Fortran and Matlab Codes

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

Geometric Integrators (Structure-Preserving Algorithms)

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.

The tar file gnimatlab.tar contain a directory with the following Matlab codes (for unfolding the directory use the command tar xvf gnimatlab.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 the methods, Kepler problem;
Computation of Poincare sections for the Henon-Heiles problem;
Driver for solving the two-body problem on the sphere.

A C++ version of these codes has been written by John Denker.

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).

Nonstiff Differential Equations

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.

Stiff Differential Equations and Differential-Algebraic Problems

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.

Stabilized explicit methods for stiff differential equations

The methods are based on explicit Runge-Kutta methods with extended stability domain along the negative real axis. They are easy to use (no linear algebra routines, ...), and they are especially suited for MOL discretizations of parabolic partial differential equations. The codes are written by Assyr Abdulle (Assyr.Abdulle (at) math.unige.ch).

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).

Delay Differential Equations

For nonstiff problems

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.

For stiff problems, including differential-algebraic and neutral delay equations with constant or state-dependent (eventually vanishing) delay. 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").

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.

For problems with distributed delay. The codes are written in collaboration with Nicola Guglielmi (nicola.guglielmi (at) gssi.it). They are free subject to the following licence agreement.

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.

Mechanical Systems

The following codes are especially adapted to the equations of motion for constrained mechanical systems: q' = T(q,t)v M(t,q)v' = f(q,v,u,t) - L(q,v,u,t)*lamda 0 = H(q,t)v + k(q,t) u' = d(q,v,u,lambda,t) They have options for projecting the numerical solution to manifolds defined by 0 = g(q,t).
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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