qmat.playgrounds.tibo.imex

Classes

DiffOpIMEX	Base class for an IMEX differential operator
CoeffSolverIMEX	Coefficient based solver class for IMEX differential operators.

Module Contents

class DiffOpIMEX(u0)

Base class for an IMEX differential operator

abstractmethod evalF2(u: numpy.ndarray, t: float, out: numpy.ndarray)

Evaluate f_EX(u,t) and store the result into out

class CoeffSolverIMEX(diffOp, tEnd=1, nSteps=1, t0=0)

Coefficient based solver class for IMEX differential operators.

diffOp: DiffOpIMEX = None

Differential Operator implementing \(f(u,t)\).

evalF2(u: numpy.ndarray, t: float, out: numpy.ndarray)

solve(QI, wI, QE, wE, uNum=None)

Solve the ODE considering lower-triangular \(Q\) coefficients.

This is equivalent to the classical implementation of a generic Runge-Kutta method using its Butcher table. For each time-step, it defines a node solution (or stage) \(u_{m}\) that is solved using previously computed node solution :

\[u_{m} - \Delta{t}q_{m,m}f(u_m,t_m) = u_0 + \Delta{t}\sum_{j=1}^{m-1}q_{m,j}f(u_j, t_j),\]

where \(t_m = t_0 + \tau_m\) and \(q_{i,j}\) are the coefficients \(Q\). Finally, the step update is done using all computed node solutions :

\[u(t_0+\Delta{t}) \simeq u_0 + \sum_{m=1}^{M} \omega_{m} f(u_m, t_m),\]

where \(\omega_{m}\) are the weights associated to the \(Q\)-coefficients. If no weights are provided, then it simply uses the last node solution for the step update :

\[u(t_0+\Delta{t}) \simeq u_M\]

Parameters:

• Q (np.2darray-like) – The lower-triangular \(Q\)-coefficients matrix.

• weights (np.1darray-like) – The associated :math:omega_{m}` weights. If not provided, use the last node solution for the update (requires \(\tau_{M} = 1\)).

• uNum (np.ndarray, optional) – Array of shape (nSteps+1,*uShape), that can be use to store the result and avoid creating it internally. The default is None.

• tInit (float, optional) – Initial time offset to be added to solver’s own t0 for successive solve calls. The default is 0.

Returns:

uNum – Array of shape (nSteps+1,*uShape) that stores the solution at each time-step.

Return type:

np.ndarray