qmat.qdelta.timestepping

Submodule for QDelta coefficients based on time-stepping scheme. Allows to build equivalent SDC sweeps as those introduced in the original paper from [Dutt, Greengard & Rokhlin, 2000].

Examples

>>> from qmat.qcoeff.collocation import Collocation
>>> coll = Collocation(nNodes=4, nodeType="LEGENDRE", quadType="RADAU-RIGHT")
>>>
>>> from qmat import genQDeltaCoeffs
>>> QDelta = genQDeltaCoeffs("IE", nodes=coll.nodes)
>>>
>>> from qmat.qdelta.timestepping import TRAP
>>> gen = TRAP(nodes=coll.nodes)
>>> SDelta, dTau = gen.genCoeffs(form="N2N", dTau=True)

Classes

TimeStepping	Base class for time-stepping based \(Q_\Delta\) approximations
BE	Approximation based on Backward Euler steps between the nodes
FE	Approximation based on Forward Euler steps between the nodes
TRAP	Approximation based on Trapezoidal Rule between the nodes
BEPAR	Approximation based on parallel Backward Euler steps from zero to nodes
TRAPAR	Approximation based on parallel Trapezoidal Rule from zero to nodes

Module Contents

class TimeStepping(nodes, tLeft=0, **kwargs)

Base class for time-stepping based \(Q_\Delta\) approximations

Parameters:

• nodes (list-like) – Normalized nodes in increasing order.

• tLeft (float, optional) – Left bound for the nodes. The default is 0.

• **kwargs – Additional parameters given in a generic call, ignored by this class.

deltas: numpy.ndarray

Differences between nodes

nodes: numpy.ndarray

Array of normalized nodes

tLeft: float = 0

Left bound for the nodes

static extractParams(qGen: qmat.qdelta.QGenerator) → dict

Extract from a \(Q\)-generator object all parameters required to instantiate the \(Q_\Delta\)-generator

property size: int

Dimension of the approximated \(Q\)-coefficients (number of nodes)

class BE(nodes, tLeft=0, **kwargs)

Approximation based on Backward Euler steps between the nodes

aliases = ['IE']

computeQDelta(k=None)

Compute and returns the \(Q_\Delta\) matrix, has to be implemented in the specialized class.

Parameters:

k (int, optional) – Iteration number of the approximation. The default is None.

Returns:

QDelta

Return type:

np.ndarray

class FE(nodes, tLeft=0, **kwargs)

Approximation based on Forward Euler steps between the nodes

aliases = ['EE']

computeQDelta(k=None)

Compute and returns the \(Q_\Delta\) matrix, has to be implemented in the specialized class.

Parameters:

k (int, optional) – Iteration number of the approximation. The default is None.

Returns:

QDelta

Return type:

np.ndarray

property dTau: numpy.ndarray

The \(\delta_\tau\) coefficients associated to \(Q_\Delta\)

class TRAP(nodes, tLeft=0, **kwargs)

Approximation based on Trapezoidal Rule between the nodes

aliases = ['CN']

computeQDelta(k=None)

Compute and returns the \(Q_\Delta\) matrix, has to be implemented in the specialized class.

Parameters:

k (int, optional) – Iteration number of the approximation. The default is None.

Returns:

QDelta

Return type:

np.ndarray

property dTau: numpy.ndarray

The \(\delta_\tau\) coefficients associated to \(Q_\Delta\)

class BEPAR(nodes, tLeft=0, **kwargs)

Approximation based on parallel Backward Euler steps from zero to nodes

aliases = ['IEpar']

computeQDelta(k=None)

Compute and returns the \(Q_\Delta\) matrix, has to be implemented in the specialized class.

Parameters:

k (int, optional) – Iteration number of the approximation. The default is None.

Returns:

QDelta

Return type:

np.ndarray

class TRAPAR(nodes, tLeft=0, **kwargs)

Approximation based on parallel Trapezoidal Rule from zero to nodes

computeQDelta(k=None)

Compute and returns the \(Q_\Delta\) matrix, has to be implemented in the specialized class.

Parameters:

k (int, optional) – Iteration number of the approximation. The default is None.

Returns:

QDelta

Return type:

np.ndarray

property dTau: numpy.ndarray

The \(\delta_\tau\) coefficients associated to \(Q_\Delta\)