Implicit discretization of a scalar equation defined on cells. More...

Topics
	Cell advection term
	Implicit advection schemes of a scalar equation defined on cells.

	Boundary condition treatment
	Boundary condition treatment of a scalar equation defined on cells.

	Cell diffusion term
	Implicit diffusion schemes of a scalar equation defined on cells.

	Cell linear system preparation
	Cell linear system preparation of a scalar equation defined on cells.

Functions
subroutine	mod_impose_cell_value::impose_cell_value (matrix, rhs, position, value, stencil_size, equation_ls_map)
	Impose the linear system to yield the given value at the given position.

Detailed Description

This directory provides a set of routines to discretize a scalar advection-diffusion equation defined on cells. The general form of the equation is:

\[ \varrho \bigg( \underbrace{ \frac {\alpha \varphi^{n+1} + \beta \varphi^n + \gamma \varphi^{n-1} } {\Delta t}}_{\text{temporal term}} + \underbrace{ \mathbf{u^{n+1}} \cdot \nabla \tilde{\varphi} }_{\text{advection term}} \bigg) = \underbrace{ \nabla \cdot \left( D \nabla \varphi^{n+1} \right)}_{\text{diffusive term}} \]

where \(\varphi\) is the scalar field, \(D\) the diffusion coefficient, and \( \varrho \) an advection coefficient. The discretization of the advection term can be done either implicitly ( \( \tilde{\varphi}=\varphi^{n+1} \)) or eventually ( \( \tilde{\varphi}=\varphi^{n} \)).

This equation becomes a linear system by the discretization process. The matrix and the right-hand side are constructed by the routines defined here in a term-by-term fashion.

Generic routines

First of all, some routines are designed to simply build-up le linear system:

add_diagonal_cell_matrix adds array elements to the diagonal terms of the matrix.
add_rhs_cell adds array elements to the right-hand side.
impose_cell_value sets the linear system to get a specific value at a specific position.
finalize_cell_linear_system should be called to complete the build-up.

Routine discretize_cell_transport_equation can be used to discretize the whole equation (see solver_energy.f90 for instance)

Todo: DOC need to be completed on this point

Specific routines

These routines are related to the terms described above.

add_cell_advection_term adds the advection term in both explicit and implicit versions. See the routine's documentation for details.

The enumerator enum_cell_advection_term_scheme is useful to provide arguments.

add_cell_diffusion_term_centered_o2 add the diffusion term.

The functions set_diffusion_flux_coef and set_diffusion_flux_coef_2 are useful to provide the arguments.

Boundary conditions

The boundary conditions are imposed by the following routines:

apply_bc_on_ghost_b_cells (explicit)
add_cell_bc (implicit)
Some more routines to apply boundary condition on extra ghost cells explicitely.
apply_immersed_boundary_condition_cell adds boundary condition to immersed boundaries, which most probably adds ghost nodes to the system.

Set-up routines

These routines contribute to the setup of the linear system:

set_cell_stencil_indices

Function Documentation

◆ impose_cell_value()

subroutine mod_impose_cell_value::impose_cell_value	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:), intent(inout)	rhs,
		double precision, dimension(3), intent(in)	position,
		double precision, intent(in)	value,
		integer, intent(in)	stencil_size,
		type(t_ls_map), intent(in)	equation_ls_map )

This routine modifies the linear system such as the solution will yield the given value at the given position. Pretty brutal, this routine has been implemented to fix ill-posed linear system such as the Laplace equation with Neumann boundary conditions everywhere.

Topics