Ghost method for implicit boundary condition treatment of a scalar equation defined on cells. More...

Functions/Subroutines
subroutine	mod_add_cell_bc_ghost::add_cell_bc_ghost_rec (matrix, rhs, boundary_condition, equation_stencil, equation_ls_map)
	Set boundary condition to a cell-discretized linear system (with ghost cells)

subroutine	mod_add_cell_bc_ghost::apply_boundary_condition_on_matrix_line (matrix, rhs, ia, l, stencil_size, cells_offset, cells_coeff, robin_coeff_1)
	Apply boundary condition on one line of the matrix.

subroutine	mod_add_cell_bc_ghost_former::add_cell_bc_ghost_former (matrix, rhs, boundary_condition, equation_stencil, equation_ls_map)
	Set boundary condition to a cell-discretized linear system (with ghost cells)

Detailed Description

Ghost method for implicit boundary condition treatment of a scalar equation defined on cells.

Function/Subroutine Documentation

◆ add_cell_bc_ghost_former()

subroutine mod_add_cell_bc_ghost_former::add_cell_bc_ghost_former	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:), intent(inout)	rhs,
		type(t_boundary_condition), intent(in)	boundary_condition,
		type(t_cell_stencil), intent(in)	equation_stencil,
		type(t_ls_map), intent(in)	equation_ls_map )

Set boundary condition to a cell-discretized linear system (with ghost cells)

Deprecated: Use mod_add_cell_bc_ghost::add_cell_bc_ghost instead.

The Robin boundary condition is discretized:

\begin{align} g = \alpha \frac {\partial S_b} {\partial n} + \beta S_b \end{align}

The first row of ghost cells outside the physical domain are used to impose boundary conditions. A second-order discretization scheme is used as follows (right boundary):

--x--x--|··x··  i: first interior cell node,  e: exterior cell node,
  j  i  b  e    j: second interior cell node, b: boundary face node.

Dirichlet "part": the ghost node value \( S_e \), located at the exterior cell node, is computed using the unknown value from the interior cell node and the given boundary value at the boundary face node as:
\begin{align} S_b = \tfrac{1}{2}(S_e + S_i) \end{align}
Dirichlet (only): the given boundary flux at the boundary face is expressed from the unknown values at the interior and exterior cell nodes as:
\begin{align} S_b = 0.375S_e + 0.750S_i - 0.125S_j \\ \alpha = 0 \\ \beta = 1 \end{align}
Neumann "part": the given boundary flux at the boundary face is expressed from the unknown values at the interior and exterior cell nodes as:
\begin{align} \alpha \frac {\partial S_b} {\partial n} = \alpha \frac {(S_e - S_i)} {\Delta x} \end{align}

The second row of ghost cells outside the physical domain may be used by a second order scheme, whose stencil is equal to 9 points (13 points in 3D). For this purpose, a second-order extrapolation is done (to be improved)

◆ add_cell_bc_ghost_rec()

subroutine mod_add_cell_bc_ghost::add_cell_bc_ghost_rec	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:), intent(inout)	rhs,
		type(t_boundary_condition), intent(in)	boundary_condition,
		type(t_cell_stencil), intent(in)	equation_stencil,
		type(t_ls_map), intent(in)	equation_ls_map )

private

Set boundary condition to a cell-discretized linear system (with ghost cells)

Parameters

[in,out]	matrix	the system matrix
[in,out]	rhs	the system right-hand side
[in]	boundary_condition	the boundary condition
[in]	equation_stencil	the equation stencil
[in]	equation_ls_map	the linear system mapping

This function applies the boundary condition discretization directly including ghost cells in the linear system matrix and rhs. It is based on the use of mod_compute_ghost_weights_table::compute_ghost_weights_table as the no ghost counterpart mod_add_cell_bc_noghost::add_cell_bc_noghost ["add_cell_bc_noghost"]. However, this routine has some restrictions:

only two ghost cells are supported;
only linear and quadratic schemes are supported;
quadratic scheme only supports Dirichlet BC.

This function directly sets the matrix lines and rhs entries according to the result of mod_compute_ghost_weights_table::compute_ghost_weights_table, using mod_add_cell_bc_ghost::apply_boundary_condition_on_matrix_line.

Note: The no ghost method should be preferred for being more accurate and efficient. mod_add_cell_bc_noghost::add_cell_bc_noghost

◆ apply_boundary_condition_on_matrix_line()

subroutine mod_add_cell_bc_ghost::apply_boundary_condition_on_matrix_line	(	double precision, dimension(:), intent(inout)	matrix,
		double precision, dimension(:), intent(inout)	rhs,
		integer, intent(in)	ia,
		integer, intent(in)	l,
		integer, intent(in)	stencil_size,
		integer, dimension(:), intent(in)	cells_offset,
		double precision, dimension(:), intent(in)	cells_coeff,
		double precision, intent(in)	robin_coeff_1 )