qmat.qcoeff.butcher

\(Q\)-coefficients based on Butcher tables of Runge-Kutta in the literature.

Examples

>>> from qmat.qcoeff.butcher import RK_SCHEMES
>>> c, b, A = RK_SCHEMES["RK4"]().genCoeffs()
>>> c, (b1, b2), A = RK_SCHEMES["ESDIRK53"]().genCoeffs(embedded=True)

>>> from qmat.qcoeff.butcher import CashKarp
>>> gen = CashKarp()
>>> c, b, A = gen.c, gen.b, gen.A

Attributes

RK_SCHEMES

Dictionary storing all the implemented RK methods

Classes

`RK`	Base class for Runge-Kutta generators
`FE`	Forward Euler method (cf Wikipedia)
`RK21`	Explicit Runge-Kutta in 2 steps of order 1 from [Wang & Spiteri, 2007]
`RK2`	Classical Runge-Kutta method of order 2 (cf Wikipedia)
`HEUN2`	Heun method of order 2 (cf Wikipedia)
`RK32`	Explicit Runge-Kutta in 3 steps of order 2 from [Wang & Spiteri, 2007]
`RK33`	Explicit Runge-Kutta in 3 steps of order 3 from [Wang & Spiteri, 2007]
`RK53`	Explicit Runge-Kutta in 5 steps of order 3 from [Wang & Spiteri, 2007]
`RK4`	Classical Runge Kutta method of order 4 (cf Wikipedia)
`RK4_38`	The 3/8-rule due to Kutta, order 4 (cf Wikipedia)
`RK65`	Explicit Runge-Kutta in 6 steps of order 5, (236a) from [Butcher, 2016]
`CashKarp`	Fifth order explicit embedded Runge-Kutta from [Cash & Karp, 1990].
`BE`	Backward Euler method (also SDIRK1, see [Alexander, 1977])
`MidPoint`	Implicit Mid-Point Rule, see Wikipedia.
`TRAP`	Trapeze method, see Wikipedia.
`SDIRK2`	First S-stable Diagonally Implicit Runge Kutta method of order 2 in two stages from [Alexander, 1977].
`SDIRK2_2`	Second S-stable Diagonally Implicit Runge Kutta method of order 2 in two stages from [Alexander, 1977].
`SDIRK3`	S-stable Diagonally Implicit Runge Kutta method of order 3 in three stages from [Alexander, 1977].
`DIRK43`	L-stable Diagonally Implicit RK method with four stages of order 3 from Wikipedia.
`GAUSS_LG`	Gauss-Legendre method of order 4 (cf Wikipedia).
`SDIRK54`	S-stable Diagonally Implicit Runge Kutta method of order 4 in five stages from [Hairer & Wanner, 1996].
`EDIRK43`	Embedded A-stable diagonally implicit RK pair of order 3 and 4 from [Norsett & Thomsen, 1984].
`EDIRK4`	Stiffly accurate, fourth-order EDIRK with four stages, taken from [Kennedy & Carpenter, 2016], second one in eq. (216).
`ESDIRK43`	A-stable embedded RK pair of orders 4 and 3, ESDIRK4(3)6L[2]SA, from [Kennedy & Carpenter, 2016].
`ESDIRK53`	A-stable embedded RK pair of orders 5 and 3, ESDIRK5(3)6L[2]SA, from [Kennedy & Carpenter, 2016].
`ARK548L2SAERK`	Explicit part of the ARK54 scheme.
`ARK548L2SAESDIRK`	Implicit part of the ARK54 scheme.
`ARK548L2SAESDIRK2`	Stiffly accurate singly diagonally L-stable implicit embedded Runge-Kutta pair of orders 5 and 4 with explicit first stage from [Kennedy & Carpenter, 2019].
`ARK548L2SAERK2`	Explicit embedded pair of Runge-Kutta methods of orders 5 and 4 from [Kennedy & Carpenter, 2019].
`ARK324L2SAERK`	Explicit part of embedded additive Runge-Kutta pair of orders 3 and 2 from [Kennedy & Carpenter, 2003].
`ARK324L2SAESDIRK`	Implicit part of embedded additive Runge-Kutta pair of orders 3 and 2 from [Kennedy & Carpenter, 2003].
`ARK222EDIRK`	2nd-order 2-stages EDIRK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.6].
`ARK222ERK`	2nd-order 2-stages ERK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.6].
`ARK443ESDIRK`	3rd-order 4-stages ESDIRK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.8].
`ARK443ERK`	3rd-order 4-stages ERK scheme [Ascher, Ruuth & Spiteri, 1997 - sec 2.8].
`ARK343ESDIRK`	3rd-order 3-stages ESDIRK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.7].
`ARK343ERK`	4rd-order 4-stages ERK scheme [Ascher, Ruuth & Spiteri, 1997 - sec 2.7].
`ARK4EDIRK`	A stable 7-stages fourth order diagonally implicit stiffly accurate Runge-Kutta method with explicit first stage.
`ARK4ERK`	7-stages fourth order explicit stiffly accurate Runge-Kutta method.

