PDESystem & arguments

[B-LIE]

Trying out MethodOfLines.jl for industrial problem
I'm trying to make MOL work for gas-lifted oil production, where I have a liquid and gas entering the distributed volume via valves, and the mixture leaving via a valve. Essentially, one could say that I consider a PDAE = Partial and Differential Algebraic Equation. And in my case, the AEs can be found both in the eqs part and in the bcs part.

Thus, I have boundary conditions typically of form:

In other words: some of my BC equations relate distributed variables (form: u(..)) to boundaries, e.g.,

u(t,0) ~ 0

while others involve temporal functions (v(t)), e.g.,

v ~ sqrt(u(t,L) - 1)

[as abstracted examples].

By trial and error, I have found that MethodOfLines, does not allow me to define equations for such intermediate variables (e.g., v(t)) in the boundary conditions.

To get around this problem, I have to eliminate these intermediate variables (e.g., v(t)) from the BCs, leading to horribly complicated BCs (see my real examples above) which are error prone to express.

---

In the core form of using PDESystem:

@named gas_lift = PDESystem(eqs,bcs,domains,ivs,dvs,params)

it seems that I can not use the vector of variables (dvs above) in the same form as when I define the variables. What I mean is:

vars = @variables begin
    # Distributed variables
    m̌_ℓ(..),            [description = "Liquid bulk density, kg/m^3"]
    ṁ_ℓ(..),            [description = "Liquid mass flow rate, kg/s"]
    m̌_g(..),            [description = "Gas bulk density, kg/m^3"]
...

can not be used as the dvs argument. So I have to recreate the same information (more or less) in a new vector, where it seems like have to:

1. strip off the description
2. replace (..) with (t,x)
3. change e.g. x(t) to x

Not a horrible problem, but somewhat inconvenient.

---

It is not clear to me what form params must/should have. Following the more modern versions of ModelingToolkit, I have defined parameters as follows:

params = @parameters begin
    g = 9.81,               [description = "Acceleration of gravity, m/s^2"]
    R = 8.31,               [description = "Ideal gas constant, J/(K*mol)"]
    T = 280,                [description = "Temperature, assumed constant, K"]
...

It seems like this doesn't work. Is this because...

1. I need to strip of the description etc.?
2. The examples in the MethodOfLines documentation uses the form [g => 9.81, ...]. Is this the cause for the problem?

Describe the solution you’d like

In ModelingToolkit, a typical working call to ODESystem is:

@named model = ODESystem(eqs, iv, dvs, params)

where iv typically is t, while dependent variables dvs can be described in the "modern" MTK form, and the same for params.

It would be very nice if PDESystem could use the same basic form. Currently:

@named gas_lift = PDESystem(eqs,bcs,domains,ivs,dvs,params)

The call structure is very similar to that of ODESystem, but where eqs is expanded with bcs. It would be very nice if:

• params in PDESystem could be used similarly as in ODESystem, i.e., with description, macro @parameters, etc. Maybe it can?? But it doesn't work in my case.

• dvs in PDESystem coulduse the variables defined via @variables. There might be a problem with that, mainly due to the way dependent variables with more than one independent variable are specified: as u(..) -- while the dvs in the PDESystem argument seem to require that one explicitly specify the independent variables u(t,x).

It is possible to add purely temporal variables in the dvs. It is not clear to me where the initial conditions for those should be given if they describe the solution of an ODE -- in the @variables macro, or in the bcs?. I have used temporal variables in a much simpler example, and equated them with external forcing functions -- that works.

---

Finally, does the functions unknowns (or formerly states), observed, etc. work with the structure created by PDESystem?

Describe alternatives you’ve considered

Some AEs that naturally relate to the BCs can be moved to eqs. But not all (at least not in a simple way) ... consider:

Here, Y_a2t and Dp_at2 are only defined at x=0. So I need to describe these at the BC using temporal variables. OK -- a possible alternative could be to define them as distributed variables, i.e., Y_a2t(..) and Dp_a2t(..) and compute the variables in the eqs section, where they are never really used, and then just apply the values Y_a2t(t,0) and Dp_a2t(t,0) in the BCs.

But this seems like a waste in that one would artificially compute Y_a2t and Dp_a2t over the entire interior (x=(0,L)), but only need the values at the boundaries.

Additional context

Add any other context or screenshots about the feature request here.