qmat.qcoeff

Defines the base abstract class to generate \(Q\)-coefficients (Butcher tables) : the QGenerator 🚀

Each submodule contains specializations of this class for many kind of methods :

collocation : Collocation based
butcher : Runge-Kutta based (Butcher tables)

Submodules

Attributes

Q_GENERATORS

Dictionary containing all specialized QGenerator classes, with all their aliases

Classes

QGenerator

Base abstract class for all \(Q\)-coefficients generators

Functions

`register`(→ type[T])	Class decorator to register a specialized `QGenerator` class in qmat
`genQCoeffs`(qType[, form, hCoeffs, embedded])	Generate \(Q\)-coefficients for a given method

Package Contents

class QGenerator[source]

Base abstract class for all \(Q\)-coefficients generators

classmethod getInstance()[source]: Provide an instance of this QGenerator using default parameters.

abstract property nodes: Nodes \(\tau\) (\(c\) coefficients in Butcher table)

abstract property Q: \(Q\) coefficients (\(A\) Butcher table)

abstract property weights: Weights \(\omega\) (\(b\) coefficients in Butcher table)

abstract property weightsEmbedded: Weights for a secondary lower order method from the same stages.

property nNodes: int: Number of nodes (or stages) for this QGenerator

property rightIsNode: bool: Wether or not the last nodes is the right boundary

property T: numpy.ndarray: Transfer matrix from zero-to-nodes to node-to-node

property S: numpy.ndarray: Quadrature matrix in node to node (N2N)

property Tinv: numpy.ndarray: Transfer matrix from node-to-node to zero-to-node

property hCoeffs: numpy.ndarray: \(h\) interpolation coefficients for the right boundary

genCoeffs(form='Z2N', hCoeffs=False, embedded=False)[source]

Generate \(Q\)-coefficients of this QGenerator object.

Parameters:

form (str, optional) – Write coefficients in zero-to-nodes (Z2N) or node-to-node (N2N). The default is “Z2N”.
hCoeffs (bool, optional) – Wether or not returning the \(h\) coefficients. The default is False.
embedded (bool, optional) – Wether or not returning the embedded \(h\) coefficients. The default is False.

Returns:

out – Contains (nodes, weights, Q). If hCoeffs=True, returns (nodes, weights, Q, hCoeffs). If embedded=True, weights is a 2xM array containing embedded weights in weights[1].

Return type:

tuple

abstract property order: Global convergence order of the method

property orderEmbedded: int: Global convergence order of the associated embedded method

solveDahlquist(lam, u0, tEnd, nSteps, useEmbeddedWeights=False)[source]

Solve the Dahlquist test problem

\[\frac{du}{dt} = \lambda u, \quad t \in [0, T], \quad u(0)=u_0\]

Parameters:

lam (complex or float) – The \(\lambda\) coefficient.
u0 (complex or float) – The initial solution \(u_0\).
tEnd (float) – Final time \(T\).
nSteps (int) – Number of time-step for the whole \([0,T]\) interval.
useEmbeddedWeights (bool, optional) – Wether or not use the embedded weights for the step update. The default is False.

Returns:

uNum – Array containing the nSteps+1 solutions \(\{u(0), ..., u(T)\}\).

Return type:

np.ndarray

errorDahlquist(lam, u0, tEnd, nSteps, uNum=None, useEmbeddedWeights=False)[source]

Compute \(L_\infty\) error in time for the Dahlquist problem

Parameters:

lam (complex or float) – The \(\lambda\) coefficient.
u0 (complex or float) – The initial solution \(u_0\).
tEnd (float) – Final time \(T\).
nSteps (int) – Number of time-step for the whole \([0,T]\) interval.
uNum (np.ndarray, optional) – Numerical solution, if not provided use the solveDahlquist method to compute the solution. The default is None.
useEmbeddedWeights (bool, optional) – Wether or not use the embedded weights for the step update. The default is False.

Returns:

The \(L_\infty\) norm.

Return type:

float

Q_GENERATORS: dict[str, type[QGenerator]]: Dictionary containing all specialized QGenerator classes, with all their aliases

register(cls: type[T]) → type[T][source]: Class decorator to register a specialized QGenerator class in qmat

genQCoeffs(qType, form='Z2N', hCoeffs=False, embedded=False, **params)[source]

Generate \(Q\)-coefficients for a given method

Parameters:

qType (str) – Name (or alias) of the QGenerator.
form (str, optional) – Write coefficients in zero-to-nodes (Z2N) or node-to-node (N2N). The default is “Z2N”.
hCoeffs (bool, optional) – Wether or not returning the \(h\) coefficients. The default is False.
embedded (bool, optional) – Wether or not returning the embedded \(h\) coefficients. The default is False.
**params – Parameters to be used to instantiate the QGenerator.

Returns:

out – Contains (nodes, weights, Q). If hCoeffs=True, returns (nodes, weights, Q, hCoeffs). If embedded=True, weights is a 2xM array containing embedded weights in weights[1].

Return type:

tuple