Functions

`checkAndStore`(→ RK)	Check that a RK-inherited class is correctly implemented (and store it into the `RK_SCHEMES` dict)
`registerRK`(→ RK)	Class decorator registering a RK method in qmat

Module Contents

class RK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK

Base class for Runge-Kutta generators

Parameters:

padding (str, optional) – Eventually add padding to the Butcher table. Can be

LEFT : add padding corresponding to an additional node at t=0
RIGHT : add padding corresponding to an additional node at t=1
BOTH : add both LEFT and RIGHT padding

The default is None.

A = None: \(A\) matrix of the Butcher table

b = None: \(b\) coefficients of the Butcher table

c = None: \(c\) coefficients of the Butcher table

b2 = None: \(b_2\) coefficients for the embedded methods

property nodes: numpy.ndarray: Nodes \(\tau\) (\(c\) coefficients in Butcher table)

property weights: numpy.ndarray: Weights \(\omega\) (\(b\) coefficients in Butcher table)

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

property Q: numpy.ndarray: \(Q\) coefficients (\(A\) Butcher table)

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

RK_SCHEMES: Dictionary storing all the implemented RK methods

checkAndStore(cls: RK) → RK[source]: Check that a RK-inherited class is correctly implemented (and store it into the RK_SCHEMES dict)

registerRK(cls: RK) → RK[source]: Class decorator registering a RK method in qmat

class FE(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.FE

Forward Euler method (cf Wikipedia)

aliases = ['EE']

A = [[0]]: \(A\) matrix of the Butcher table

b = [1]: \(b\) coefficients of the Butcher table

c = [0]: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class RK21(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK21

Explicit Runge-Kutta in 2 steps of order 1 from [Wang & Spiteri, 2007]

aliases = ['ERK21']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class RK2(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK2

Classical Runge-Kutta method of order 2 (cf Wikipedia)

aliases = ['ERK2', 'ExplicitMidPoint', 'EMP']

A: \(A\) matrix of the Butcher table

b = [0, 1]: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class HEUN2(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.HEUN2

Heun method of order 2 (cf Wikipedia)

aliases = ['HEUN', 'HeunEuler']

A = [[0, 0], [1, 0]]: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c = [0, 1.0]: \(c\) coefficients of the Butcher table

b2 = [1, 0]: \(b_2\) coefficients for the embedded methods

property order: int: Global convergence order of the method

class RK32(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK32

Explicit Runge-Kutta in 3 steps of order 2 from [Wang & Spiteri, 2007]

aliases = ['ERK32', 'RK32-SSP']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class RK33(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK33

Explicit Runge-Kutta in 3 steps of order 3 from [Wang & Spiteri, 2007]

aliases = ['ERK33', 'RK33-SSP']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class RK53(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK53

Explicit Runge-Kutta in 5 steps of order 3 from [Wang & Spiteri, 2007]

aliases = ['ERK53']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class RK4(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK4

Classical Runge Kutta method of order 4 (cf Wikipedia)

aliases = ['ERK4']

