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

Implicit discretization of a scalar equation defined on cells.

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 )

Impose the linear system to yield the given value at the given position.

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