Manage coefficients of a (extended) Butcher tableau. More...

Public Types
using	value_type = real_type

Public Member Functions
	ButcherTableau ()=default
	No memory allocation. More...

	ButcherTableau (unsigned s, unsigned order, const real_type a, const real_type b, const real_type *c)
	Construct a classic non-embedded tableau. More...

	ButcherTableau (unsigned s, unsigned embedded_order, unsigned order, const real_type a, const real_type b, const real_type bt, const real_type c)
	Construct an embedded tableau. More...

	ButcherTableau (unsigned s, real_type *data)
	Construct from ARKode standard format. More...

real_type	a (unsigned i, unsigned j) const
	Read the a_ij coefficients. More...

real_type	c (unsigned i) const
	Read the c_i coefficients. More...

real_type	b (unsigned j) const
	Read the b_j coefficients. More...

real_type	bt (unsigned j) const
	Read the embedded bt_j coefficients. More...

real_type	d (unsigned j) const
	Return the coefficients for the error estimate Equivalent to b(j)-bt(j) More...

unsigned	num_stages () const
	The number of stages s. More...

unsigned	order () const
	global order of accuracy for the method represented by b More...

unsigned	embedded_order () const
	global order of accuracy for the embedded method represented by bt More...

bool	isEmbedded () const
	True if the method has an embedding. More...

bool	isImplicit () const
	True if an element on or above the diagonal in a is non-zero. More...

bool	isFsal () const

Detailed Description

template<class real_type>
struct dg::ButcherTableau< real_type >

Manage coefficients of a (extended) Butcher tableau.

The goal of this class is to represent a Butcher tableau for the use in Runge Kutta type ODE integrators. See https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods for an introduction. The coefficients of the tableau are easily constructible and accessible. Furthermore, we provide utilities like the number of stages, whether the tableau is embedded or not and the order of the method.

Currently available are

Explicit methods

We follow the naming convention of the ARKode library https://sundials.readthedocs.io/en/latest/arkode/Butcher_link.html (They also provide nice stability plots for their methods) as NAME-S-P-Q or NAME-S-Q, where

NAME is the author or name of the method
S is the number of stages in the method
P is the global order of the embedding
Q is the global order of the method

Name	Identifier	Description
Euler	dg::EXPLICIT_EULER_1_1	Explicit Euler
Midpoint-2-2	dg::MIDPOINT_2_2	Midpoint method
Kutta-3-3	dg::KUTTA_3_3	Kutta-3-3
Runge-Kutta-4-4	dg::CLASSIC_4_4	"The" Runge-Kutta method
SSPRK-2-2	dg::SSPRK_2_2	SSPRK (2,2) CFL_eff = 0.5 "Shu-Osher-Form"
SSPRK-3-2	dg::SSPRK_3_2	SSPRK (3,2) CFL_eff = 0.66 "Shu-Osher-Form"
SSPRK-3-3	dg::SSPRK_3_3	SSPRK (3,3) CFL_eff = 0.33 "Shu-Osher-Form"
SSPRK-5-3	dg::SSPRK_5_3	SSPRK (5,3) CFL_eff = 0.5 "Shu-Osher-Form"
SSPRK-5-4	dg::SSPRK_5_4	SSPRK (5,4) CFL_eff = 0.37 "Shu-Osher-Form"
Heun-Euler-2-1-2	dg::HEUN_EULER_2_1_2	Heun-Euler-2-1-2
Cavaglieri-3-1-2 (explicit)	dg::CAVAGLIERI_3_1_2	Low-storage implicit/explicit Runge-Kutta schemes for the simulation of stiff high-dimensional ODE systems IMEXRKCB2 scheme
Fehlberg-3-2-3	dg::FEHLBERG_3_2_3	The original uses the embedding as the solution [Hairer, Noersett, Wanner, Solving ordinary differential Equations I, 1987]
Fehlberg-4-2-3	dg::FEHLBERG_4_2_3	The original uses the embedding as the solution [Hairer, Noersett, Wanner, Solving ordinary differential Equations I, 1987] (fsal)
Bogacki-Shampine-4-2-3	dg::BOGACKI_SHAMPINE_4_2_3	Bogacki-Shampine (fsal)
Cavaglieri-4-2-3 (explicit)	dg::CAVAGLIERI_4_2_3	Low-storage implicit/explicit Runge-Kutta schemes for the simulation of stiff high-dimensional ODE systems The SSP scheme IMEXRKCB3c (explicit part)
ARK-4-2-3 (explicit)	dg::ARK324L2SA_ERK_4_2_3	ARK-4-2-3 (explicit)
Zonneveld-5-3-4	dg::ZONNEVELD_5_3_4	Zonneveld-5-3-4
ARK-6-3-4 (explicit)	dg::ARK436L2SA_ERK_6_3_4	ARK-6-3-4 (explicit)
Sayfy_Aburub-6-3-4	dg::SAYFY_ABURUB_6_3_4	Sayfy_Aburub_6_3_4
Cash_Karp-6-4-5	dg::CASH_KARP_6_4_5	Cash-Karp
Fehlberg-6-4-5	dg::FEHLBERG_6_4_5	Runge-Kutta-Fehlberg
Dormand-Prince-7-4-5	dg::DORMAND_PRINCE_7_4_5	Dormand-Prince method (fsal)
Tsitouras09-7-4-5	dg::TSITOURAS09_7_4_5	Tsitouras 5(4) method from 2009 (fsal), The default method in Julia
Tsitouras11-7-4-5	dg::TSITOURAS11_7_4_5	Tsitouras 5(4) method from 2011 (fsal) Further improves Tsitouras09 (Note that in the paper b-bt is printed instead of bt)
ARK-8-4-5 (explicit)	dg::ARK548L2SA_ERK_8_4_5	Kennedy and Carpenter (2019) Optimum ARK_2 method (explicit part)
Verner-9-5-6	dg::VERNER_9_5_6	Verner-9-5-6 (fsal)
Verner-10-6-7	dg::VERNER_10_6_7	Verner-10-6-7
Fehlberg-13-7-8	dg::FEHLBERG_13_7_8	Fehlberg-13-7-8
Dormand-Prince-13-7-8	dg::DORMAND_PRINCE_13_7_8	[Hairer, Noersett, Wanner, Solving ordinary differential Equations I, 1987]
Feagin-17-8-10	dg::FEAGIN_17_8_10	Feagin (The RK10(8) method)

