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‎.github/workflows/CompatHelper.yml

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- name: CompatHelper.main()
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env:
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GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
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run: julia -e 'using CompatHelper; CompatHelper.main(;subdirs=["", "docs", "test/down", "diagrams"])'
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run: julia -e 'using CompatHelper; CompatHelper.main(;subdirs=["", "docs", "test/downstream"])'

‎Project.toml

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name = "OrdinaryDiffEq"
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uuid = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
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authors = ["Chris Rackauckas <accounts@chrisrackauckas.com>", "Yingbo Ma <mayingbo5@gmail.com>"]
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version = "6.87.0"
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version = "6.89.0"
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[deps]
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ADTypes = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
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Adapt = "3.0, 4"
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ArrayInterface = "7"
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DataStructures = "0.18"
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DiffEqBase = "6.147"
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DiffEqBase = "6.154"
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DocStringExtensions = "0.9"
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EnumX = "1"
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ExponentialUtilities = "1"
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MacroTools = "0.5"
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MuladdMacro = "0.2.1"
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NonlinearSolve = "3"
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OrdinaryDiffEqAdamsBashforthMoulton = "1"
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OrdinaryDiffEqBDF = "1"
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OrdinaryDiffEqCore = "1"
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OrdinaryDiffEqDefault = "1"
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OrdinaryDiffEqDifferentiation = "1"
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OrdinaryDiffEqExplicitRK = "1"
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OrdinaryDiffEqExponentialRK = "1"
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OrdinaryDiffEqExtrapolation = "1"
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OrdinaryDiffEqFIRK = "1"
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OrdinaryDiffEqFeagin = "1"
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OrdinaryDiffEqFunctionMap = "1"
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OrdinaryDiffEqHighOrderRK = "1"
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OrdinaryDiffEqIMEXMultistep = "1"
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OrdinaryDiffEqLinear = "1"
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OrdinaryDiffEqLowOrderRK = "1"
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OrdinaryDiffEqLowStorageRK = "1"
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OrdinaryDiffEqNonlinearSolve = "1"
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OrdinaryDiffEqNordsieck = "1"
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OrdinaryDiffEqPDIRK = "1"
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OrdinaryDiffEqPRK = "1"
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OrdinaryDiffEqQPRK = "1"
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OrdinaryDiffEqRKN = "1"
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OrdinaryDiffEqRosenbrock = "1"
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OrdinaryDiffEqSDIRK = "1"
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OrdinaryDiffEqStabilizedIRK = "1"
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OrdinaryDiffEqSSPRK = "1"
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OrdinaryDiffEqStabilizedRK = "1"
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OrdinaryDiffEqSymplecticRK = "1"
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OrdinaryDiffEqTsit5 = "1"
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OrdinaryDiffEqVerner = "1"
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Polyester = "0.7"
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PreallocationTools = "0.4"
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PrecompileTools = "1"
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Preferences = "1.3"
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RecursiveArrayTools = "2.36, 3"
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Reexport = "1.0"
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SciMLBase = "2.27.1"
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SciMLBase = "2.53"
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SciMLOperators = "0.3"
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SciMLStructures = "1"
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SimpleNonlinearSolve = "1"

‎docs/common_first_steps.jl

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using Markdown
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function first_steps(name, solver)
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Markdown.parse("""## Installation
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To be able to access the solvers in `$name`, you must first install them use the Julia package manager:
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```julia
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using Pkg
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Pkg.add("$name")
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```
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This will only install the solvers listed at the bottom of this page.
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If you want to explore other solvers for your problem,
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you will need to install some of the other libraries listed in the navigation bar on the left.
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## Example usage
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```julia
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using $name
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function lorenz!(du, u, p, t)
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du[1] = 10.0 * (u[2] - u[1])
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u[2] = u[1] * (28.0 - u[3]) - u[2]
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du[3] = u[1] * u[2] - (8 / 3) * u[3]
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end
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u0 = [1.0; 0.0; 0.0]
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tspan = (0.0, 100.0)
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prob = ODEProblem(lorenz!, u0, tspan)
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sol = solve(prob, $solver())
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```""")
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end

