# ODE with Jacobian

This example shows how to use ComponentArrays for composing Jacobian update functions as well as ODE functions. For most practical purposes, it is generally easier to use automatic differentiation libraries like ForwardDiff.jl, ReverseDiff.jl, or Zygote.jl for calculating Jacobians. Although those libraries all work with ComponentArrays, this is a nice way to handle it if you already have derived analytical Jacobians.

Note using plain symbols to index into ComponentArrays is still pretty slow. For speed, all symbolic indices should be wrapped in a Val like D[Val(:x), Val(:y)].

using ComponentArrays
using DifferentialEquations
using Parameters: @unpack

tspan = (0.0, 20.0)

## Lorenz system
function lorenz!(D, u, p, t; f=0.0)
@unpack σ, ρ, β = p
@unpack x, y, z = u

D.x = σ*(y - x)
D.y = x*(ρ - z) - y - f
D.z = x*y - β*z
return nothing
end
function lorenz_jac!(D, u, p, t)
@unpack σ, ρ, β = p
@unpack x, y, z = u

D[:x,:x] = -σ
D[:x,:y] = σ

D[:y,:x] = ρ
D[:y,:y] = -1
D[:y,:z] = -x

D[:z,:x] = y
D[:z,:y] = x
D[:z,:z] = -β
return nothing
end

lorenz_p = (σ=10.0, ρ=28.0, β=8/3)
lorenz_ic = ComponentArray(x=0.0, y=0.0, z=0.0)
lorenz_fun = ODEFunction(lorenz!, jac=lorenz_jac!)
lorenz_prob = ODEProblem(lorenz_fun, lorenz_ic, tspan, lorenz_p)

## Lotka-Volterra system
function lotka!(D, u, p, t; f=0.0)
@unpack α, β, γ, δ = p
@unpack x, y = u

D.x =  α*x - β*x*y + f
D.y = -γ*y + δ*x*y
return nothing
end
function lotka_jac!(D, u, p, t)
@unpack α, β, γ, δ = p
@unpack x, y = u

D[:x,:x] = α - β*y
D[:x,:y] = -β*x

D[:y,:x] = δ*y
D[:y,:y] = -γ + δ*x
return nothing
end

lotka_p = (α=2/3, β=4/3, γ=1.0, δ=1.0)
lotka_ic = ComponentArray(x=1.0, y=1.0)
lotka_fun = ODEFunction(lotka!, jac=lotka_jac!)
lotka_prob = ODEProblem(lotka_fun, lotka_ic, tspan, lotka_p)

## Composed Lorenz and Lotka-Volterra system
function composed!(D, u, p, t)
c = p.c #coupling parameter
@unpack lorenz, lotka = u

lorenz!(D.lorenz, lorenz, p.lorenz, t, f=c*lotka.x)
lotka!(D.lotka, lotka, p.lotka, t, f=c*lorenz.x)
return nothing
end
function composed_jac!(D, u, p, t)
c = p.c
@unpack lorenz, lotka = u

lorenz_jac!(@view(D[:lorenz,:lorenz]), lorenz, p.lorenz, t)
lotka_jac!(@view(D[:lotka,:lotka]), lotka, p.lotka, t)

@view(D[:lorenz,:lotka])[:y,:x] = -c
@view(D[:lotka,:lorenz])[:x,:x] = c
return nothing
end

comp_p = (lorenz=lorenz_p, lotka=lotka_p, c=0.01)
comp_ic = ComponentArray(lorenz=lorenz_ic, lotka=lotka_ic)
comp_fun = ODEFunction(composed!, jac=composed_jac!)
comp_prob = ODEProblem(comp_fun, comp_ic, tspan, comp_p)

## Solve problem
# We can solve the composed system...
comp_sol = solve(comp_prob, Rodas5())

# ...or we can unit test one of the component systems
lotka_sol = solve(lotka_prob, Rodas5())