A = [[0, 0, 0, 0], [0.5, 0, 0, 0], [0, 0.5, 0, 0], [0, 0, 1, 0]]: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class RK4_38(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK4_38

The 3/8-rule due to Kutta, order 4 (cf Wikipedia)

aliases = ['ERK4_38']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class RK65(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.RK65

Explicit Runge-Kutta in 6 steps of order 5, (236a) from [Butcher, 2016]

aliases = ['ERK65']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c = [0, 0.25, 0.25, 0.5, 0.75, 1]: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class CashKarp(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.CashKarp

Fifth order explicit embedded Runge-Kutta from [Cash & Karp, 1990].

aliases = ['Cash_Karp']

c: \(c\) coefficients of the Butcher table

b: \(b\) coefficients of the Butcher table

b2: \(b_2\) coefficients for the embedded methods

A: \(A\) matrix of the Butcher table

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [32, 64, 128]

class BE(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.BE

Backward Euler method (also SDIRK1, see [Alexander, 1977])

aliases = ['IE', 'SDIRK1']

A = [[1]]: \(A\) matrix of the Butcher table

b = [1]: \(b\) coefficients of the Butcher table

c = [1]: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class MidPoint(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.MidPoint

Implicit Mid-Point Rule, see Wikipedia.

aliases = ['IMP', 'ImplicitMidPoint']

A: \(A\) matrix of the Butcher table

b = [1]: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class TRAP(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.TRAP

Trapeze method, see Wikipedia.

aliases = ['TRAPZ', 'CN', 'CrankNicolson']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c = [0, 1]: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class SDIRK2(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.SDIRK2

First S-stable Diagonally Implicit Runge Kutta method of order 2 in two stages from [Alexander, 1977].

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class SDIRK2_2(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.SDIRK2_2

Second S-stable Diagonally Implicit Runge Kutta method of order 2 in two stages from [Alexander, 1977].

aliases = ['SDIRK2-2']

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [64, 128, 256]

class SDIRK3(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.SDIRK3

S-stable Diagonally Implicit Runge Kutta method of order 3 in three stages from [Alexander, 1977].

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c = [0.43586652150845967, 0.7179332607542298, 1.0]: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class DIRK43(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.DIRK43

L-stable Diagonally Implicit RK method with four stages of order 3 from Wikipedia.

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class GAUSS_LG(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.GAUSS_LG

Gauss-Legendre method of order 4 (cf Wikipedia).

aliases = ['GAUSS-LG']

A: \(A\) matrix of the Butcher table

b = [0.5, 0.5]: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class SDIRK54(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.SDIRK54

S-stable Diagonally Implicit Runge Kutta method of order 4 in five stages from [Hairer & Wanner, 1996].

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class EDIRK43(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.EDIRK43

Embedded A-stable diagonally implicit RK pair of order 3 and 4 from [Norsett & Thomsen, 1984].

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

b2: \(b_2\) coefficients for the embedded methods

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [32, 64, 128]

class EDIRK4(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.EDIRK4

Stiffly accurate, fourth-order EDIRK with four stages, taken from [Kennedy & Carpenter, 2016], second one in eq. (216).

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [32, 64, 128]

class ESDIRK43(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ESDIRK43

A-stable embedded RK pair of orders 4 and 3, ESDIRK4(3)6L[2]SA, from [Kennedy & Carpenter, 2016].

s2 = 1.4142135623730951

c: \(c\) coefficients of the Butcher table

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

b2: \(b_2\) coefficients for the embedded methods

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [64, 128, 256]

class ESDIRK53(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ESDIRK53

A-stable embedded RK pair of orders 5 and 3, ESDIRK5(3)6L[2]SA, from [Kennedy & Carpenter, 2016].

c: \(c\) coefficients of the Butcher table

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

b2: \(b_2\) coefficients for the embedded methods

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

property order: int: Global convergence order of the method

class ARK548L2SAERK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK548L2SAERK

Explicit part of the ARK54 scheme.

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

c: \(c\) coefficients of the Butcher table

b2: \(b_2\) coefficients for the embedded methods

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [32, 64, 128]

class ARK548L2SAESDIRK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK548L2SAESDIRK

Implicit part of the ARK54 scheme. Be careful with the embedded scheme : it seems that both schemes are order 5 as opposed to 5 and 4 as claimed. This may cause issues when doing adaptive time-stepping.

