OrdinaryDiffEq.jl

OrdinaryDiffEq.jl is a component package in the DifferentialEquations ecosystem. It holds the ordinary differential equation solvers and utilities. While completely independent and usable on its own, users interested in using this functionality should check out DifferentialEquations.jl.

Installation

Assuming that you already have Julia correctly installed, it suffices to import OrdinaryDiffEq.jl in the standard way:

import Pkg;
Pkg.add("OrdinaryDiffEq");

API

OrdinaryDiffEq.jl is part of the SciML common interface, but can be used independently of DifferentialEquations.jl. The only requirement is that the user passes an OrdinaryDiffEq.jl algorithm to solve. For example, we can solve the ODE tutorial from the docs using the Tsit5() algorithm:

using OrdinaryDiffEq
f(u, p, t) = 1.01 * u
u0 = 1 / 2
tspan = (0.0, 1.0)
prob = ODEProblem(f, u0, tspan)
sol = solve(prob, Tsit5(), reltol = 1e-8, abstol = 1e-8)
using Plots
plot(sol, linewidth = 5, title = "Solution to the linear ODE with a thick line",
    xaxis = "Time (t)", yaxis = "u(t) (in μm)", label = "My Thick Line!") # legend=false
plot!(sol.t, t -> 0.5 * exp(1.01 * t), lw = 3, ls = :dash, label = "True Solution!")

That example uses the out-of-place syntax f(u,p,t), while the in-place syntax (more efficient for systems of equations) is shown in the Lorenz example:

using OrdinaryDiffEq
function lorenz!(du, u, p, t)
    du[1] = 10.0 * (u[2] - u[1])
    du[2] = u[1] * (28.0 - u[3]) - u[2]
    du[3] = u[1] * u[2] - (8 / 3) * u[3]
end
u0 = [1.0; 0.0; 0.0]
tspan = (0.0, 100.0)
prob = ODEProblem(lorenz!, u0, tspan)
sol = solve(prob, Tsit5())
using Plots;
plot(sol, idxs = (1, 2, 3))

Very fast static array versions can be specifically compiled to the size of your model. For example:

using OrdinaryDiffEq, StaticArrays
function lorenz(u, p, t)
    SA[10.0 * (u[2] - u[1]), u[1] * (28.0 - u[3]) - u[2], u[1] * u[2] - (8 / 3) * u[3]]
end
u0 = SA[1.0; 0.0; 0.0]
tspan = (0.0, 100.0)
prob = ODEProblem(lorenz, u0, tspan)
sol = solve(prob, Tsit5())

For "refined ODEs", like dynamical equations and SecondOrderODEProblems, refer to the DiffEqDocs. For example, the harmonic oscillator equations can be solved using symplectic methods. The harmonic oscillator is described by:

$$\ddot{x} + \omega^2 x = 0$$

which is equivalent to the first-order system:

$$\dot{x} = v$$ $$\dot{v} = -\omega^2 x$$

using OrdinaryDiffEq
function harmonic_oscillator!(dv, v, u, p, t)
    ω = p[1]
    dv[1] = -ω^2 * u[1]
end
ω = 2.0  # angular frequency
initial_position = [1.0]
initial_velocity = [0.0]
tspan = (0.0, 10.0)
prob = SecondOrderODEProblem(harmonic_oscillator!, initial_velocity, initial_position, tspan, [ω])
sol = solve(prob, VelocityVerlet(), dt = 1 / 100)
using Plots
plot(sol, idxs = (1, 2), label = "Phase space", xaxis = "Position", yaxis = "Velocity")

For more complex dynamical systems, such as the Hénon-Heiles potential, symplectic integrators preserve the structure of Hamiltonian dynamics. In DiffEqTutorials.jl we show how to solve these equations of motion:

function HH_acceleration!(dv, v, u, p, t)
    x, y = u
    dx, dy = dv
    dv[1] = -x - 2 * x * y
    dv[2] = y^2 - y - x^2
end
initial_positions = [0.0, 0.1]
initial_velocities = [0.5, 0.0]
prob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)
sol2 = solve(prob, KahanLi8(), dt = 1 / 10);

Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).

Available Solvers

For the list of available solvers, please refer to the DifferentialEquations.jl ODE Solvers, Dynamical ODE Solvers, and the Split ODE Solvers pages.