Partial Differential Equations on cells

Functions
subroutine	mod_discretize_cell_advection_term::discretize_cell_advection_term (matrix, coefficient, divergence, equation_advection_term_discretization_type, equation_advection_term_scheme, equation_has_div_u_advection_term, equation_has_immersed_boundaries, equation_ib_has_one_sided_inner_discretization, equation_ib_inner_discretization_order, equation_stencil, time_step_n, time_step, equation_cfl_cell_advection, scalar_field_n, equation_explicit_time_order_discretization, flux_type, boundary_condition, equation_ls_map, velocity_nm1, velocity_n, velocity_np1, work_cell_field, equation_advection_term_splitting_scheme, equation_splitting_time_coef, directional_splitting, ibc_variable, equation_isd_target)
	Discretize the advection term, explicitly or implicitly. More...
subroutine	mod_discretize_cell_diffusion_term::discretize_cell_diffusion_term (matrix, equation_diffusion_term_discretization_type, equation_diffusion_term_scheme, equation_diffusion_explicit_time_order_discretization, equation_has_immersed_boundaries, ibc_variable, equation_isd_target, equation_stencil, equation_ls_map, time_step, equation_cfl_cell_diffusion, scalar_field_n, work_cell_field, boundary_condition, temporal_coefficient, diffusion_coefficient, face_diffusion_coefficient, extra_inside_gradient_coef_diff)
	Discretize the advection term. More...
subroutine	mod_discretize_cell_poisson_equation::discretize_cell_poisson_equation (matrix, rhs, diffusion_coefficient, equation_source_term, linear_term, ibc_variable, equation_isd_target, boundary_condition, equation_diffusion_term_discretization_type, equation_diffusion_term_scheme, equation_has_linear_term, equation_has_ghost_boundary_cells, equation_has_immersed_boundaries, equation_stencil, equation_ls_map, equation_name)
	Discretize the poisson (or advection/diffusion) equation defined on cells. More...
subroutine	mod_discretize_cell_transport_equation::discretize_cell_transport_equation (matrix, rhs, coefficient, diffusion_coefficient, velocity_nm1, velocity_n, velocity_np1, divergence, equation_source_term, scalar_field_n, scalar_field_nm1, scalar_field_nm2, linear_term, boundary_condition, ibc_variable, equation_isd_target, equation_time_step, equation_time_step_n, equation_time_step_nm1, equation_time_order_discretization, equation_cfl_advection, equation_cfl_diffusion, equation_explicit_time_order_discretization, equation_flux_type, equation_has_source_term, equation_has_temporal_term, equation_has_advection_term, equation_has_div_u_advection_term, equation_has_diffusion_term, equation_has_linear_term, equation_has_ghost_boundary_cells, equation_has_immersed_boundaries, equation_ib_has_one_sided_inner_discretization, equation_ib_inner_discretization_order, equation_fully_explicit, equation_use_explicit_terms_accumulator, equation_advection_term_discretization_type, equation_advection_term_scheme, equation_diffusion_term_discretization_type, equation_diffusion_term_scheme, equation_diffusion_explicit_time_order_discretization, equation_advection_term_splitting_scheme, equation_splitting_time_coef, equation_advection_term_dir_split, equation_stencil, equation_ls_map, is_matrix_invertible_local, equation_name, equation_explicit_terms_accumulator_n, equation_explicit_terms_accumulator_nm1, equation_explicit_terms_accumulator_nm2)
	Discretize the transport (or advection/diffusion) equation defined on cells. More...

Detailed Description

In this directory you will find the first level of routines to solve PDE term by term on cell based variables.

