A data type representing a matrix finite difference operator. It can be useful when preconditioning systems which use a spectral discretisation. It is inherently 1-D in its implementation. Note that multiplication of a field will simply call that field's differentiation operator, which may or may not use a finite difference method.
When constructing an instance of this type, a template field must be passed which has the same grid as any other fields which will be operated upon. Additionally, types and locations of boundary conditions must be passed.
| Type | Visibility | Attributes | Name | Initial | |||
|---|---|---|---|---|---|---|---|
| real(kind=r8), | private, | dimension(:), allocatable | :: | diagonal | The diagonal of the tridiagonal matrix representation of this block. |
||
| real(kind=r8), | private, | dimension(:), allocatable | :: | super_diagonal | The super-diagonal of the tridiagonal matrix representation of this block. |
||
| real(kind=r8), | private, | dimension(:), allocatable | :: | sub_diagonal | The sub-diagonal of the tridiagonal matrix representation of this block. |
||
| real(kind=r8), | private, | dimension(:), allocatable | :: | l_multipliers | Multipliers defining the L matrix in the LU factorisation of the tridiagonal matrix representation of this block. |
||
| real(kind=r8), | private, | dimension(:), allocatable | :: | u_diagonal | The diagonal of the U matrix in the LU factorisation of the tridiagonal matrix representation of this block. |
||
| real(kind=r8), | private, | dimension(:), allocatable | :: | u_superdiagonal1 | The first superdiagonal of the U matrix in the LU factorisation of the tridiagonal matrix representation of this block. |
||
| real(kind=r8), | private, | dimension(:), allocatable | :: | u_superdiagonal2 | The second superdiagonal of the U matrix in the LU factorisation of the tridiagonal matrix representation of this block. |
||
| integer, | private, | dimension(:), allocatable | :: | pivots | Pivot indicies from the LU factorisation of the tridiagonal matrix representation of this block. |
||
| integer, | private, | dimension(:), allocatable | :: | boundary_locs | Locations in the raw arrays which are used to specify boundary conditions. |
||
| integer, | private, | dimension(:), allocatable | :: | boundary_types | The types of boundary conditions, specified using the parameters found in boundary_types_mod. |
||
| logical, | private | :: | had_offset | = | .true. | True if the factorisation was computed from a tridiagonal system in which an offset was added to the diagonal. |
Build a tridiagonal matrix block for finite differences. See the end of the documentation of the fin_diff_block type for a description of how boundary conditions are treated.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| class(abstract_field), | intent(in) | :: | template | A scalar field with the same grid as any fields passed as arguments to the solve_for method. |
||
| integer, | intent(in), | optional | dimension(:) | :: | boundary_locs | The locations in the raw representation of |
| integer, | intent(in), | optional | dimension(:) | :: | boundary_types | Integers specifying the type of boundary condition. The type
of boundary condition corresponding to a given integer is
specified in boundary_types_mod. Only Dirichlet and
Neumann conditions are supported. Defaults to Dirichlet. The
order in which they are stored must match that of
|
A new finite difference operator
Solves the linear(ised) system represented by this finite difference block, for a given right hand side state vector (represented by a scalar field). Optionally, the differential operator can be augmented by adding an offset, i.e. a scalar field which is added to the operator.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| class(fin_diff_block), | intent(inout) | :: | this | |||
| class(scalar_field), | intent(in) | :: | rhs | The right hand side of the linear(ised) system. |
||
| class(scalar_field), | intent(in), | optional | :: | offset | An offset to add to the differential operator |
Solves the linear(ised) system represented by this finite difference block, for a given right hand side state vector (represented by a vector field). Optionally, the differential operator can be augmented by adding an offset, i.e. a vector field which is added to the operator.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| class(fin_diff_block), | intent(inout) | :: | this | |||
| class(cheb1d_vector_field), | intent(in) | :: | rhs | The right hand side of the linear(ised) system. |
||
| class(cheb1d_vector_field), | intent(in), | optional | :: | offset | An offset to add to the differential operator |
Solves the linear(ised) system represented by this finite difference block, for a given right hand side state vector (represented by a scalar field). Optionally, the differential operator can be augmented by adding an offset, i.e. a scalar field which is added to the operator.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| class(fin_diff_block), | intent(inout) | :: | this | |||
| class(scalar_field), | intent(in) | :: | rhs | The right hand side of the linear(ised) system. |
||
| class(scalar_field), | intent(in), | optional | :: | offset | An offset to add to the differential operator |
Solves the linear(ised) system represented by this finite difference block, for a given right hand side state vector (represented by a vector field). Optionally, the differential operator can be augmented by adding an offset, i.e. a vector field which is added to the operator.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| class(fin_diff_block), | intent(inout) | :: | this | |||
| class(cheb1d_vector_field), | intent(in) | :: | rhs | The right hand side of the linear(ised) system. |
||
| class(cheb1d_vector_field), | intent(in), | optional | :: | offset | An offset to add to the differential operator |
type, public :: fin_diff_block
!* Author: Chris MacMackin
! Date: December 2016
!
! A data type representing a matrix finite difference operator. It
! can be useful when preconditioning systems which use a spectral
! discretisation. It is inherently 1-D in its implementation. Note
! that multiplication of a field will simply call that field's
! differentiation operator, which may or may not use a finite
! difference method.
!
! When constructing an instance of this type, a template field
! must be passed which has the same grid as any other fields which
! will be operated upon. Additionally, types and locations of
! boundary conditions must be passed.
!
private
real(r8), dimension(:), allocatable :: diagonal
!! The diagonal of the tridiagonal matrix representation of this
!! block.
real(r8), dimension(:), allocatable :: super_diagonal
!! The super-diagonal of the tridiagonal matrix representation
!! of this block.
real(r8), dimension(:), allocatable :: sub_diagonal
!! The sub-diagonal of the tridiagonal matrix representation of
!! this block.
real(r8), dimension(:), allocatable :: l_multipliers
!! Multipliers defining the L matrix in the LU factorisation of
!! the tridiagonal matrix representation of this block.
real(r8), dimension(:), allocatable :: u_diagonal
!! The diagonal of the U matrix in the LU factorisation of
!! the tridiagonal matrix representation of this block.
real(r8), dimension(:), allocatable :: u_superdiagonal1
!! The first superdiagonal of the U matrix in the LU
!! factorisation of the tridiagonal matrix representation of
!! this block.
real(r8), dimension(:), allocatable :: u_superdiagonal2
!! The second superdiagonal of the U matrix in the LU
!! factorisation of the tridiagonal matrix representation of
!! this block.
integer, dimension(:), allocatable :: pivots
!! Pivot indicies from the LU factorisation of the tridiagonal
!! matrix representation of this block.
integer, dimension(:), allocatable :: boundary_locs
!! Locations in the raw arrays which are used to specify
!! boundary conditions.
integer, dimension(:), allocatable :: boundary_types
!! The types of boundary conditions, specified using the
!! parameters found in [[boundary_types_mod]].
logical :: had_offset = .true.
!! True if the factorisation was computed from a tridiagonal
!! system in which an offset was added to the diagonal.
! real(r8) :: magnitude
! !! The norm of the superdiagonal of the matrix. If an iterative
! !! method is used, this is needed to decide how to do so.
contains
private
procedure :: fin_diff_block_solve_scalar
procedure :: fin_diff_block_solve_vector
generic, public :: solve_for => fin_diff_block_solve_scalar, &
fin_diff_block_solve_vector
end type fin_diff_block