Custom Butcher Tableau for IMEX Schemes

[jbigmanUT]

Do you have any interest in allowing for user-specified Butcher tableau in DIRKIMEXMethod? I have created the following class

    class Custom_DIRK_IMEX(ButcherTableau):
        """Allows for custom tableau to be used in IMEX schemes"""
        def __init__(self, A: np.ndarray, b: np.ndarray, c: np.ndarray, A_hat: np.ndarray,
                     b_hat: np.ndarray, c_hat: np.ndarray, order = None):

            assert A.shape == A_hat.shape, "A_hat and A must have the same shape"
            assert len(b) == len(b_hat) - 1, "b_hat must have one more element than b"
            assert len(c) == len(c_hat) - 1, "c_hat must have one more element than c"
            assert np.all(c == c_hat[1:]), "Implicit and explicit schemes must share abscissae"

            super(Custom_DIRK_IMEX, self).__init__(A, b, None, c, order, None, None)
            self.A_hat = A_hat
            self.b_hat = b_hat
            self.c_hat = c_hat
            self.is_dirk_imex = True  # Mark this as a DIRK-IMEX scheme

which I then use in the following manner

    gamma = 1 - np.sqrt(2)/2
    delta = 1 - 1/(2*gamma)
    a_expl = np.array([[gamma, 0], [delta, 1 - delta]], dtype=float)
    a_impl = np.array([[gamma, 0], [1 - gamma, gamma]], dtype=float)
    b_expl = np.array([delta, 1 - delta, 0], dtype=float)
    b_impl =  np.array([1 - gamma, gamma], dtype=float)
    c_expl = np.array([0, gamma, 1], dtype=float)
    c_impl = np.array([gamma, 1], dtype=float)

    tab_imex = Custom_DIRK_IMEX(a_impl, b_impl, c_impl, a_expl, b_expl, c_expl)

    imex_ts = DIRKIMEXMethod(F_impl, F_expl, tab_imex, t_, dt_, u_initial)

[ScottMacLachlan]

Modulo naming things consistently, having a more generic constructor for DIRK-IMEX methods seems like good functionality, especially since we've been having internal discussions about adding methods that don't conform to the Ascher-Ruuth-Spiteri naming convention.

If this method is the only one you're interested in adding, I would say that you could just PR an addition to dirk_imex_tableaux.py, since this looks like just the (2,2,2) Tableau from Ascher-Ruuth-Spiteri, so it would fit in the factory that we already have.

Longer-term (or if you have more methods of interest), I'd probably suggest that we use DIRK_IMEX as the generic constructor, and rename what we have there now as ARS_DIRK_IMEX or similar, as we're already thinking towards implementing a wider selection of methods (such as those considered here).

[rckirby]

I agree with @ScottMacLachlan here -- The idea that our DIRK/IMEX tableaux should fit in some class hierarchy good one. To make things more parallel with how we do standard RK methods, I recommend putting in a base DIRKIMEX class (or some such name) that just holds the various tableaux, renaming the existing ones based on Ascher, Ruuth, and Spiteri and having them inherit from that. Then, you should put in at least one other DIRK-type IMEX to prove you needed to do that and then chase down the name change through demos & tests.