This is a modern Fortran (2003/2008) implementation of Hairer's DOP853 ODE solver. The original FORTRAN 77 code has been extensively refactored, and is now object-oriented and thread-safe, with an easy-to-use class interface. DOP853 is an explicit Runge-Kutta method of order 8(5,3) due to Dormand & Prince (with stepsize control and dense output).

This project is hosted on GitHub.

Status

Example

Basic use of the solver is shown here. The main methods in the dop853_class are initialize() and integrate().

  program dop853_example

  use dop853_module, wp => dop853_wp
  use iso_fortran_env, only: output_unit

  implicit none

  integer,parameter               :: n   = 2               !! dimension of the system
  real(wp),parameter              :: tol = 1.0e-12_wp      !! integration tolerance
  real(wp),parameter              :: x0  = 0.0_wp          !! initial x value
  real(wp),parameter              :: xf  = 100.0_wp        !! endpoint of integration
  real(wp),dimension(n),parameter :: y0  = [0.0_wp,0.1_wp] !! initial y value

  type(dop853_class)    :: prop
  real(wp),dimension(n) :: y
  real(wp),dimension(1) :: rtol,atol
  real(wp)              :: x
  integer               :: idid
  logical               :: status_ok

  x    = x0   ! initial conditions
  y    = y0   !
  rtol = tol  ! set tolerances
  atol = tol  !

  !initialize the integrator:
  call prop%initialize(fcn=fvpol,n=n,status_ok=status_ok)
  if (.not. status_ok) error stop 'initialization error'

  !now, perform the integration:
  call prop%integrate(x,y,xf,rtol,atol,iout=0,idid=idid)

  !print solution:
  write (output_unit,'(1X,A,F6.2,A,2E18.10)') &
              'x =',x ,' y =',y(1),y(2)

contains

  subroutine fvpol(me,x,y,f)
  !! Right-hand side of van der Pol's equation

  implicit none

  class(dop853_class),intent(inout) :: me
  real(wp),intent(in)               :: x
  real(wp),dimension(:),intent(in)  :: y
  real(wp),dimension(:),intent(out) :: f

  real(wp),parameter :: mu  = 0.2_wp

  f(1) = y(2)
  f(2) = mu*(1.0_wp-y(1)**2)*y(2) - y(1)

  end subroutine fvpol

end program dop853_example

The result is:

x =100.00 y = -0.1360372426E+01  0.1325538438E+01

For dense output, see the example in the src/tests directory.

Building DOP853

A Fortran Package Manager manifest file 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 use dop853 within your FPM project, add the following to your fpm.toml file:

[dependencies]
dop853 = { git="https://github.com/jacobwilliams/dop853.git" }

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:

Preprocessor flag | Kind | Number of bytes ----------------- | ----- | --------------- REAL32 | real(kind=real32) | 4 REAL64 | real(kind=real64) | 8 REAL128 | real(kind=real128) | 16

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

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

To generate the documentation using FORD, run:

ford dop853.md

3rd Party Dependencies

The unit tests require pyplot-fortran which will be automatically downloaded by FPM.

Documentation

The latest API documentation for the master branch can be found here. This is generated by processing the source files with FORD.

References

1. E. Hairer, S.P. Norsett and G. Wanner, "Solving ordinary Differential Equations I. Nonstiff Problems", 2nd edition. Springer Series in Computational Mathematics, Springer-Verlag (1993).
2. Ernst Hairer's website: Fortran and Matlab Codes

License

• Original license for Hairer's codes.
• The updates are released under a similar BSD-style license.

See also

• numbalsoda Python wrapper to this code.