foodie_integrator_runge_kutta_low_storage Module

  1. foodie_integrator_runge_kutta_low_storage.f90
  2. foodie_integrator_runge_kutta_low_storage
module~~foodie_integrator_runge_kutta_low_storage~~UsesGraph module~foodie_integrator_runge_kutta_low_storage foodie_integrator_runge_kutta_low_storage module~foodie_integrator_multistage_object foodie_integrator_multistage_object module~foodie_integrator_multistage_object->module~foodie_integrator_runge_kutta_low_storage penf penf penf->module~foodie_integrator_runge_kutta_low_storage penf->module~foodie_integrator_multistage_object module~foodie_integrator_object foodie_integrator_object penf->module~foodie_integrator_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_integrator_object->module~foodie_integrator_runge_kutta_low_storage module~foodie_integrator_object->module~foodie_integrator_multistage_object module~foodie_error_codes->module~foodie_integrator_runge_kutta_low_storage module~foodie_integrand_object->module~foodie_integrator_runge_kutta_low_storage module~foodie_integrand_object->module~foodie_integrator_multistage_object iso_fortran_env iso_fortran_env iso_fortran_env->module~foodie_integrator_multistage_object iso_fortran_env->module~foodie_integrator_object
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FOODIE integrator: provide an explicit class of low storage Runge-Kutta schemes, from 1st to 4th order accurate.

The integrators provided have the low storage property allowing for an efficient use of the memory. Following Williamson approach [1], the LSRK(5,4)2N (solution 3) scheme of Carpenter et al. [2] is implemented.

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 class of schemes implemented are written in the form:

$$\begin{matrix} K_1 = U^n \\ K_2 = 0 \\ \left.\begin{matrix} K_2 = A_s K_2 + \Delta t R(t^n + C_s \Delta t, K_1) \\ K_1 = K_1 + B_s K_2 \end{matrix}\right\} s=1,2,...N_s\\ U^{n+1} = K_1 \end{matrix}$$

where Ns is the number of stages used and \(K_1, K_2\) are the 2 registers used for stages computation.

The schemes are explicit thus \(A_1=C_1=0\). The coefficients \(A_s, B_s, C_s\) are given in the Williamson low storage table form.

The different schemes are selected accordingly to the number of stages used. Currently the following schemes are available:

1 stage, Explicit Forward Euler, 1st order

This scheme is TVD and reverts to Explicit Forward Euler, it being 1st order. It is not a real low storage scheme, this being meaningless for a first order scheme. However it is added for safety reason.

Stage A B C
1 0 1 0

5 stages, SSP, 4th order

This scheme is a low storage RK(5, 4), based on the solution 3 proposed in [2].

Stage A B C
1 0 1432997174477/9575080441755 0
2 -567301805773 /1357537059087 5161836677717/13612068292357 1432997174477/9575080441755
3 -2404267990393/2016746695238 1720146321549/2090206949498 2526269341429/6820363962896
4 -3550918686646/2091501179385 3134564353537/4481467310338 2006345519317/3224310063776
5 -1275806237668/842570457699 2277821191437/14882151754819 2802321613138/2924317926251

6 stages, 4th order

This scheme is a low storage RK(6, 4), by [3].

Stage A B C
1 0 0.122000000000 0
2 -0.691750960670 0.477263056358 0.122000000000
3 -1.727127405211 0.381941220320 0.269115878630
4 -0.694890150986 0.447757195744 0.447717183551
5 -1.039942756197 0.498614246822 0.749979795490
6 -1.531977447611 0.186648570846 0.898555413085

7 stages, 4th order

This scheme is a low storage RK(7, 4), by [3].

Stage A B C
1 0 0.117322146869 0
2 -0.647900745934 0.503270262127 0.117322146869
3 -2.704760863204 0.233663281658 0.294523230758
4 -0.460080550118 0.283419634625 0.305658622131
5 -0.500581787785 0.540367414023 0.582864148403
6 -1.906532255913 0.371499414620 0.858664273599
7 -1.450000000000 0.136670099385 0.868664273599

12 stages, 4th order

This scheme is a low storage RK(12, 4), by [4].

Stage A B C
1 0 0.0650008435125904 0
2 -0.0923311242368072 0.0161459902249842 0.0650008435125904
3 -0.9441056581158819 0.5758627178358159 0.0796560563081853
4 -4.3271273247576394 0.1649758848361671 0.1620416710085376
5 -2.1557771329026072 0.3934619494248182 0.2248877362907778
6 -0.9770727190189062 0.0443509641602719 0.2952293985641261
7 -0.7581835342571139 0.2074504268408778 0.3318332506149405
8 -1.7977525470825499 0.6914247433015102 0.4094724050198658
9 -2.6915667972700770 0.3766646883450449 0.6356954475753369
10 -4.6466798960268143 0.0757190350155483 0.6806551557645497
11 -0.1539613783825189 0.2027862031054088 0.7143773712418350
12 -0.5943293901830616 0.2167029365631842 0.9032588871651854

13 stages, 4th order

This scheme is a low storage RK(13, 4), by [4].

