Status

Description

This is a modern object-oriented Fortran implementation of the DDEABM Adams-Bashforth-Moulton ODE solver. The original Fortran 77 code was obtained from the SLATEC library. It has been extensively refactored.

DDEABM uses the Adams-Bashforth-Moulton predictor-corrector formulas of orders 1 through 12 to integrate a system of first order ordinary differential equations of the form dx/dt = f(t,x). Also included is an event-location capability, where the equations can be integrated until a specified function g(t,x) = 0. Dense output is also supported.

This project is hosted on GitHub.

Examples

The ddeabm_module provides a thread-safe and object-oriented interface to the DDEABM method. Some example use cases are presented below:

Basic integration

This example shows how to integrate a conic orbit (6 state equations) around the Earth from an initial time t0 to a final time tf:

program ddeabm_example

use ddeabm_module, wp => ddeabm_rk

implicit none

real(wp),parameter :: mu = 398600.436233_wp !! Earth gravitational parameter (km^3/s^2)
integer,parameter  :: n = 6                 !! number of state variables

type(ddeabm_class) :: s
real(wp),dimension(n) :: x0,x
real(wp) :: t0,tf,t
integer :: idid

call s%initialize(n,maxnum=10000,df=twobody,rtol=[1.0e-12_wp],atol=[1.0e-12_wp])

!initial conditions:
x0 = [10000.0_wp,10000.0_wp,10000.0_wp,&   !initial state [r,v] (km,km/s)
        1.0_wp,2.0_wp,3.0_wp]
t0 = 0.0_wp       !initial time (sec)
tf = 1000.0_wp    !final time (sec)

write(*,'(A/,*(F15.6/))') 'Initial time: ',t0
write(*,'(A/,*(F15.6/))') 'Initial state:',x0
t = t0
x = x0
call s%integrate(t,x,tf,idid=idid)
write(*,'(A/,*(F15.6/))') 'Final time: ',t
write(*,'(A/,*(F15.6/))') 'Final state:',x

contains

    subroutine twobody(me,t,x,xdot)

        !! derivative routine for two-body orbit propagation

        implicit none

        class(ddeabm_class),intent(inout) :: me
        real(wp),intent(in)               :: t
        real(wp),dimension(:),intent(in)  :: x
        real(wp),dimension(:),intent(out) :: xdot

        real(wp),dimension(3) :: r,v,a_grav
        real(wp) :: rmag

        r = x(1:3)
        v = x(4:6)
        rmag = norm2(r)
        a_grav = -mu/rmag**3 * r ! acceleration due to gravity

        xdot(1:3) = v
        xdot(4:6) = a_grav

    end subroutine twobody

end program ddeabm_example

It produces the following output:

Initial time:
       0.000000

Initial state:
   10000.000000
   10000.000000
   10000.000000
       1.000000
       2.000000
       3.000000

Final time:
    1000.000000

Initial time:
   10667.963305
   11658.055962
   12648.148619
       0.377639
       1.350074
       2.322509

Reporting of intermediate points

The intermediate integration points can also be reported to a user-defined procedure. For the above example, the following subroutine could be defined:

subroutine twobody_report(me,t,x)

    !! report function - write time,state to console

    implicit none

    class(ddeabm_class),intent(inout)    :: me
    real(wp),intent(in)                  :: t
    real(wp),dimension(:),intent(in)     :: x

    write(*,'(*(F15.6,1X))') t,x

end subroutine twobody_report

Which can be added to the class on initialization:

call s%initialize(n,maxnum=10000,df=twobody,&
                  rtol=[1.0e-12_wp],atol=[1.0e-12_wp],&
                  report=twobody_report)

This function is then called at each time step if the equations are integrated using the integration_mode=2 option like so:

call s%integrate(t,x,tf,idid=idid,integration_mode=2)

