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:

julia 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:

julia 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(1.01t),lw=3,ls=:dash,label="True Solution!")

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

julia 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:

julia 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, in DiffEqTutorials.jl we show how to solve equations of motion using symplectic methods:

julia function HH_acceleration!(dv,v,u,p,t) x,y = u dx,dy = dv dv[1] = -x - 2x*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.