‎docs/make.jl

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clean = true,
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doctest = false,
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modules = [OrdinaryDiffEq,
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OrdinaryDiffEq.OrdinaryDiffEqAdamsBashforthMoulton,
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OrdinaryDiffEq.OrdinaryDiffEqBDF,
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OrdinaryDiffEq.OrdinaryDiffEqDefault,
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OrdinaryDiffEq.OrdinaryDiffEqExplicitRK,
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OrdinaryDiffEq.OrdinaryDiffEqExponentialRK,
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OrdinaryDiffEq.OrdinaryDiffEqExtrapolation,
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OrdinaryDiffEq.OrdinaryDiffEqStabilizedRK,
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OrdinaryDiffEq.OrdinaryDiffEqStabilizedIRK,
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OrdinaryDiffEq.OrdinaryDiffEqLowStorageRK,
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OrdinaryDiffEq.OrdinaryDiffEqSSPRK,
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OrdinaryDiffEq.OrdinaryDiffEqFeagin,
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OrdinaryDiffEq.OrdinaryDiffEqSymplecticRK,
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OrdinaryDiffEq.OrdinaryDiffEqFIRK,
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OrdinaryDiffEq.OrdinaryDiffEqHighOrderRK,
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OrdinaryDiffEq.OrdinaryDiffEqIMEXMultistep,
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OrdinaryDiffEq.OrdinaryDiffEqLinear,
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OrdinaryDiffEq.OrdinaryDiffEqLowOrderRK,
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OrdinaryDiffEq.OrdinaryDiffEqLowStorageRK,
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OrdinaryDiffEq.OrdinaryDiffEqNordsieck,
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OrdinaryDiffEq.OrdinaryDiffEqPDIRK,
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OrdinaryDiffEq.OrdinaryDiffEqPRK,
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OrdinaryDiffEq.OrdinaryDiffEqQPRK,
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OrdinaryDiffEq.OrdinaryDiffEqRKN,
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OrdinaryDiffEq.OrdinaryDiffEqVerner,
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OrdinaryDiffEq.OrdinaryDiffEqRosenbrock,
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OrdinaryDiffEq.OrdinaryDiffEqSDIRK,
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OrdinaryDiffEq.OrdinaryDiffEqBDF,
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OrdinaryDiffEq.OrdinaryDiffEqDefault,
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OrdinaryDiffEq.OrdinaryDiffEqFIRK],
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OrdinaryDiffEq.OrdinaryDiffEqSSPRK,
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OrdinaryDiffEq.OrdinaryDiffEqStabilizedIRK,
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OrdinaryDiffEq.OrdinaryDiffEqStabilizedRK,
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OrdinaryDiffEq.OrdinaryDiffEqSymplecticRK,
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OrdinaryDiffEq.OrdinaryDiffEqTsit5,
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OrdinaryDiffEq.OrdinaryDiffEqVerner
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],
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warnonly = [:docs_block, :missing_docs, :eval_block],
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format = Documenter.HTML(analytics = "UA-90474609-3",
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assets = ["assets/favicon.ico"],
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canonical = "https://ordinarydiffeq.sciml.ai/stable/"),
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canonical = "https://ordinarydiffeq.sciml.ai/stable/",
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size_threshold_ignore = [joinpath("semiimplicit", "Rosenbrock.md"),
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joinpath("massmatrixdae", "Rosenbrock.md")]),
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pages = [
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"OrdinaryDiffEq.jl: ODE solvers and utilities" => "index.md",
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"Usage" => "usage.md",
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"Standard Non-Stiff ODEProblem Solvers" => [
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"nonstiff/explicitrk.md",
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"nonstiff/lowstorage_ssprk.md",
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"nonstiff/explicit_extrapolation.md",
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"nonstiff/nonstiff_multistep.md"
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"Explicit Solvers" => [
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"explicit/Tsit5.md",
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"explicit/Verner.md",
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"explicit/AdamsBashforthMoulton.md",
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"explicit/LowStorageRK.md",
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"explicit/SSPRK.md",
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"explicit/LowOrderRK.md",
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"explicit/HighOrderRK.md",
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"explicit/Feagin.md",
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"explicit/PRK.md",
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"explicit/QPRK.md",
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"explicit/Extrapolation.md"
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],
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"Standard Stiff ODEProblem Solvers" => [
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"stiff/firk.md",
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"stiff/rosenbrock.md",
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"stiff/stabilized_rk.md",
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"stiff/sdirk.md",
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"stiff/stiff_multistep.md",
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"stiff/implicit_extrapolation.md"
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"Semi-Implicit Solvers" => [
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"semiimplicit/Rosenbrock.md",
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"semiimplicit/StabalizedRK.md",
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"semiimplicit/ExponentialRK.md"
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],
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"Second Order and Dynamical ODE Solvers" => [
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"dynamical/nystrom.md",
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"dynamical/symplectic.md"
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"Implicit Solvers" => [
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"implicit/SDIRK.md",
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"implicit/FIRK.md",
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"implicit/BDF.md",
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"implicit/Extrapolation.md",
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"implicit/PDIRK.md",
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"implicit/Nordsieck.md"
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],
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"IMEX Solvers" => [
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"imex/imex_multistep.md",
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"imex/imex_sdirk.md"
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"imex/IMEXMultistep.md",
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"imex/StabalizedIRK.md",
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"imex/IMEXBDF.md"
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],
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"Dynamical ODE Explicit Solvers" => [
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"dynamicalodeexplicit/RKN.md",
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"dynamicalodeexplicit/SymplecticRK.md"
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],
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"Semilinear ODE Solvers" => [
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"semilinear/exponential_rk.md",
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"semilinear/magnus.md"
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"semilinear/ExponentialRK.md",
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"semilinear/Linear.md"
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],
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"Mass Matrix DAE Solvers" => [
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"massmatrixdae/Rosenbrock.md",
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"massmatrixdae/BDF.md"
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],
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"DAEProblem Solvers" => [
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"dae/fully_implicit.md"
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"Fully Implicit DAE Solvers" => [
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"fullyimplicitdae/BDF.md"
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],
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"Misc Solvers" => [
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"misc.md"

