PETSc interface fails with Lagrange-Legendre collocation

[dallan-keylogic]

Presently, the PETSc interface throws an error when one attempts to integrate a DAE discretized with Lagrange-Legendre discretization:

    import pyomo.environ as pyo
    import pyomo.dae as pyodae
    from idaes.core.solvers import petsc

    m = pyo.ConcreteModel()

    m.time = pyodae.ContinuousSet(initialize=(0.0, 1.0, 2.0))
    m.x = pyo.Var(m.time)
    m.u = pyo.Var(m.time)
    m.dxdt = pyodae.DerivativeVar(m.x, wrt=m.time)

    def diff_eq_rule(m, t):
        return m.dxdt[t] == m.x[t] ** 2 - m.u[t]

    m.diff_eq = pyo.Constraint(m.time, rule=diff_eq_rule)

    discretizer = pyo.TransformationFactory("dae.collocation")
    discretizer.apply_to(m, nfe=2, scheme="LAGRANGE-LEGENDRE")

    m.u.fix(1.0)
    m.x[0].fix(0.0)

    res = petsc.petsc_dae_by_time_element(
        m,
        time=m.time,
        between=[0.0, 2.0],
        ts_options={
            "--ts_type": "beuler",
            "--ts_dt": 3e-2,
        },
        skip_initial=True,  # With u and x fixed, no variables to solve for at t0
        calculate_derivatives=True,
    )

produces a log with a degrees-of-freedom error:

2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Called fg_read, err: 0 (0 is good)
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: ---------------------------------------------------
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: DAE: 1
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Reading nl file: C:\Users\dallan\AppData\Local\Temp\tmpwptc4we3.pyomo.nl
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of constraints: 2
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of nonlinear constraints: 1
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of linear constraints: 1
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of inequalities: 0
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of variables: 6
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of integers: 0
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of binary: 0
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of objectives: 0 (Ignoring)
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of non-zeros in Jacobian: 7
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Explicit time variable: 0
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of derivatives: 1
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of differential vars: 1
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of algebraic vars: 4
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of state vars: 5
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: Number of degrees of freedom: 3
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: ---------------------------------------------------
2024-02-19 14:06:27 [INFO] idaes.solve.petsc-dae: ERROR: Degrees of freedom not equal to 0

I suspect the code @eslickj wrote to collect differential equations and discretization equations is failing because this discretization method has continuity equations in addition to normal discretization equations. How exactly we want to handle this case is up for discussion.