Function Documentation

discretize_cell_advection_term()

subroutine mod_discretize_cell_advection_term::discretize_cell_advection_term	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:,:,:), intent(in)	coefficient,
		double precision, dimension(:,:,:), intent(in)	divergence,
		integer, intent(in)	equation_advection_term_discretization_type,
		integer, intent(in)	equation_advection_term_scheme,
		logical, intent(in)	equation_has_div_u_advection_term,
		logical, intent(in)	equation_has_immersed_boundaries,
		logical, intent(in)	equation_ib_has_one_sided_inner_discretization,
		integer, dimension (:), intent(in)	equation_ib_inner_discretization_order,
		type(t_cell_stencil), intent(in)	equation_stencil,
		double precision, intent(in)	time_step_n,
		double precision, intent(in)	time_step,
		double precision, intent(inout)	equation_cfl_cell_advection,
		double precision, dimension(:,:,:), intent(in)	scalar_field_n,
		integer, intent(in)	equation_explicit_time_order_discretization,
		type(t_fv_flux), intent(in)	flux_type,
		type(t_boundary_condition), intent(in)	boundary_condition,
		type(t_ls_map), intent(in)	equation_ls_map,
		type(t_face_field), intent(in)	velocity_nm1,
		type(t_face_field), intent(in)	velocity_n,
		type(t_face_field), intent(in)	velocity_np1,
		double precision, dimension(:,:,:), intent(inout), allocatable	work_cell_field,
		integer, intent(in)	equation_advection_term_splitting_scheme,
		double precision, dimension (3), intent(inout)	equation_splitting_time_coef,
		logical, intent(in)	directional_splitting,
		type(t_immersed_boundary_condition), dimension(:), intent(in)	ibc_variable,
		integer, dimension(:), intent(in)	equation_isd_target
	)

The term writes:

\[ \varrho \mathbf{u}^{n+1} \cdot \nabla \varphi^{n+1} = \varrho \nabla \cdot (\mathbf{u}^{n+1} \varphi^{n+1}) - \varphi^{n+1} \nabla \cdot \mathbf{u}^{n+1} \]

This term can be treated implicitly or explicitly. In the first case, discretization coefficients are added to the matrix of the linear system. In the other case, the term is added to the right-hand side.

Since the Navier-Stokes equation are solved before other advection and diffusion equations, velocity at time \( t^{n+1} \) is known at this stage.

Parameters

[in,out]	matrix	matrix of the linear system
[in]	coefficient	the coefficient field
[in]	divergence	the velocity divergence \( \nabla \cdot \mathbf{u}^{n} \)
[in]	equation_advection_term_discretization_type	the temporal discretization type implicit_discretization or explicit_discretization
[in]	equation_advection_term_scheme	the temporal discrete scheme, see enum_cell_advection_term_scheme
[in]	equation_has_div_u_advection_term	if true, add \( -c \nabla \cdot u \) to the RHS for pure advection
[in]	equation_has_immersed_boundaries	if true, adapt discretization to immersed boundaries
[in]	equation_ib_has_one_sided_inner_discretization	if true, uncentered discretization on first inner node
[in]	equation_ib_inner_discretization_order	set the convergence order of the inner discretization method
[in]	equation_stencil	equation stencil structure
[in]	time_step_n	the previous time step \( \delta t^{n-1} = t^{n} - t^{n-1} \)
[in]	time_step	the current time step \( \delta t^{n} = t^{n+1} - t^{n} \)
[in]	equation_cfl_cell_advection
[in]	scalar_field_n	the scalar field at time \( t^{n} \).
[in]	equation_explicit_time_order_discretization	the time discretization order for computing the advection with the explicit scheme (Euler, NSSP32, NSSP53)
[in]	flux_type	the flux type to use for the explicit advection (type_fv_flux)
[in]	boundary_condition	the boundary conditions of the cell scalar
[in]	equation_ls_map
[in]	velocity_nm1	velocity field at \( t^{n-1} \)
[in]	velocity_n	velocity field at \( t^n \)
[in]	velocity_np1	velocity field at \( t^{n+1} \)
[in]	work_cell_field
[in]	equation_advection_term_splitting_scheme
[in]	equation_splitting_time_coef
[in]	directional_splitting
[in]	ibc_variable
[in]	equation_isd_target

Todo:

Consider velocity interpolation when STRETCHED RECTILINEAR MESH are used.