‎docs/src/dae/fully_implicit.md

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‎docs/src/dynamical/nystrom.md

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‎docs/src/dynamical/symplectic.md

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‎docs/src/dynamicalodeexplicit/RKN.md

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```@meta
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CollapsedDocStrings = true
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```
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# OrdinaryDiffEqRKN
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Second order solvers.
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To be able to access the solvers in `OrdinaryDiffEqRKN`, you must first install them use the Julia package manager:
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```julia
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using Pkg
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Pkg.add("OrdinaryDiffEqRKN")
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```
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This will only install the solvers listed at the bottom of this page.
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If you want to explore other solvers for your problem,
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you will need to install some of the other libraries listed in the navigation bar on the left.
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## Example usage
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```julia
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using OrdinaryDiffEqOrdinaryDiffEqRKN
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function HH_acceleration!(dv, v, u, p, t)
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x, y = u
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dx, dy = dv
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dv[1] = -x - 2x * y
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dv[2] = y^2 - y - x^2
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end
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initial_positions = [0.0, 0.1]
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initial_velocities = [0.5, 0.0]
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tspan = (0.0, 1.0)
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prob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)
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sol = solve(prob, Nystrom4(), dt = 1 / 10)
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```
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## Full list of solvers
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```@docs
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IRKN3
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IRKN4
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Nystrom4
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Nystrom4VelocityIndependent
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Nystrom5VelocityIndependent
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FineRKN4
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FineRKN5
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DPRKN4
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DPRKN5
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DPRKN6
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DPRKN6FM
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DPRKN8
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DPRKN12
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ERKN4
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ERKN5
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ERKN7
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RKN4
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```

‎docs/src/dynamicalodeexplicit/SymplecticRK.md

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```@meta
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CollapsedDocStrings = true
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```
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# OrdinaryDiffEqSymplecticRK
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A symplectic integrator is an integrator whose solution resides on a symplectic manifold.
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Because of discretization error, when it is solving a Hamiltonian system it doesn't get exactly the correct trajectory on the manifold.
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Instead, that trajectory itself is perturbed `O(Δtn)` for the order n from the true trajectory.
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Then there's a linear drift due to numerical error of this trajectory over time
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Normal integrators tend to have a quadratic (or more) drift, and do not have any good global guarantees about this phase space path (just local).
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What means is that symplectic integrators tend to capture the long-time patterns better than normal integrators because of this lack of drift and this almost guarantee of periodicity.
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## Installation
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To be able to access the solvers in `OrdinaryDiffEqSymplecticRK`, you must first install them use the Julia package manager:
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```julia
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using Pkg
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Pkg.add("OrdinaryDiffEqSymplecticRK")
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```
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This will only install the solvers listed at the bottom of this page.
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If you want to explore other solvers for your problem,
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you will need to install some of the other libraries listed in the navigation bar on the left.
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## Example usage
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```julia
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using OrdinaryDiffEqSymplecticRK
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function HH_acceleration!(dv, v, u, p, t)
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x, y = u
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dx, dy = dv
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dv[1] = -x - 2x * y
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dv[2] = y^2 - y - x^2
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end
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initial_positions = [0.0, 0.1]
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initial_velocities = [0.5, 0.0]
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tspan = (0.0, 1.0)
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prob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)
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sol = solve(prob, KahanLi8(), dt = 1 / 10)
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```
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## Full list of solvers
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```@docs
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SymplecticEuler
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VelocityVerlet
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VerletLeapfrog
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PseudoVerletLeapfrog
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McAte2
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Ruth3
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McAte3
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CandyRoz4
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McAte4
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CalvoSanz4
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McAte42
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McAte5
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Yoshida6
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KahanLi6
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McAte8
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KahanLi8
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SofSpa10
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```

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