If you have any problems with the file transfer, please, contact Ernst.Hairer (at) unige.ch

We hope the programs will be of use for you and your group. We appreciate, if you communicate us your experiences (bad and good) with our codes.

A testset of stiff differential equations with drivers for some of our codes is also at your disposal.

The software of this page (with the exception of Radar5 and dc_sumexp below) is free subject to the following licence agreement

Symplectic methods for Hamiltonian systems and symmetric methods for reversible problems show an improved qualitative and quantitative behaviour, especially for long-time integrations.

For a description see: E. Hairer, C. Lubich and G. Wanner (2002): Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential equations. Springer Series in Comput. Math., vol. 31 as well as the article E. Hairer and M. Hairer (2002): GniCodes - Matlab programs for geometric numerical integration.

The tar file gnicodes.tar
contains a directory with the following Fortran 77 codes (for
unfolding the directory use the command **tar xvf gnicodes.tar**)

- GNI_IRK2, symplectic and symmetric implicit Runge-Kutta code for second order differential equations;
- GNI_COMP, composition methods with basic method as external subroutine;
- GNI_LMM2, symmetric linear multistep methods for second order differential equations;
- Driver examples for all three methods;
- Comparison of three 8th order methods at the Kepler problem and the outer solar system;
- sophisticated use of composition methods: rigid-body simulation (top) based on splitting or on Rattle; two-body problem on the sphere.

- GNI_IRK2, symplectic and symmetric implicit Runge-Kutta code for second order differential equations;
- GNI_COMP, composition methods with basic method as external subroutine;
- GNI_LMM2, symmetric linear multistep methods for second order differential equations;
- Driver examples for the methods, Kepler problem;
- Computation of Poincare sections for the Henon-Heiles problem;
- Driver for solving the two-body problem on the sphere.

**Rigid body integrator**, which is described in
E. Hairer and G. Vilmart (2006): Preprocessed Discrete Moser-Veselov
algorithm for the full dynamics of the rigid body,
J. Phys. A: Math. Gen. 39 (2006) 13225-13235.

- DMV10 Fortran subroutine for the modification of order 10 of the DMV algorithm (subroutines for the versions of orders 2,4,6,8 are included).
- DR_DMV10 Driver for DMV10 (asymmetric rigid body).

For a description see: Hairer, Norsett and Wanner (1993): Solving Ordinary Differential Equations. Nonstiff Problems. 2nd edition. Springer Series in Comput. Math., vol. 8.

- DOPRI5 explicit Runge-Kutta method of order 5(4) for problems y'=f(x,y); with dense output of order 4
- DR_DOPRI5 Driver for DOPRI5
- DOP853 explicit Runge-Kutta method of order 8(5,3) for problems y'=f(x,y); with dense output of order 7
- DR_DOP853 Driver for DOP853
- ODEX Extrapolation method (GBS) for problems y'=f(x,y); with dense output
- DR_ODEX Driver for ODEX
- ODEX2 Extrapolation method (Stoermers rule) for second order differential equations y''=f(x,y); with dense output
- DR_ODEX2
Driver for ODEX2
- There is a .tar file of a folder which contains
C-versions
of DOPRI5, DOP853 and RETARD.
A Unix command-line version of DOP853 is available from
Keith Briggs.
A modification of the C-version that can handle discontinuites is available from
Frederico Bergero
- There is a folder, written by Blake Ashby
"bmashby (at) stanford.edu", which contains
C++ versions
of the nonstiff integrator DOPRI5 and of the stiff integrator RADAU5.
It can be unfolded with "tar xzf IntegratorT.tgz".
- There are Matlab and Octave files
MatlabNonStiff,
OctaveNonStiff,
written by Denis Bichsel
"dbichsel (at) infomaniak.ch", for the
nonstiff integrators DOPRI5 and DOP853.
- Concerning a
Matlab Interface
for the nonstiff integrators DOPRI5, DOP853, and ODEX consult the
homepage at the Technische Universität München.
- There is a gzip-ped nonstiff.tar.gz file containing all this programs.

Description: Hairer and Wanner (1996): Solving Ordinary Differential Equations. Stiff and Differential-Algebraic Problems. 2nd edition. Springer Series in Comput. Math., vol. 14.