Implicit methods

We follow the naming convention of the ARKode library https://sundials.readthedocs.io/en/latest/arkode/Butcher_link.html (They also provide nice stability plots for their methods) as NAME-S-P-Q or NAME-S-Q, where

NAME is the author or name of the method
S is the number of stages in the method
P is the global order of the embedding
Q is the global order of the method

Name	Identifier	Description
Euler (implicit)	dg::IMPLICIT_EULER_1_1	backward Euler \(a_{11} = 1\)
Midpoint (implicit)	dg::IMPLICIT_MIDPOINT_1_2	implicit Midpoint \( a_{11} = 0.5 \)
Trapezoidal-2-2	dg::TRAPEZOIDAL_2_2	Crank-Nicolson method \( a_{11} = 0\ a_{22} = 0.5 \)
SDIRK-2-1-2	dg::SDIRK_2_1_2	generic 2nd order A and L-stable \( a_{ii} = 0.29 \)
Cavaglieri-3-1-2 (implicit)	dg::CAVAGLIERI_IMPLICIT_3_1_2	Low-storage implicit/explicit Runge-Kutta schemes for the simulation of stiff high-dimensional ODE systems IMEXRKCB2 scheme \( a_{11} = 0 \)
Billington-3-3-2	dg::BILLINGTON_3_3_2	Billington-3-3-2 \( a_{ii} = 0.29\)
TRBDF2-3-3-2	dg::TRBDF2_3_3_2	TRBDF2-3-3-2 \( a_{11} = 0\)
Sanchez-3-3	dg::SANCHEZ_3_3	symplectic DIRK
Kvaerno-4-2-3	dg::KVAERNO_4_2_3	Kvaerno-4-2-3 \( a_{11} = 0\ a_{ii} = 0.44 \)
SDIRK-4-2-3	dg::SDIRK_4_2_3	Cameron2002 \( a_{ii}=0.25\)
Cavaglieri-4-2-3 (implicit)	dg::CAVAGLIERI_IMPLICIT_4_2_3	Low-storage implicit/explicit Runge-Kutta schemes for the simulation of stiff high-dimensional ODE systems The SSP scheme IMEXRKCB3c (implicit part) \( a_{11} = 0 \)
ARK-4-2-3 (implicit)	dg::ARK324L2SA_DIRK_4_2_3	ARK-4-2-3 (implicit) \( a_{11} = 0\ a_{ii} = 0.44 \)
Sanchez-3-4	dg::SANCHEZ_3_4	symplectic DIRK
Cash-5-2-4	dg::CASH_5_2_4	Cash-5-2-4 \( a_{ii} = 0.44 \)
Cash-5-3-4	dg::CASH_5_3_4	Cash-5-3-4 \( a_{ii} = 0.44 \)
SDIRK-5-3-4	dg::SDIRK_5_3_4	SDIRK-5-3-4 \( a_{ii} = 0.25\)
Kvaerno-5-3-4	dg::KVAERNO_5_3_4	Kvaerno-5-3-4 \( a_{11} = 0\ a_{ii} = 0.44\)
ARK-6-3-4 (implicit)	dg::ARK436L2SA_DIRK_6_3_4	ARK-6-3-4 (implicit) \( a_{11} = 0\ a_{ii} = 0.25\)
Sanchez-6-5	dg::SANCHEZ_6_5	symplectic DIRK
Kvaerno-7-4-5	dg::KVAERNO_7_4_5	Kvaerno-7-4-5 \( a_{11} = 0 \ a_{ii} = 0.26\)
ARK-8-4-5 (implicit)	dg::ARK548L2SA_DIRK_8_4_5	Kennedy and Carpenter (2019) Optimum ARK_2 method (implicit part) \( a_{11} = 0\ a_{ii} = 0.22\)
Sanchez-7-6	dg::SANCHEZ_7_6	symplectic DIRK

Note: Schemes marked with \( a_{11} = 0\) call the implicit_rhs instead of a solve once per step (the first stage is explicit, typically named EDIRK), all others never call implicit_rhs. Schemes marked with \(a_{ii} = ...\) (lower is better) have all diagonal elements equal (typically named SDIRK for "singly"). