qmat.solvers.sdc
================

.. py:module:: qmat.solvers.sdc

.. autoapi-nested-parse::

   Functions to run SDC and evaluate its numerical error on simple problems.



Functions
---------

.. autoapisummary::

   qmat.solvers.sdc.solveDahlquistSDC
   qmat.solvers.sdc.errorDahlquistSDC
   qmat.solvers.sdc.getOrderSDC


Module Contents
---------------

.. py:function:: solveDahlquistSDC(lam, u0, tEnd, nSteps: int, nSweeps: int, Q: numpy.ndarray, QDelta: numpy.ndarray, weights=None, monitors=None)

   Solve the Dahlquist problem with SDC.

   :Parameters: * **lam** (*complex or float*) -- The :math:`\lambda` coefficient.
                * **u0** (*complex or float*) -- The initial solution :math:`u_0`.
                * **tEnd** (*float*) -- Final time :math:`T`.
                * **nSteps** (*int*) -- Number of time-step for the whole :math:`[0,T]` interval.
                * **nSweeps** (*int*) -- Number of SDC sweeps.
                * **Q** (*np.ndarray*) -- Quadrature matrix :math:`Q` used for SDC.
                * **QDelta** (*np.ndarray*) -- Approximate quadrature matrix :math:`Q_\Delta` used for SDC.
                  If three dimensional, use the first dimension for the sweep index.
                * **weights** (*np.ndarray, optional*) -- Quadrature weights to use for the step update.
                  If None, step update is not performed. The default is None.

   :returns: **uNum** -- Array containing the `nSteps+1` solutions :math:`\{u(0), ..., u(T)\}`.
   :rtype: np.ndarray


.. py:function:: errorDahlquistSDC(lam, u0, tEnd, nSteps, nSweeps, Q, QDelta, weights=None, uNum=None)

   Compute the time :math:`L_\infty` error of SDC.

   :Parameters: * **lam** (*complex or float*) -- The :math:`\lambda` coefficient.
                * **u0** (*complex or float*) -- The initial solution :math:`u_0`.
                * **tEnd** (*float*) -- Final time :math:`T`.
                * **nSteps** (*int*) -- Number of time-step for the whole :math:`[0,T]` interval.
                * **nSweeps** (*int*) -- Number of SDC sweeps.
                * **Q** (*np.ndarray*) -- Quadrature matrix :math:`Q` used for SDC.
                * **QDelta** (*np.ndarray*) -- Approximate quadrature matrix :math:`Q_\Delta` used for SDC.
                * **weights** (*np.ndarray, optional*) -- Quadrature weights to use for the step update.
                  If None, step update is not performed. The default is None.
                * **uNum** (*np.ndarray, optional*) -- Numerical solution, if not provided use the `solveDahlquist` method
                  to compute the solution. The default is None.

   :returns: The :math:`L_\infty` norm.
   :rtype: float


.. py:function:: getOrderSDC(coll, nSweeps, qDelta, stepUpdate)

   Give the expected order of SDC after a fixed number of iterations.

   :Parameters: * **coll** (:class:`qmat.qcoeff.collocation.Collocation`) -- The underlying `Collocation` class.
                * **nSweeps** (*int*) -- Number of sweeps for SDC.
                * **qDelta** (*str*) -- Type of the :math:`Q_\Delta` approximation used.
                * **stepUpdate** (*bool*) -- Wether or not the stepUpdate is done at the end.

   :returns: **order** -- Expected order of the SDC time-integration.
   :rtype: int