jacobian_block_mod

Provides a derived type which is useful for representing blocks in a Jacobian matrix. This follows the abstract calculus pattern, providing operators for matrix multiplication and for solving the linear(ised) system. See the documentation for the jacobian_block type for more details.

Uses

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Used by

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Variables

no_extra_derivative

Interfaces

jacobian_block

Derived Types

jacobian_block

Functions

constructor jacobian_block_multiply jacobian_block_add_real jacobian_block_add_field jacobian_block_add_block jacobian_block_solve

Subroutines

jacobian_block_assign jacobian_block_get_tridiag jacobian_block_bounds

Variables

Type	Visibility	Attributes		Name		Initial
integer,	private,	parameter	::	no_extra_derivative	=	-1

Interfaces

public interface jacobian_block

private function constructor(source_field, direction, extra_derivative, boundary_locs, boundary_types, boundary_operations, coef) result(this)

Author: Chris MacMackin
Date: December 2016

Build a block in a Jacobian matrix, with the form $\frac{\partial F}{\partial x_i} + F\Delta_i,$ where $F$ is a scalar field and $\Delta_i$ is the differentiation operator in the $i$ -direction. Additionally, a further differentiation operator may be added to the right hand side of this matrix block. Optional arguments allow for handling of boundary conditions. See the end of the documentation of the jacobian_block type for a description of how boundary conditions are treated.

Arguments

Type	Intent	Optional	Attributes		Name
class(scalar_field),	intent(in)			::	source_field	A scalar field ( $F$ ) making up this block of the Jacobian
integer,	intent(in)			::	direction	The direction in which field derivatives are taken.
integer,	intent(in),	optional		::	extra_derivative	If present, specifies the direction of a differentiation operator to be added to the right hand side of this matrix block.
integer,	intent(in),	optional	dimension(:)	::	boundary_locs	The locations in the raw representation of `rhs` for which boundary conditions are specified. Defaults to there being none.
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 `boundary_locs`.
procedure(jacobian_block_bounds),		optional		::	boundary_operations	A function specifying the values to place at the boundaries of the result when using the Jacobian block for multiplication. By default, all boundaries are set to 0. The order in which the resulting values are stored should match that of `boundary_locs`.
real(kind=r8),	intent(in),	optional		::	coef	An optional coefficient by which the the $\partial F/\partial x$ term in the operator will be multipled. Default value is 1.

Return Value type(jacobian_block)

A new Jacobian block

Derived Types

type, public :: jacobian_block

A data type representing a submatrix of a Jacobian. Specifically, it represents the commonly occurring operation $\frac{\partial F}{\partial x_i} + F\Delta_i,$ where $\Delta_i$ is the differentiation operator in the $i$ -direction. Optionally, there can be an additional differentiation operator on the right-hand-side of this.

Components

Type	Visibility	Attributes		Name		Initial
integer,	private		::	direction			The direction in which any derivatives are taken.
integer,	private		::	extra_derivative	=	no_extra_derivative	The direction in which to apply a differentiation on the right-hand-side of the Jacobian block operator. Defaults none.
class(scalar_field),	private,	allocatable	::	contents			The value, $A$ , to which the Jacobian block operation is beiing applied.
real(kind=r8),	private		::	coef	=	1._r8	Optional coefficient by which the the $\partial F/\partial x$ term in the operator will be multiplied.
procedure(jacobian_block_bounds),	private,	pointer, nopass	::	get_boundaries			A subroutine which determines the expected boundary conditions (and their location in the raw array) for the solution of the Jacobian block.
class(scalar_field),	private,	allocatable	::	derivative			The cached derivative of `contents`
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.
real(kind=r8),	private,	dimension(:), allocatable	::	boundary_vals			Expected boundary values for the solution to the Jacobian 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.
real(kind=r8),	private		::	real_increment			A scalar value which is to be added to this Jacobian block (i.e. to the diagonal).
class(scalar_field),	private,	allocatable	::	field_increment			A scalar field which is to be added to this Jacobian block (i.e. to the diagonal).
type(jacobian_block),	private,	pointer	::	block_increment	=>	null()	Another Jacobian block which is to be added to this one
integer,	private		::	has_increment	=	0	Indicates whether or not there has been an increment added to this block. If not, then 0. If a scalar real value has been added, then 1. If a scalar value has been added, then 2.

