Expressing equations as the sum of a linear and a nonlinear operator.

[ali-ramadhan]

Building on what @jm-c said, I think we want to design an interface in which an equation is expressed as something like

φ_t = Lφ + Ν(φ)

where φ is the solution (an array of fields in this case, probably), L is a linear operator that can be treated implicitly (depending on the time-stepping scheme), and N(φ) is a nonlinear operator.

The challenge is including the pressure correction in this paradigm. I think it should be possible --- we may just have to consider only time-steppers that invoke the special pressure correction substep.

I think the design should be about more than just 'marking' the terms. We can tackle the problem of designing a time-stepper to step forward an equation of a certain form completely independent from the development of an algorithm to calculate nonlinear terms, pressure corrections, and linear terms for a specific kind of equation. Then we put the pieces together to solve the problem.

Originally posted by @glwagner in https://github.com/ali-ramadhan/Oceananigans.jl/issues/49#issuecomment-463209211

[ali-ramadhan]

Superseded by #259.