Stage A B C
1 0 0.0271990297818803 0
2 -0.6160178650170565 0.1772488819905108 0.0271990297818803
3 -0.4449487060774118 0.0378528418949694 0.0952594339119365
4 -1.0952033345276178 0.6086431830142991 0.1266450286591127
5 -1.2256030785959187 0.2154313974316100 0.1825883045699772
6 -0.2740182222332805 0.2066152563885843 0.3737511439063931
7 -0.0411952089052647 0.0415864076069797 0.5301279418422206
8 -0.1797084899153560 0.0219891884310925 0.5704177433952291
9 -1.1771530652064288 0.9893081222650993 0.5885784947099155
10 -0.4078831463120878 0.0063199019859826 0.6160769826246714
11 -0.8295636426191777 0.3749640721105318 0.6223252334314046
12 -4.7895970584252288 1.6080235151003195 0.6897593128753419
13 -0.6606671432964504 0.0961209123818189 0.9126827615920843

14 stages, 4th order

This scheme is a low storage RK(14, 4), by [4].

Stage A B C
1 0 0.0367762454319673 0
2 -0.7188012108672410 0.3136296607553959 0.0367762454319673
3 -0.7785331173421570 0.1531848691869027 0.1249685262725025
4 -0.0053282796654044 0.0030097086818182 0.2446177702277698
5 -0.8552979934029281 0.3326293790646110 0.2476149531070420
6 -3.9564138245774565 0.2440251405350864 0.2969311120382472
7 -1.5780575380587385 0.3718879239592277 0.3978149645802642
8 -2.0837094552574054 0.6204126221582444 0.5270854589440328
9 -0.7483334182761610 0.1524043173028741 0.6981269994175695
10 -0.7032861106563359 0.0760894927419266 0.8190890835352128
11 0.0013917096117681 0.0077604214040978 0.8527059887098624
12 -0.0932075369637460 0.0024647284755382 0.8604711817462826
13 -0.9514200470875948 0.0780348340049386 0.8627060376969976
14 -7.1151571693922548 5.5059777270269628 0.8734213127600976

Bibliography

[1] Low-Storage Runge-Kutta Schemes, J. H. Williamson, Journal of Computational Physics, vol. 35, 1980, pp. 48–56.

[2] Fourth-Order 2N-Storage Runge-Kutta Schemes, Mark H. Carpenter, Christopher A. Kennedy, NASA Technical Memorandum 109112, June 1994.

[3] High-accuracy large-step explicit Runge–Kutta (HALE-RK) schemes for computational aeroacoustics, Vasanth Allampalli and Ray Hixon and M. Nallasamy and Scott D. Sawyer, Journal of Computational Physics, vol. 228, 2009, pp. 3837–3850.

[4] Efficient low-storage Runge–Kutta schemes with optimized stability regions, Jens Niegemann and Richard Diehl and Kurt Busch, Journal of Computational Physics, vol. 231, 2012, pp. 364–372.

Used By

module~~foodie_integrator_runge_kutta_low_storage~~UsedByGraph module~foodie_integrator_runge_kutta_low_storage foodie_integrator_runge_kutta_low_storage module~foodie foodie module~foodie_integrator_runge_kutta_low_storage->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_ ='runge_kutta_ls'

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

Flag to check if integrator provides fast mode integrate.
integer(kind=I_P), private, parameter:: registers =2

Registers used (2N schemes).
character(len=99), private, parameter:: supported_schemes_(1:7) =[trim(class_name_)//'_stages_1_order_1 ', trim(class_name_)//'_stages_5_order_4 ', trim(class_name_)//'_stages_6_order_4 ', trim(class_name_)//'_stages_7_order_4 ', trim(class_name_)//'_stages_12_order_4', trim(class_name_)//'_stages_13_order_4', trim(class_name_)//'_stages_14_order_4']

List of supported schemes.

Derived Types

type, public, extends(integrator_multistage_object) :: integrator_runge_kutta_ls

FOODIE integrator: provide an explicit class of low storage Runge-Kutta schemes, from 1st to 4th order accurate.

Components

TypeVisibility AttributesNameInitial
real(kind=R_P), private, allocatable:: A(:)

Low storage A coefficients.
real(kind=R_P), private, allocatable:: B(:)

Low storage B coefficients.
real(kind=R_P), private, allocatable:: C(:)

Low storage C coefficients.
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 :: registers

Number of registers used for stages.
class(integrand_object), public, allocatable:: stage(:)

Stages.
integer(kind=I_P), public :: stages

Number of stages.

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_multistage

Assign members of integrator_multistage_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_multistage

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, nopass :: registers_number

Return the number of registers used.

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.

Functions

private pure function class_name(self)

Return the class name of schemes.

Arguments

Type IntentOptional AttributesName
class(integrator_runge_kutta_ls), 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_runge_kutta_ls), 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_runge_kutta_ls), intent(in) :: self

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

Selected scheme.

Return Value logical

Inquire result.

private pure function registers_number()

Return the number of registers used.

Arguments

None

Return Value integer(kind=I_P)

Number of registers used.

private pure function supported_schemes(self) result(schemes)

Return the list of supported schemes.

Arguments

Type IntentOptional AttributesName
class(integrator_runge_kutta_ls), 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_runge_kutta_ls), intent(inout) :: self

Integrator.

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

Create the actual RK integrator: initialize the Butcher' low storage table coefficients.

Arguments

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

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

Selected scheme.
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_runge_kutta_ls), intent(inout) :: lhs

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

Right hand side.

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

Integrate field with explicit low storage Runge-Kutta scheme.

Arguments

Type IntentOptional AttributesName
class(integrator_runge_kutta_ls), 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.
real(kind=R_P), intent(out), optional :: new_Dt

New adapted time step.

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

Integrate field with explicit low storage Runge-Kutta scheme, fast mode.

Arguments

Type IntentOptional AttributesName
class(integrator_runge_kutta_ls), 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.
real(kind=R_P), intent(out), optional :: new_Dt

New adapted time step.

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