Separate differential states that can be explicitly integrated

[valentinsulzer]

In all of the battery models, there is the state Q=Variable("Discharge capacity [A.h]"), which is just the integral of the current. This variable does not appear in any of the rhs or algebraic equations.

This might be making the problem unnecessarily stiff. In particular, this is causing problems for the "Rest" steps, where dQ/dt = 0 which makes the first line of the Jacobian all zero and hence the Jacobian singular.

We should not be solving for Q at the same time as the rest of the model, since we can just do a post-processing step where we integrate the current. However, we still want to be able to access Q just like the other variables with solution["Discharge capacity [A.h]"]. I can think of two approaches:

1. Do this implicitly behind the scenes. When we discretize, notice that Q does not appear anywhere in the rhs or algebraic equations and moving it to a different dictionary rhs_explicit_integration
2. Do this explicitly by putting Q in a different dictionary rhs_explicit_integration when creating the model

Then the rhs_explicit_integration dictionary can be handled by the solver after the other variables have been solved for, e.g. using cumtrapz. Ideally, this wouldn't even be done until the user calls solution["Discharge capacity [A.h]"] (i.e. the Solution object handles it, maybe by adding some functionality to ProcessedVariable)

[valentinsulzer]

@abillscmu

[valentinsulzer]

This is a specific case of a more general feature, which would be to notice when blocks of the jacobian are independent of each other (e.g. the two particles in the SPM with constant current, the two particles and the electrolyte in the SPMe with constant current) and solve them separately in parallel. I don't know if any of the solvers are already doing this automatically

[valentinsulzer]

Option 1 is better. More specifically the model should be solved in 3 steps:

1. Identify variables where the rhs is a constant. These should be replaced with explicit functions of time, e.g.

model.rhs = {Q: 4}
model.initial_conditions = {Q: 1}

gets replaced with Q = 1 + 4*(t-t0) where t0 is an input parameter
2. Identify variables that do not appear in any rhs (e.g. the discharge capacity) and put those aside. Solve other variables first
3. Solve the remaining variables by explicit integration (e.g. trapz)

A preliminary implementation could skip step 1 and just do steps 2 and 3 first.