Skip to content

High performance ordinary differential equation (ODE) and differential-algebraic equation (DAE) solvers, including neural ordinary differential equations (neural ODEs) and scientific machine learning (SciML)

License

Notifications You must be signed in to change notification settings

SciML/OrdinaryDiffEq.jl

Folders and files

NameName
Last commit message
Last commit date

Latest commit

891622a · Dec 28, 2022
Oct 15, 2022
Dec 28, 2022
Nov 19, 2022
Dec 28, 2022
Dec 25, 2022
Aug 29, 2022
Oct 22, 2016
Apr 7, 2022
Nov 21, 2019
May 29, 2020
Dec 28, 2022
Nov 19, 2022

Repository files navigation

OrdinaryDiffEq.jl

Join the chat at https://julialang.zulipchat.com #sciml-bridged Global Docs

codecov Build Status Build status

ColPrac: Contributor's Guide on Collaborative Practices for Community Packages SciML Code Style

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(1.01t),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:

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

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.