A: \(A\) matrix of the Butcher table

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

class ARK548L2SAESDIRK2(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK548L2SAESDIRK2

Stiffly accurate singly diagonally L-stable implicit embedded Runge-Kutta pair of orders 5 and 4 with explicit first stage from [Kennedy & Carpenter, 2019]. This method is part of the IMEX method ARK548L2SA.

gamma = 0.2222222222222222

c: \(c\) coefficients of the Butcher table

b: \(b\) coefficients of the Butcher table

A: \(A\) matrix of the Butcher table

b2: \(b_2\) coefficients for the embedded methods

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [16, 32, 64]

class ARK548L2SAERK2(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK548L2SAERK2

Explicit embedded pair of Runge-Kutta methods of orders 5 and 4 from [Kennedy & Carpenter, 2019]. This method is part of the IMEX method ARK548L2SA.

A: \(A\) matrix of the Butcher table

class ARK324L2SAERK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK324L2SAERK

Explicit part of embedded additive Runge-Kutta pair of orders 3 and 2 from [Kennedy & Carpenter, 2003].

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

b2: \(b_2\) coefficients for the embedded methods

c: \(c\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class ARK324L2SAESDIRK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK324L2SAESDIRK

Implicit part of embedded additive Runge-Kutta pair of orders 3 and 2 from [Kennedy & Carpenter, 2003].

A: \(A\) matrix of the Butcher table

CONV_TEST_NSTEPS = [120, 100, 80]

class ARK222EDIRK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK222EDIRK

2nd-order 2-stages EDIRK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.6]. Use as implicit part for ARK scheme in combination with ARK222ERK.

gamma

delta

c: \(c\) coefficients of the Butcher table

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class ARK222ERK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK222ERK

2nd-order 2-stages ERK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.6]. Use as explicit part for ARK scheme in combination with ARK222EDIRK.

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

class ARK443ESDIRK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK443ESDIRK

3rd-order 4-stages ESDIRK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.8]. Use as implicit part for ARK scheme in combination with ARK443ERK.

c: \(c\) coefficients of the Butcher table

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class ARK443ERK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK443ERK

3rd-order 4-stages ERK scheme [Ascher, Ruuth & Spiteri, 1997 - sec 2.8]. Use as explicit part for ARK scheme in combination with ARK443ESDIRK.

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

class ARK343ESDIRK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK343ESDIRK

3rd-order 3-stages ESDIRK scheme from [Ascher, Ruuth & Spiteri, 1997 - sec 2.7]. Use as implicit part for ARK scheme in combination with ARK443ERK.

c: \(c\) coefficients of the Butcher table

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

property order: int: Global convergence order of the method

class ARK343ERK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK343ERK

4rd-order 4-stages ERK scheme [Ascher, Ruuth & Spiteri, 1997 - sec 2.7]. Use as explicit part for ARK scheme in combination with ARK343ESDIRK.

A: \(A\) matrix of the Butcher table

property order: int: Global convergence order of the method

class ARK4EDIRK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK4EDIRK

A stable 7-stages fourth order diagonally implicit stiffly accurate Runge-Kutta method with explicit first stage. Implicit part of Additive RK.4.A.2 from [Liu & Zou, 2006]. Use with ARK4ERK to get fourth order stiffly accurate IMEX method.

c: \(c\) coefficients of the Butcher table

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table

property order: int: Global convergence order of the method

CONV_TEST_NSTEPS = [60, 40, 20, 10]

class ARK4ERK(padding=None)[source]

Inheritance diagram of qmat.qcoeff.butcher.ARK4ERK

7-stages fourth order explicit stiffly accurate Runge-Kutta method. Explicit part of Additive RK.4.A.2 from [Liu & Zou, 2006]. Use with ARK4EDIRK to get fourth order stiffly accurate IMEX method.

A: \(A\) matrix of the Butcher table

b: \(b\) coefficients of the Butcher table