IMEXEuler() is a multistep version, not the classical additive Runge-Kutta method version

[ranocha]

I ran into this issue while working on ranocha/BSeries.jl#53. Given an additive partition of an ODE of the form

u'(t) = f1(u) + f2(u)

there are two different versions of "the" IMEX Euler method. What I would have expected is the classical additive Runge-Kutta method version as described by Araújo, Murua, Sanz-Serna (1997). This version reads as

y1   = uold + dt * f1(y1)
unew = uold + dt * (f1(y1) + f2(y1))

In OrdinaryDiffEq.jl, IMEXEuler() is the single-step version of the multistep methods of Ascher, Ruuth, and Wetton (1995), which can be written as

unew = uold + dt * (f1(unew) + f2(uold))

I guess it would be good to document this more explicitly - I will prepare a PR later today.

I think it would also make sense to implement an IMEXEulerARK or AdditiveIMEXEuler method (or however it would be called). What do you think?

[ChrisRackauckas]

Yes, you're right. And I actually completely forgot we had done this one in the style of SBDF1. Thanks. And you mean:

y1   = uold + dt * f1(y1)
unew = uold + dt * (f1(unew) + f2(y1))

yeah, it's whether the explicit portion is evaluated at an approximate unew or whether it's evaluated at uold. Both can probably share the same implementation with just a conditional in there to cut down on code, but agreed it's a bit confusing that there's these two "the IMEXEuler" and we should have both and clarify it.

[ranocha]

I agree with you that this is confusing and it would be great to have both variants 👍