- RADAU5
implicit Runge-Kutta method of order 5
(Radau IIA) for problems of the form
My'=f(x,y) with possibly singular matrix M;
with dense output (collocation solution).
Concerning the linear algebra routines the
user has the choice to link the program either with
DC_DECSOL and DECSOL
or with
DC_LAPACK and LAPACK and LAPACKC

Hints for the choice:

a) the LAPACK routines use BLAS

b) the compiler has to accept COMPLEX*16 for the LAPACK routines; the DECSOL routines only use REAL*8 declarations - C-Interface to radau/radau5 code, written by Michael Hauth (Michael.Hauth (at) wsi-gris.uni-tuebingen.de). This package allows to call the radau/radau5-codes from C.
- DC_DECSOL subroutines for the decomposition and backsubstitution of linear systems; uses the subroutines of DECSOL
- DECSOL linear algebra routines, used by DC_DECSOL,SDIRK4,ROS4,SEULEX,SODEX
- DC_LAPACK subroutines for the decomposition and backsubstitution of linear systems; uses the subroutines of LAPACK and LAPACKC
- LAPACK linear algebra routines, used by DC_LAPACK
- LAPACKC linear algebra routines for complex arithmetic, used by DC_LAPACK
- DR1_RADAU5 Driver for RADAU5 (stiff Van der Pol equation)
- DR2_RADAU5 Driver for RADAU5 (Amplifier Problem, differential-algebraic of index 1)
- RADAU implicit Runge-Kutta method (Radau IIA) of variable order (switches automatically between orders 5, 9, and 13) for problems of the form My'=f(x,y) with possibly singular matrix M; For the choices IWORK(11)=3 and IWORK(12)=3, the code is mathematically equivalent to RADAU5 (in general a little bit slower than RADAU5).
- DR_RADAU Driver for RADAU (stiff Van der Pol equation)
- RODAS Rosenbrock method of order 4(3), for problems of the form My'=f(x,y) with possibly singular matrix M; with dense output; algebraic order conditions are considered Concerning the linear algebra routines the user has the choice to link the code with DC_DECSOL and DECSOL or with DC_LAPACK and LAPACK and LAPACKC (actually, the file LAPACKC and the routines of DC_LAPACK which need COMPLEX*16 are not used)
- DR_RODAS Driver for RODAS (stiff Van der Pol equation)
- SEULEX Extrapolation method based on linearly implicit Euler for problems of the form My'=f(x,y) with possibly singular matrix M; with dense output; concerning the linear algebra the user has the same choice as for RODAS
- DR_SEULEX
Driver for SEULEX
(stiff Van der Pol equation)
**Stiff problems with distributed delays (integro-differential equations):**such problems can be solved with RADAU5 by replacing DC_DECSOL with DC_SUMEXP. A description can be found in: N. Guglielmi and E. Hairer, Applying stiff integrators for ODEs and DDEsto problems with distributed delays. The codes are written in collaboration with Nicola Guglielmi (nicola.guglielmi (at) gssi.it). They are free subject to the following licence agreement (for details see the copyright notice (Section 9) of the "Users' guide" for RADAR5). - DC-SUMEXP linear algebra subroutines that have to be used with an application of RADAU5
- DR-GAMMA-SUMEXP driver for a test problem with distributed delay (gamma distribution)
- DR-MYELO-SUMEXP
driver for a distributed delay model of chemotherapy-induced
myelosuppression (gamma distribution)
**Oldies:**Some further available codes which, however, do not have the same linear algebra options are: - SDIRK4 a diagonally-implicit Runge-Kutta method of order 4 for problems of the form My'=f(x,y) with possibly singular matrix M; with dense output, uses DECSOL; A driver is also available.
- ROS4 Classical Rosenbrock methods of order 4(3), for problems of the form My'=f(x,y) with possibly singular matrix M; no dense output; uses DECSOL; A driver is also available.
- SODEX Extrapolation method based on linearly implicit mid-point rule for problems of the form My'=f(x,y); uses DECSOL; A driver is also available.
- RADAUP
implicit Runge-Kutta method of order 5, 9, or 13
(Radau IIA) for problems of the form
My'=f(x,y) with possibly singular matrix M;
superseded by RADAU.
- There is a folder, written by Blake Ashby
"bmashby (at) stanford.edu", which contains
C++ versions
of the nonstiff integrator DOPRI5 and of the stiff integrator RADAU5.
It can be unfolded with "tar xzf IntegratorT.tgz".
- There are Matlab and Octave files
MatlabStiff,
OctaveStiff,
written by Denis Bichsel
"dbichsel (at) infomaniak.ch", for the
stiff integrator RADAU.
- Concerning a Matlab Interface for the stiff integrators RADAU5 and SEULEX consult the homepage at the Technische Universität München.
- Matlab versions of RADAU5 can be downloaded from
Universität Tübingen
(Ch. Engstler).
- An interactive pdf file
Universidad de Zaragoza
is provided by (Luis Randez).
It is an academic implementation of RADAU5, and permits to
integrate bidimensional differential systems.
- There is a gzip-ped stiff.tar.gz file containing all this programs.

