Allowing discrete Difference equations in ODESystem

[sharanry]

The proposed plan is to support difference equations containing DIfference operator in ODESystems using PeriodicCallbacks.

This could involve the following workflow:

Difining difference and differential equations together:

using ModelingToolkit
@parameters t σ ρ β
@variables x(t) y(t) z(t)
δ = Differential(t)
D = Difference(t; dt=0.01)

eqs = [
       δ(x) ~ σ*(y-x),
       δ(y) ~ x*(ρ-z)-y,
       δ(z) ~ x*y - β*z,
       D(y) ~ x
       ]

Manually mentioning which are the difference variables while defining the ODESystem using an optional kwarg:

sys = ODESystem(eqs,t,[x,y,z],[σ,ρ,β]; differencevars=[y])

Producing the ODEProblem and the callback

prob = ODEProblem(sys,u0,tspan,p,jac=true)
difference_cb = generate_difference_cb(sys)

Mention the callback while solving the problem:

solve(prob, Rodas5(); cb=difference_cb)

@ChrisRackauckas @YingboMa
Does this kind of workflow seem reasonable to begin with?
OR do we want more automation that might involve API changes?
OR did you have something else in mind?

[ChrisRackauckas]

No, just put the callback in the problem. The user shouldn't have to deal with it. And automatically detect the difference equations and move them from eqs.