Constructor

private function constructor(source_field, direction, extra_derivative, boundary_locs, boundary_types, boundary_operations, coef)	Build a block in a Jacobian matrix, with the form $\frac{\partial F}{\partial x_i} + F\Delta_i,$ where $F$ is a scalar field and $\Delta_i$ is the differentiation operator in the $i$ -direction. Additionally, a further differentiation operator may be added to the right hand side of this matrix block. Optional arguments allow for handling of boundary conditions. See the end of the documentation of the jacobian_block type for a description of how boundary conditions are treated.

Type-Bound Procedures

procedure, private :: jacobian_block_multiply
procedure, private :: jacobian_block_add_real
procedure, private :: jacobian_block_add_field
procedure, private :: jacobian_block_add_block
procedure, private :: jacobian_block_assign
procedure, private :: get_tridiag => jacobian_block_get_tridiag
generic, public :: *operator()** => jacobian_block_multiply
generic, public :: operator(+) => jacobian_block_add_real, jacobian_block_add_field, jacobian_block_add_block
generic, public :: assignment(=) => jacobian_block_assign
procedure, public :: solve_for => jacobian_block_solve

Functions

private function constructor(source_field, direction, extra_derivative, boundary_locs, boundary_types, boundary_operations, coef) result(this)

Author: Chris MacMackin
Date: December 2016

Arguments

Type	Intent	Optional	Attributes		Name
class(scalar_field),	intent(in)			::	source_field	A scalar field ( $F$ ) making up this block of the Jacobian
integer,	intent(in)			::	direction	The direction in which field derivatives are taken.
integer,	intent(in),	optional		::	extra_derivative	If present, specifies the direction of a differentiation operator to be added to the right hand side of this matrix block.
integer,	intent(in),	optional	dimension(:)	::	boundary_locs	The locations in the raw representation of `rhs` for which boundary conditions are specified. Defaults to there being none.
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 `boundary_locs`.
procedure(jacobian_block_bounds),		optional		::	boundary_operations	A function specifying the values to place at the boundaries of the result when using the Jacobian block for multiplication. By default, all boundaries are set to 0. The order in which the resulting values are stored should match that of `boundary_locs`.
real(kind=r8),	intent(in),	optional		::	coef	An optional coefficient by which the the $\partial F/\partial x$ term in the operator will be multipled. Default value is 1.

Return Value type(jacobian_block)

A new Jacobian block

private recursive function jacobian_block_multiply(this, rhs) result(product)

Author: Chris MacMackin
Date: December 2016

Provides a matrix multiplication operator between a Jacobian block and a scalar field (which corresponds to a state vector).

Arguments

Type	Intent	Optional	Attributes		Name
class(jacobian_block),	intent(in)			::	this
class(scalar_field),	intent(in)			::	rhs	A field corresponding to a state vector being multiplied by the Jacobian block.

Return Value class(scalar_field), pointer

private function jacobian_block_add_real(this, rhs) result(sum)

Author: Chris MacMackin
Date: December 2016

Produces a Jacobian block which has been offset by some constant increment.

Arguments

Type	Intent	Optional	Attributes		Name
class(jacobian_block),	intent(in)			::	this
real(kind=r8),	intent(in)			::	rhs	A scalar which should be added to this block

Return Value type(jacobian_block)

private function jacobian_block_add_field(this, rhs) result(sum)

Author: Chris MacMackin
Date: May 2017

Produces a Jacobian block which has been offset by a scalar field.

Arguments

Type	Intent	Optional	Attributes		Name
class(jacobian_block),	intent(in)			::	this
class(scalar_field),	intent(in)			::	rhs	A scalar which should be added to this block

Return Value type(jacobian_block)

private function jacobian_block_add_block(this, rhs) result(sum)

Author: Chris MacMackin
Date: May 2017

Produces a Jacobian block which is the sum of two existing blocks. Boundary conditions are set by the first operand (this).