The methods are based on explicit Runge-Kutta methods with extended stability domain along the negative real axis. They are

- the tar file rock.tar
contains a directory with the Fortran 77 codes ROCK2 (order 2)
and ROCK4 (order 4), and subdirectories for the following three examples. For
unfolding the directory use the command
**tar xvf rock.tar** - BURGER, a MOL discretization of Burger's equation (dimension 500);
- BRUSS-2D, a MOL discretization of a reaction-diffusion problem in 2 space dimensions (dimension of the ODE is 32768);
- FINAG, the FitzHug and Nagumo nerve conduction equation (dimension 400).

For

- RETARD Extension of the code DOPRI5 to delay differential equations y'(t)=f(t,y(t),y(t-a),...)
- DR_RETARD
Driver for RETARD
There is a folder with Matlab codes, written by Denis Bichsel "dbichsel (at) infomaniak.ch", for adaptations of the nonstiff integrators DOPRI5 and DOP853 to delay differential equations.

- the tar file radar5-v2.2.tar (version 2.2 of February 2024) contains a directory with the Fortran 90 code RADAR5, the necessary linear algebra routines, and subdirectories for the following nine examples. A user's guide for the version 2.2 is available.
- ENZYME kinetics with an inhibitor molecule, dimension 4, one constant delay;
- HAYASHI an almost singular state dependent neutral problem, dimension 2 as differential-algebraic problem, one vanishing delay;
- HEPATITIS acute hepatitis B virus infection, dimension 10, 5 constant delays;
- OREGONATOR chemical kinetics, dimension 2, one constant delay;
- ROBERTSON chemical reaction with steady state solution, dimension 3, one constant delay, very large time interval (step sizes larger than the delay);
- SDISC, non smooth artificial problem, dimension 1, 1 constant delay;
- WALTMAN, threshold model for antibody production, dimension 6, 2 state-dependent delays tending to zero, discontinuities in the right-hand side of the equations;
- PAUL, state-dependent artificial problem;
- ELSNOR, problem with solution termination.

- RADAR5 (version 2.2 of February 2024)
- DC_SUMEXPDEL replaces the file DC_DECSOLDEL for probems with distributed delays
- DR_PARETO Driver for RADAR5 for a problem with a Pareto-distributed integral term.

The following codes are especially adapted to the equations of motion for constrained mechanical systems:

A dense output is available.

- HEM5 Half-explicit Runge-Kutta method of order 5(3); uses DECSOL, LAPACK, LINSP (code by V. Brasey)
- HEM5.PS PostScript file containing a user's guide for HEM5.
- LINSP Sparse Linear Algebra routines containing MA28 of the Harwell library
- DR_ISO Driver for HEM5 (chain of insulators, non-sparse version)
- DR_ISOSP Driver for HEM5 (chain of insulators, sparse version)
- PHEM56 Partitioned half-explicit Runge-Kutta method of order 5(4); uses DECSOL, LAPACK, LINSP (code by A. Murua ander(at)si.ehu.es)
- PHEM56.PS PostScript file containing a user's guide for PHEM56.
- DR_PHEM56
Driver for PHEM56 (chain of insulators, sparse version)
- There is a gzip-ped mechanic.tar.gz file containing all this programs.

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