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JuliaDiffEq / ODE.jl


Assorted basic Ordinary Differential Equation solvers


Various basic Ordinary Differential Equation solvers implemented in Julia.

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Pull requests are always highly welcome to fix bugs, add solvers, or anything else!

API discussions

There are currently discussions about how the Julian API for ODE solvers should look like, and the current documentation is more like a wishlist than a documentation. The API has changed considerably since the initial v0.1 release, so be carefull when you upgrade to v0.2 or later versions.

Current status of the project

The first release, v0.1, contains the basic functionality that was moved here when the package was originally moved from Base. Although quite poorly tested, at least some of the functionality is quite reliable. However, that version is almost entirely undocumented, and will probably stay that way.

Since then, quite a lot has happened in the package, and the best way to use current ODE.jl is by checking out the latest master with Pkg.checkout("ODE"). By doing so, you get access to a new, better API -- but be careful; several breaking changes have been introduced since v0.1. Therefore, the best way to learn the current API is to read the source. (The documentation in is to be regarded as a wishlist, where some but not all of the features have been implemented as of yet).

Currently, ODE exports the following adaptive solvers:

  • ode23: 2nd order adaptive solver with 3rd order error control, using the Bogacki–Shampine coefficients

  • ode45: 4th order adaptive solver with 5th order error control, using the Dormand Prince coefficients. Fehlberg and Cash-Karp coefficients are also available.

  • ode78: 7th order adaptive solver with 8th order error control, using the Fehlberg coefficients.

  • ode23s: 2nd/3rd order adaptive solver for stiff problems, using a modified Rosenbrock triple.

all of which have the following basic API:

tout, yout = odeXX(F, y0, tspan; keywords...)

to solve the explicitly defined ODE by dy/dt = F(t,y). A few other solvers are also exported, see the source code for details.

The adaptive solvers accept the following keywords

  • norm: user-supplied norm for determining the error E (default Base.vecnorm),
  • abstol and/or reltol: an integration step is accepted if E <= abstol || E <= reltol*abs(y) (defaults reltol = 1e-5, abstol = 1e-8),
  • maxstep, minstep and initstep: determine the maximal, minimal and initial integration step (defaults minstep=|tspan[end] - tspan[1]|/1e9, maxstep=|tspan[end] - tspan[1]|/2.5 and automatic initial step estimation).
  • points=:all (default): output is given for each value in tspan as well as for each intermediate point the solver used.
  • points=:specified: output is given only for each value in tspan.

Additionally, ode23s solver supports

  • jacobian = G(t,y): user-supplied Jacobian G(t,y) = dF(t,y)/dy (default estimate by finite-difference method).

There are also fixed step Runge-Kutta and Rosenbrock solvers available.


The examples directory contain a few notebooks that show how to get started. You can also see them here: