Add "voltage as a state" option

[rtimms]

Description

Adds an option "voltage as a state" that can be "false" (default) or "true". If "true" adds an explicit algebraic equation for the voltage. This can be used to reformulate the SPM(e) as a DAE rather than and ODE. This increases the solve time but evaluating the voltage as an explicit state rather than as a function of other state variables can be faster if the time eval is dense.

import pybamm
import numpy as np

t = np.linspace(0, 3600, 100000)
for value in ["true", "false"]:
    model = pybamm.lithium_ion.SPMe({"voltage as a state": value})
    sim = pybamm.Simulation(model, solver=pybamm.IDAKLUSolver())
    sim.solve([0, 3600])
    print("voltage as a state:", value)
    solve_time = sim.solution.solve_time
    print("solve time:", solve_time)
    timer = pybamm.Timer()
    sim.solution["Voltage [V]"](t)
    eval_time = timer.time()
    print("eval time:", eval_time)
    print("total time:", solve_time + eval_time, "\n")

returns

voltage as a state: true
solve time: 3.293 ms
eval time: 8.836 ms
total time: 12.129 ms 

voltage as a state: false
solve time: 2.311 ms
eval time: 687.892 ms
total time: 690.204 ms 

Fixes # (issue)

Type of change

Please add a line in the relevant section of CHANGELOG.md to document the change (include PR #) - note reverse order of PR #s. If necessary, also add to the list of breaking changes.

• New feature (non-breaking change which adds functionality)
• Optimization (back-end change that speeds up the code)
• Bug fix (non-breaking change which fixes an issue)

Key checklist:

• No style issues: $ pre-commit run (or $ nox -s pre-commit) (see CONTRIBUTING.md for how to set this up to run automatically when committing locally, in just two lines of code)
• All tests pass: $ python run-tests.py --all (or $ nox -s tests)
• The documentation builds: $ python run-tests.py --doctest (or $ nox -s doctests)

You can run integration tests, unit tests, and doctests together at once, using $ python run-tests.py --quick (or $ nox -s quick).

Further checks:

• Code is commented, particularly in hard-to-understand areas
• Tests added that prove fix is effective or that feature works

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.42%. Comparing base (d48f8c6) to head (0648a91).
Report is 160 commits behind head on develop.

Additional details and impacted files

@@             Coverage Diff             @@
##           develop    #4507      +/-   ##
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- Coverage    99.42%   99.42%   -0.01%     
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  Files          301      299       -2     
  Lines        22736    22715      -21     
===========================================
- Hits         22605    22584      -21     
  Misses         131      131              

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[valentinsulzer]

Looks good, thanks. Those timings are crazy - is that even with the latest changes from Marc?

[rtimms]

Yep. Evaluating all those asinh terms is slow if you have a large t_eval.