Scientific Computing FAQ: Integration and Ordinary Differential Equations (ODEs)

Integration and Ordinary Differential Equations (ODEs)

Most of the general NA texts, such as Stoer & Bulirsch 1980, have chapters on integration techniques.

See also: Davis, Philip J.; Rabinowitz, Philip, 1984, Methods of Numerical Integration, Academic Press, ISBN 0122063600

ODEPACK, from netlib (see Netlib ). [Dave Linder]: Well known and widely used. Allows error-based adaptive step-size control.

A systematized collection of ODE solvers. Handles stiff and nonstiff systems, sparse systems, explicit and linearly implicit forms. Fortran IV, double precision. A single precision version may be found in sodepack, also in netlib (see Netlib ).

RKSUITE, from netlib ode/rksuite (see Netlib ) [Dave Lindner]: Also good. Allows error-based adaptive step-size control and easy to use for explicit systems.

For initial value problem for first order ordinary differential equations. A suite of codes for solving IVPs in ODEs. A choice of RK methods is available. Includes an error assessment facility and a sophisticated stiffness checker. Template programs and example results provided. Supersedes RKF45, DDERKF, D02PAF.

LSODE and DVODE, from netlib (see Netlib ): [Jean Claude Boettner] The algorithm is the same but the jacobian matrix is stored in DVODE allowing better efficiency, especially for big systems of ODE (100-300 typically in my case). So expect a slow down (2 ?) when using LSODE. It might be interesting to test VODEPK (on NETLIB) with Krylov methods.

See also many other packages in the directory ode, in Netlib .

Ernst/Wanner package: Separate Fortran codes for stiff, nonstiff, and mechanical systems. For the description see the book by the authors: Hairer, Ernst; Norsett; Wanner, Gerhard (1993): Solving Ordinary Differential Equations. Nonstiff Problems. 2nd edition. Springer Series in Comput. Math., vol. 8.

Phil Brubaker's ODEcalc ODEcalc for Windows: An Ordinary Differential Equation Calculator.

SLDRIVER (Interactive Sturm-Liouville Package)

The classical Sturm-Liouville problem (SLP) is

(-p(x) y')' + q(x) y = lambda w(x) y (1)

where y is a real function of x on a real interval a "Regular" SLPs generally pose few difficulties, but "singular" SLPs show great variety of behaviour: the spectrum can comprise both eigenvalues and continuous spectrum; the eigenvalues can form a finite set, or an infinite sequence that may be increasing, decreasing or two-sided, bounded or unbounded. Eigenvalues can lie within continuous spectrum. Good library software has been available for some years and has gradually enlarged the range of SLPs that can be reliably handled.

SLDRIVER is an interactive Fortran 90 program which supports exploration of a set of Sturm-Liouville problems with the four SL-solvers (whose source comes with the package): SLEIGN, SLEDGE, SL02F and SLEIGN2.