discretize_cell_diffusion_term()

subroutine mod_discretize_cell_diffusion_term::discretize_cell_diffusion_term	(	double precision, dimension(:), intent(inout)	matrix,
		integer, intent(in)	equation_diffusion_term_discretization_type,
		integer, intent(in)	equation_diffusion_term_scheme,
		integer, intent(in)	equation_diffusion_explicit_time_order_discretization,
		logical, intent(in)	equation_has_immersed_boundaries,
		type(t_immersed_boundary_condition), dimension(:), intent(in)	ibc_variable,
		integer, dimension(:), intent(in)	equation_isd_target,
		type(t_cell_stencil), intent(in)	equation_stencil,
		type(t_ls_map), intent(in)	equation_ls_map,
		double precision, intent(in)	time_step,
		double precision, intent(inout)	equation_cfl_cell_diffusion,
		double precision, dimension(:,:,:), intent(in)	scalar_field_n,
		double precision, dimension(:,:,:), intent(inout)	work_cell_field,
		type(t_boundary_condition), intent(in)	boundary_condition,
		double precision, dimension(:,:,:), intent(in), optional	temporal_coefficient,
		double precision, dimension(:,:,:), intent(in), optional	diffusion_coefficient,
		type(t_face_field), intent(inout), optional	face_diffusion_coefficient,
		double precision, dimension(:,:,:), intent(in), optional	extra_inside_gradient_coef_diff
	)

The term writes:

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

Either cell or face diffusion coefficients must be given as an argument. When cell diffusion coefficient is given, they are interpolated on face nodes using interpolate_diffusion_coef (harmonic interpolation).

Note

• As described in Cell diffusion term, a central scheme of order 2 (or 4) is used (regular mesh).
• No treatment of boundary condition is done here for implicit formulation.
• Boundary condition is also done here for explicit sub iteration formulation.
• First order Euler time discretization is used for explicit diffusion sub iterations if \(\delta t > \delta t_{max cfl}\).
• \( \nabla \cdot (D \nabla p) \) for constant coefficient Poisson equation, hereby called Pconst, (A fast pressure-correction methdo for incompressible two-fluid flows, Dodd & Ferrante 2014) is also performed.
• Pconst is discretized with a second order explicit central difference.
• Pconst must be given face diffusion coefficients (hard coded before calling this subroutine).

Parameters

[in,out]	matrix	matrix of linear system
[in]	equation_diffusion_term_discretization_type
[in]	equation_diffusion_term_scheme
[in]	equation_diffusion_explicit_time_order_discretization
[in]	equation_has_immersed_boundaries
[in]	ibc_variable	instance of immersed_boundary_condition associated to \( \phi^{n} \)
[in]	equation_isd_target	immersed subdomain target array of the equation
[in]	equation_stencil
[in]	equation_ls_map
[in]	time_step
[in]	equation_cfl_cell_diffusion
[in]	scalar_field_n
[in]	work_cell_field
[in]	boundary_condition
[in]	diffusion_coefficient
[in]	face_diffusion_coefficient	(optional)

discretize_cell_poisson_equation()

subroutine mod_discretize_cell_poisson_equation::discretize_cell_poisson_equation	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:), intent(inout)	rhs,
		double precision, dimension(:,:,:), intent(in)	diffusion_coefficient,
		double precision, dimension(:,:,:), intent(in)	equation_source_term,
		double precision, dimension(:,:,:), intent(in)	linear_term,
		type(t_immersed_boundary_condition), dimension(:), intent(inout), allocatable	ibc_variable,
		integer, dimension(:), intent(in), allocatable	equation_isd_target,
		type(t_boundary_condition), intent(in)	boundary_condition,
		integer, intent(in)	equation_diffusion_term_discretization_type,
		integer, intent(in)	equation_diffusion_term_scheme,
		logical, intent(in)	equation_has_linear_term,
		logical, intent(in)	equation_has_ghost_boundary_cells,
		logical, intent(in)	equation_has_immersed_boundaries,
		type(t_cell_stencil), intent(in)	equation_stencil,
		type(t_ls_map), intent(in)	equation_ls_map,
		character(len=*), intent(in)	equation_name
	)

