the coefficient field
[in]	velocity	the velocity field
[in,out]	matrix_line	the resulting stencil matrix
[in]	ns	the stencil's bound
[in]	i,j,k	the cell indices

Compute the second order upwind scheme with variable steps. For uniform steps, in 1D, with a positive velocity the right and left fluxes are computed as:

\[ F_l = u_l \phi_l \simeq u_l (3 \phi_L - \phi_{LL})/2 \\ F_r = u_r \phi_r \simeq u_r (3 \phi_C - \phi_L)/2 \]

where \(u_l\) and \(u_r\) are the interpolated velocity at the left/right faces.

Todo

The coefficient argument could be factorized.

apply_cell_advection_quick_scheme()

subroutine mod_apply_cell_advection_scheme::apply_cell_advection_quick_scheme	(	double precision, dimension(:,:,:), intent(in)	coefficient,
		type(t_face_field), intent(in)	velocity,
		double precision, dimension(-ns:ns,-ns:ns,-ns:ns), intent(inout)	matrix_line,
		integer, intent(in)	ns,
		integer, intent(in)	i,
		integer, intent(in)	j,
		integer, intent(in)	k )

Compute the implicit second/third order QUICK advection scheme stencil matrix.

Parameters

[in]	coefficient	the coefficient field
[in]	velocity	the velocity field
[in,out]	matrix_line	the resulting stencil matrix
[in]	ns	the stencil's bound
[in]	i,j,k	the cell indices

Compute the second/third order QUICK scheme with variable steps.

Todo

The coefficient argument could be factorized.