foodie_integrator_backward_differentiation_formula Module

  1. foodie_integrator_backward_differentiation_formula.f90
  2. foodie_integrator_backward_differentiation_formula
module~~foodie_integrator_backward_differentiation_formula~~UsesGraph module~foodie_integrator_backward_differentiation_formula foodie_integrator_backward_differentiation_formula module~foodie_integrator_object foodie_integrator_object module~foodie_integrator_object->module~foodie_integrator_backward_differentiation_formula module~foodie_integrator_multistep_object foodie_integrator_multistep_object module~foodie_integrator_object->module~foodie_integrator_multistep_object module~foodie_integrator_multistep_object->module~foodie_integrator_backward_differentiation_formula penf penf penf->module~foodie_integrator_backward_differentiation_formula penf->module~foodie_integrator_object penf->module~foodie_integrator_multistep_object module~foodie_error_codes foodie_error_codes penf->module~foodie_error_codes module~foodie_integrand_object foodie_integrand_object penf->module~foodie_integrand_object module~foodie_error_codes->module~foodie_integrator_backward_differentiation_formula module~foodie_integrand_object->module~foodie_integrator_backward_differentiation_formula module~foodie_integrand_object->module~foodie_integrator_multistep_object iso_fortran_env iso_fortran_env iso_fortran_env->module~foodie_integrator_object iso_fortran_env->module~foodie_integrator_multistep_object
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FOODIE integrator: provide an implicit class of Backward Differentiation Formula schemes, from 1st to 6th order accurate.

Considering the following ODE system:

$$ U_t = R(t,U) $$

where \(U_t = \frac{dU}{dt}\), U is the vector of state variables being a function of the time-like independent variable t, R is the (vectorial) residual function, the Backward Differentiation Formula class scheme implemented is:

$$ U^{n+N_s} + \sum_{s=1}^{N_s} \alpha_s U^{n+N_s-s} = \Delta t \left[ \beta R(t^{n+N_s}, U^{n+N_s}) \right] $$

where \(N_s\) is the number of previous steps considered.

The schemes are implicit. The coefficients \(\alpha_s\) and \(\beta\) define the actual scheme, that is selected accordingly to the number of steps used.

Currently, the following schemes are available:

Step beta alpha 1 alpha 2 alpha 3 alpha 4 alpha 5 alpha 6
1 1 -1
2 2/3 -4/3 1/3
3 6/11 -18/11 9/11 -2/11
4 12/25 -48/25 36/25 -16/25 3/25
5 60/137 -300/137 300/137 -200/137 75/137 -12/137
6 60/147 -360/147 450/147 -400/147 225/147 -72/147 10/147