Parameters

[in]	matrix	matrix of the linear system.
[in]	rhs	right-hand side of the linear system.
[in]	diffusion_coefficient	Coefficient of diffusion.
[in]	equation_source_term	Source term added to the right-hand side.
[in]	linear_term	Contains the coefficient in front of the linear term.
[in]	ibc_variable	Boundary conditions on immersed boundaries.
[in]	equation_isd_target	Subset of immersed boundary list.
[in]	boundary_condition	the boundary conditions of the cell scalar
[in]	equation_diffusion_term_discretization_type	Type of discretization (implicit/explicit) the diffusion term.
[in]	equation_diffusion_term_scheme	Numerical scheme of the diffusion term
[in]	equation_has_linear_term	Flag to enable the linear term.
[in]	equation_has_ghost_boundary_cells	Flag to enable the ghost cells into the linear system
[in]	equation_has_immersed_boundaries	Flag to enable immersed boundaries.
[in]	equation_stencil	Stencil description.
[in]	equation_ls_map	Structure used to map the general numbering to the linear system numbering.
[in]	equation_name	Name of the equation used to display a message.

discretize_cell_transport_equation()

subroutine mod_discretize_cell_transport_equation::discretize_cell_transport_equation	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:), intent(inout)	rhs,
		double precision, dimension(:,:,:), intent(in)	coefficient,
		double precision, dimension(:,:,:), intent(in)	diffusion_coefficient,
		type(t_face_field), intent(in)	velocity_nm1,
		type(t_face_field), intent(in)	velocity_n,
		type(t_face_field), intent(in)	velocity_np1,
		double precision, dimension(:,:,:), intent(in)	divergence,
		double precision, dimension(:,:,:), intent(in)	equation_source_term,
		double precision, dimension(:,:,:), intent(in)	scalar_field_n,
		double precision, dimension(:,:,:), intent(in)	scalar_field_nm1,
		double precision, dimension(:,:,:), intent(in)	scalar_field_nm2,
		double precision, dimension(:,:,:), intent(in)	linear_term,
		type(t_boundary_condition), intent(in)	boundary_condition,
		type(t_immersed_boundary_condition), dimension(:), intent(in)	ibc_variable,
		integer, dimension(:), intent(in)	equation_isd_target,
		double precision, intent(in)	equation_time_step,
		double precision, intent(in)	equation_time_step_n,
		double precision, intent(in)	equation_time_step_nm1,
		integer, intent(in)	equation_time_order_discretization,
		double precision, intent(inout)	equation_cfl_advection,
		double precision, intent(inout)	equation_cfl_diffusion,
		integer, intent(in)	equation_explicit_time_order_discretization,
		type(t_fv_flux), intent(in)	equation_flux_type,
		logical, intent(in)	equation_has_source_term,
		logical, intent(in)	equation_has_temporal_term,
		logical, intent(in)	equation_has_advection_term,
		logical, intent(in)	equation_has_div_u_advection_term,
		logical, intent(in)	equation_has_diffusion_term,
		logical, intent(in)	equation_has_linear_term,
		logical, intent(in)	equation_has_ghost_boundary_cells,
		logical, intent(in)	equation_has_immersed_boundaries,
		logical, intent(in)	equation_ib_has_one_sided_inner_discretization,
		integer, dimension(:), intent(in)	equation_ib_inner_discretization_order,
		logical, intent(in)	equation_fully_explicit,
		logical, intent(in)	equation_use_explicit_terms_accumulator,
		integer, intent(in)	equation_advection_term_discretization_type,
		integer, intent(in)	equation_advection_term_scheme,
		integer, intent(in)	equation_diffusion_term_discretization_type,
		integer, intent(in)	equation_diffusion_term_scheme,
		integer, intent(in)	equation_diffusion_explicit_time_order_discretization,
		integer, intent(in)	equation_advection_term_splitting_scheme,
		double precision, dimension (3), intent(inout)	equation_splitting_time_coef,
		logical, intent(in)	equation_advection_term_dir_split,
		type(t_cell_stencil), intent(in)	equation_stencil,
		type(t_ls_map), intent(in)	equation_ls_map,
		logical, intent(in)	is_matrix_invertible_local,
		character(len=*), intent(in)	equation_name,
		double precision, dimension(:,:,:), intent(inout)	equation_explicit_terms_accumulator_n,
		double precision, dimension(:,:,:), intent(inout)	equation_explicit_terms_accumulator_nm1,
		double precision, dimension(:,:,:), intent(inout)	equation_explicit_terms_accumulator_nm2
	)