Event location

A user-defined event function g(t,x) can also be defined in order to stop the integration at a specified event (i.e., when g(t,x)=0). In the above example, say it is desired that the integration stop when z = x(3) = 12,000 km. The event function for this would be:

subroutine twobody_event(me,t,x,g)

    !! event function for z = 12,000 km

    implicit none

    class(ddeabm_with_event_class),intent(inout) :: me
    real(wp),intent(in)                          :: t
    real(wp),dimension(:),intent(in)             :: x
    real(wp),intent(out)                         :: g

    g = 12000.0_wp - x(3)

end subroutine twobody_event

For event finding, the ddeabm_with_event_class type is used (which is an extension of the main ddeabm_class). For example:

type(ddeabm_with_event_class) :: s
...
call s%initialize_event(n,maxnum=10000,df=twobody,&
                        rtol=[1.0e-12_wp],atol=[1.0e-12_wp],&
                        g=twobody_event,root_tol=1.0e-12_wp)
...
call s%integrate_to_event(t,x,tf,idid=idid,gval=gval)

In this case, root_tol is the tolerance for the event location, and gval is the value of the event function at the final time (note that the integration will stop when g(t,x)=0 or at t=tf, whichever occurs first).

A vector event function is also supported (in which case, the integration stops if any of the roots are found). This is done using the ddeabm_with_event_class_vec type.

Fixed time step

All of the integration methods have an optional argument (tstep) to enable a fixed time step, which can be used for dense output, or to specify a fixed step used for event finding (since the default step may be too large). For example, for performing a root-finding integration with the event function evaluated every 100 seconds:

call s%integrate_to_event(t,x,tf,idid=idid,gval=gval,tstep=100.0_wp)

Building DDEABM

DDEABM and the test programs will build with any modern Fortran compiler. A Fortran Package Manager (FPM) manifest file (fmp.toml) is included, so that the library and tests cases can be compiled with FPM. For example:

fpm build --profile release
fpm test --profile release

To generate the documentation using ford, run:

ford ddeabm.md

To use ddeabm within your fpm project, add the following to your fpm.toml file:

[dependencies]
ddeabm = { git="https://github.com/jacobwilliams/ddeabm.git" }

A specific version can also be specified:

[dependencies]
ddeabm = { git="https://github.com/jacobwilliams/ddeabm.git", rev = "2.1.0" }

By default, the library is built with double precision (real64) real values. Explicitly specifying the real kind can be done using the following processor flags:

For example, to build a single precision version of the library, use:

fpm build --profile release --flag "-DREAL32"

Dependencies

Building the library requires the roots-fortran module. Building the tests requires the pyplot-fortran module. FPM will automatically download the correct versions of both (see fpm.toml).

Documentation

The latest API documentation can be found here. This was generated from the source code using FORD.

License

The DDEABM source code and related files and documentation are distributed under a permissive free software license (BSD-style). The original DDEABM Fortran 77 code is public domain.

Keywords

Adams-Bashforth-Moulton Method, DEPAC, Initial Value Problems, ODE, Ordinary Differential Equations, Predictor-Corrector, SLATEC, Modern Fortran

References

1. L. F. Shampine, M. K. Gordon, "Solving ordinary differential equations with ODE, STEP, and INTRP", Report SLA-73-1060, Sandia Laboratories, 1973.
2. L. F. Shampine, M. K. Gordon, "Computer solution of ordinary differential equations, the initial value problem", W. H. Freeman and Company, 1975.
3. L. F. Shampine, H. A. Watts, "DEPAC - Design of a user oriented package of ode solvers", Report SAND79-2374, Sandia Laboratories, 1979.
4. H. A. Watts, "A smoother interpolant for DE/STEP, INTRP and DEABM: II", Report SAND84-0293, Sandia Laboratories, 1984.
5. R. P. Brent, "An algorithm with guaranteed convergence for finding a zero of a function", The Computer Journal, Vol 14, No. 4., 1971.
6. R. P. Brent, "Algorithms for minimization without derivatives", Prentice-Hall, Inc., 1973.