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Used By

module~~foodie_integrator_backward_differentiation_formula~~UsedByGraph module~foodie_integrator_backward_differentiation_formula foodie_integrator_backward_differentiation_formula module~foodie foodie module~foodie_integrator_backward_differentiation_formula->module~foodie program~integrate_burgers integrate_burgers module~foodie->program~integrate_burgers program~integrate_lorenz integrate_lorenz module~foodie->program~integrate_lorenz module~type_euler_1d type_euler_1D module~foodie->module~type_euler_1d module~foodie_test_integrand_tester_object foodie_test_integrand_tester_object module~foodie->module~foodie_test_integrand_tester_object module~foodie_test_lcce_test foodie_test_lcce_test module~foodie->module~foodie_test_lcce_test module~type_euler_1d_openmp type_euler_1D_openmp module~foodie->module~type_euler_1d_openmp program~integrate_euler_1d_caf~2 integrate_euler_1D_caf module~foodie->program~integrate_euler_1d_caf~2 module~foodie_test_oscillation_test foodie_test_oscillation_test module~foodie->module~foodie_test_oscillation_test program~integrate_euler_1d integrate_euler_1D module~foodie->program~integrate_euler_1d module~foodie_test_object foodie_test_object module~foodie->module~foodie_test_object module~foodie_test_integrand_oscillation foodie_test_integrand_oscillation module~foodie->module~foodie_test_integrand_oscillation module~foodie_test_integrand_lcce foodie_test_integrand_lcce module~foodie->module~foodie_test_integrand_lcce module~type_euler_1d_caf type_euler_1D_caf module~foodie->module~type_euler_1d_caf module~type_burgers type_burgers module~foodie->module~type_burgers module~foodie_test_integrand_ladvection foodie_test_integrand_ladvection module~foodie->module~foodie_test_integrand_ladvection program~integrate_euler_1d_openmp integrate_euler_1D_openmp module~foodie->program~integrate_euler_1d_openmp module~type_lorenz type_lorenz module~foodie->module~type_lorenz module~type_euler_1d->program~integrate_euler_1d module~foodie_test_integrand_tester_object->module~foodie_test_object module~foodie_test_integrand_tester_object->module~foodie_test_integrand_oscillation module~foodie_test_integrand_tester_object->module~foodie_test_integrand_lcce module~foodie_test_integrand_tester_object->module~foodie_test_integrand_ladvection program~foodie_test_lcce foodie_test_lcce module~foodie_test_lcce_test->program~foodie_test_lcce module~type_euler_1d_openmp->program~integrate_euler_1d_openmp program~foodie_test_oscillation foodie_test_oscillation module~foodie_test_oscillation_test->program~foodie_test_oscillation program~foodie_tester foodie_tester module~foodie_test_object->program~foodie_tester module~foodie_test_integrand_oscillation->module~foodie_test_oscillation_test module~foodie_test_integrand_oscillation->module~foodie_test_object module~foodie_test_integrand_lcce->module~foodie_test_lcce_test module~foodie_test_integrand_lcce->module~foodie_test_object module~type_euler_1d_caf->program~integrate_euler_1d_caf~2 module~type_burgers->program~integrate_burgers module~foodie_test_integrand_ladvection->module~foodie_test_object module~type_lorenz->program~integrate_lorenz
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Variables

Derived Types

Functions

Subroutines


Variables

TypeVisibility AttributesNameInitial
character(len=99), private, parameter:: class_name_ ='back_df'

Name of the class of schemes.
logical, private, parameter:: has_fast_mode_ =.true.

Flag to check if integrator provides fast mode integrate.
character(len=99), private, parameter:: supported_schemes_(1:6) =[trim(class_name_)//'_1', trim(class_name_)//'_2', trim(class_name_)//'_3', trim(class_name_)//'_4', trim(class_name_)//'_5', trim(class_name_)//'_6']

List of supported schemes.

Derived Types

type, public, extends(integrator_multistep_object) :: integrator_back_df

FOODIE integrator: provide an implicit class of Backward-Differentiation-Formula multi-step schemes, from 1st to 6th order accurate.

Components

TypeVisibility AttributesNameInitial
real(kind=R_P), public, allocatable:: Dt(:)

Previous time steps.
real(kind=R_P), private, allocatable:: a(:)

\(\alpha\) coefficients.
logical, public :: autoupdate

Perform cyclic autoupdate of previous time steps buffers.
real(kind=R_P), private :: b =0.0_R_P

\(\beta\) coefficient.
class(integrand_object), public, allocatable:: buffer

Buffer used for fast integration.
character(len=:), public, allocatable:: description_

Informative description of the integrator.
integer(kind=I_P), public :: error =0

Error status code.
character(len=:), public, allocatable:: error_message

Error message, hopefully meaningful.
integer(kind=I_P), public :: iterations

Implicit iterations.
class(integrand_object), public, allocatable:: previous(:)

Previous steps.
integer(kind=I_P), public :: registers

Number of registers used for steps.
integer(kind=I_P), public :: steps

Number of time steps.
real(kind=R_P), public, allocatable:: t(:)

Previous times.