Parameters

[in]	matrix	matrix of the linear system.
[in]	rhs	right-hand side of the linear system.
[in]	coefficient	Coefficient in front of the time and advection term.
[in]	diffusion_coefficient	Coefficient of diffusion.
[in]	velocity_nm1	velocity field at \( t^{n-1} \)
[in]	velocity_n	velocity field at \( t^n \)
[in]	velocity	velocity field at \( t^{n+1} \)
[in]	divergence	Divergence of the velocity.
[in]	equation_source_term	Source term added to the right-hand side.
[in]	scalar_field_n	Scalar field at previous time iteration.
[in]	scalar_field_nm1	Scalar field at time iteration n-1.
[in]	scalar_field_nm2	Scalar field at time iteration n-2.
[in]	linear_term	Contains the coefficient in front of the linear term.
[in]	boundary_condition	the boundary conditions of the cell scalar
[in]	ibc_variable	Boundary conditions on immersed boundaries.
[in]	equation_isd_target	Subset of immersed boundary list.
[in]	equation_time_step	the current time step \( \delta t^{n} = t^{n+1} - t^{n} \)
[in]	equation_time_step_n	the previous time step \( \delta t^{n-1} = t^{n} - t^{n-1} \)
[in]	equation_time_step_nm1	the previous time step \( \delta t^{n-2} = t^{n-1} - t^{n-2} \)
[in]	equation_time_order_discretization	the time discretization order for the equation (SBDF), 1 or 2
[in,out]	equation_cfl_advection
[in,out]	equation_cfl_diffusion
[in]	equation_explicit_time_order_discretization	the time discretization order for computing the advection with the explicit scheme (Euler, NSSP32, NSSP53)
[in]	equation_flux_type	the flux type to use for the explicit advection (type_fv_flux)
[in]	equation_has_source_term	Flag to enable the source term.
[in]	equation_has_temporal_term	Flag to enable the temporal term.
[in]	equation_has_advection_term	Flag to enable the advection term.
[in]	equation_has_div_u_advection_term	Flag to enable the div(u⃗) advection term
[in]	equation_has_diffusion_term	Flag to enable the diffusion term.
[in]	equation_has_linear_term	Flag to enable the linear term.
[in]	equation_has_ghost_boundary_cells	Flag to enable the ghost cells into the linear system
[in]	equation_has_immersed_boundaries	Flag to enable immersed boundaries.
[in]	equation_ib_has_one_sided_inner_discretization
[in]	equation_ib_inner_discretization_order
[in]	equation_fully_explicit
[in]	equation_use_explicit_terms_accumulator
[in]	equation_advection_term_discretization_type	Type of discretization (implicit/explicit) the advection term.
[in]	equation_advection_term_scheme	Numerical scheme of the advection term (upwind, etc.).
[in]	equation_diffusion_term_discretization_type	Type of discretization (implicit/explicit) the diffusion term.
[in]	equation_diffusion_term_scheme	Numerical scheme of the diffusion term
[in]	equation_diffusion_explicit_time_order_discretization
[in]	equation_advection_term_splitting_scheme
[in,out]	equation_splitting_time_coef
[in]	equation_advection_term_dir_split
[in]	equation_stencil	Stencil description.
[in]	equation_ls_map	Structure used to map the general numbering to the linear system numbering.
[in]	is_matrix_invertible_local	Flag to require a null right-hand side integral.
[in]	equation_name	Name of the equation used to display a message.
[in,out]	equation_explicit_terms_accumulator_n	the current accumulator of explicit terms at time \( t^{n+1/2} \) (mean)
[in]	equation_explicit_terms_accumulator_nm1	the accumulator of explicit terms at time \( t^{n-1/2} \) (mean)