Arguments

Type	Intent	Optional	Attributes		Name
class(jacobian_block),	intent(in)			::	this
class(jacobian_block),	intent(in),		target	::	rhs	A second block which should be added to this block

Return Value type(jacobian_block)

private function jacobian_block_solve(this, rhs) result(solution)

Author: Chris MacMackin
Date: December 2016

Solves the linear(ised) system represented by this Jacobian block, for a given right hand side state vector (represented by a scalar field).

Arguments

Type	Intent	Optional	Attributes		Name
class(jacobian_block),	intent(inout)			::	this
class(scalar_field),	intent(in)			::	rhs	The right hand side of the linear(ised) system.

Return Value class(scalar_field), pointer

Subroutines

private subroutine jacobian_block_assign(this, rhs)

Author: Chris MacMackin
Date: December 2016

Copies the contents of the rhs Jacobian block into this one. It will safely deallocate any data necessary.

Arguments

Type	Intent	Optional	Attributes		Name
class(jacobian_block),	intent(out)			::	this
type(jacobian_block),	intent(in)			::	rhs

private recursive subroutine jacobian_block_get_tridiag(this, diagonal, subdiagonal, superdiagonal)

Author: Chris MacMackin
Date: May 2017

Computes the tridiagonal matrix used to solve for this Jacobian block.

Arguments

Type	Intent	Attributes		Name
class(jacobian_block),	intent(in)		::	this
real(kind=r8),	intent(out),	dimension(:), allocatable	::	diagonal
real(kind=r8),	intent(out),	dimension(:), allocatable	::	subdiagonal
real(kind=r8),	intent(out),	dimension(:), allocatable	::	superdiagonal

private subroutine jacobian_block_bounds(contents, derivative, rhs, boundary_locs, boundary_types, boundary_values)

Author: Chris MacMackin
Date: January 2016

A default implementation of the get_boundaries procedure pointer for the jacobian_block type. It corresponds to setting all boundaries to 0 when multiplying a field by the Jacobian block.

Arguments

Type	Intent	Attributes		Name
class(scalar_field),	intent(in)		::	contents	The field used to construct the Jacobian block
class(scalar_field),	intent(in)		::	derivative	The first spatial derivative of the field used to construct the Jacobian block, in the direction specified
class(scalar_field),	intent(in)		::	rhs	The scalar field representing the vector being multiplied by Jacobian
integer,	intent(in),	dimension(:), allocatable	::	boundary_locs	The locations in the raw representation of `rhs` containing the boundaries.
integer,	intent(in),	dimension(:), allocatable	::	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. The storage order must correspond to that of `boundary_locs`.
real(kind=r8),	intent(out),	dimension(:), allocatable	::	boundary_values	The values to go at the boundaries when multiplying a field by the Jacobian block. The storage order must be the same as for `boundary_locs`.

jacobian_block_mod Module

Contents

Uses

Used by

Contents

Variables

Interfaces

public interface jacobian_block

private function constructor(source_field, direction, extra_derivative, boundary_locs, boundary_types, boundary_operations, coef) result(this)

Arguments

Return Value type(jacobian_block)

Derived Types

type, public :: jacobian_block

Components

Constructor

Type-Bound Procedures

Functions

private function constructor(source_field, direction, extra_derivative, boundary_locs, boundary_types, boundary_operations, coef) result(this)

Arguments

Return Value type(jacobian_block)

private recursive function jacobian_block_multiply(this, rhs) result(product)

Arguments

Return Value class(scalar_field), pointer

private function jacobian_block_add_real(this, rhs) result(sum)

Arguments

Return Value type(jacobian_block)

private function jacobian_block_add_field(this, rhs) result(sum)

Arguments

Return Value type(jacobian_block)

private function jacobian_block_add_block(this, rhs) result(sum)

Arguments

Return Value type(jacobian_block)

private function jacobian_block_solve(this, rhs) result(solution)

Arguments

Return Value class(scalar_field), pointer

Subroutines

private subroutine jacobian_block_assign(this, rhs)

Arguments

private recursive subroutine jacobian_block_get_tridiag(this, diagonal, subdiagonal, superdiagonal)

Arguments

private subroutine jacobian_block_bounds(contents, derivative, rhs, boundary_locs, boundary_types, boundary_values)

Arguments