Type-Bound Procedures

procedure, public, pass(self) :: allocate_integrand_members

Allocate integrand members.

procedure, public, pass(lhs) :: assign_abstract

Assign ony members of abstract integrator_object type.

procedure, public, pass(lhs) :: assign_multistep

Assign members of integrator_multistep_object and parents.

generic, public :: assignment(=) => integr_assign_integr

Overload =.

procedure, public, pass(self) :: check_error

Check for error occurrencies.

procedure, public, pass(self) :: class_name

Return the class name of schemes.

procedure, public, pass(self) :: description

Return informative integrator description.

procedure, public, pass(self) :: destroy

Destroy the integrator.

procedure, public, pass(self) :: destroy_abstract

Destroy only members of abstract integrator_object type.

procedure, public, pass(self) :: destroy_multistep

Destroy the integrator.

procedure, public, pass(self) :: has_fast_mode

Return .true. if the integrator class has fast mode integrate.

procedure, public, pass(self) :: initialize

Initialize (create) the integrator.

procedure, public, pass(lhs) :: integr_assign_integr

Operator =.

procedure, public, pass(self) :: integrate

Integrate integrand field.

procedure, public, pass(self) :: integrate_fast

Integrate integrand field, fast mode.

procedure, public, pass(self) :: is_multistage

Return .true. for multistage integrator.

procedure, public, pass(self) :: is_multistep

Return .true. for multistep integrator.

procedure, public, pass(self) :: is_supported

Return .true. if the integrator class support the given scheme.

procedure, public, pass(self) :: stages_number

Return number of stages used.

procedure, public, pass(self) :: steps_number

Return number of steps used.

procedure, public, pass(self) :: supported_schemes

Return the list of supported schemes.

procedure, public, pass(self) :: trigger_error

Trigger an error.

procedure, public, nopass :: update_previous

Cyclic update previous time steps.

Functions

private pure function class_name(self)

Return the class name of schemes.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(in) :: self

Integrator.

Return Value character(len=99)

Class name.

private elemental function has_fast_mode(self)

Return .true. if the integrator class has fast mode integrate.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(in) :: self

Integrator.

Return Value logical

Inquire result.

private elemental function is_supported(self, scheme)

Return .true. if the integrator class support the given scheme.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(in) :: self

Integrator.
character(len=*), intent(in) :: scheme

Selected scheme.

Return Value logical

Inquire result.

private pure function supported_schemes(self) result(schemes)

Return the list of supported schemes.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(in) :: self

Integrator.

Return Value character(len=99), allocatable, (:)

Queried scheme.


Subroutines

private elemental subroutine destroy(self)

Destroy the integrator.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(inout) :: self

Integrator.

private subroutine initialize(self, scheme, iterations, autoupdate, U, stop_on_fail)

Create the actual BDF integrator: initialize the alpha and beta coefficients.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(inout) :: self

Integrator.
character(len=*), intent(in) :: scheme

Selected scheme.
integer(kind=I_P), intent(in), optional :: iterations

Implicit iterations.
logical, intent(in), optional :: autoupdate

Enable cyclic autoupdate of previous time steps.
class(integrand_object), intent(in), optional :: U

Integrand molding prototype.
logical, intent(in), optional :: stop_on_fail

Stop execution if initialization fail.

private subroutine integr_assign_integr(lhs, rhs)

Operator =.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(inout) :: lhs

Left hand side.
class(integrator_object), intent(in) :: rhs

Right hand side.

private subroutine integrate(self, U, Dt, t)

Integrate field with BDF class scheme.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(inout) :: self

Integrator.
class(integrand_object), intent(inout) :: U

Field to be integrated.
real(kind=R_P), intent(in) :: Dt

Time step.
real(kind=R_P), intent(in) :: t

Time.

private subroutine integrate_fast(self, U, Dt, t)

Integrate field with BDF class scheme.

Arguments

Type IntentOptional AttributesName
class(integrator_back_df), intent(inout) :: self

Integrator.
class(integrand_object), intent(inout) :: U

Field to be integrated.
real(kind=R_P), intent(in) :: Dt

Time step.
real(kind=R_P), intent(in) :: t

Time.

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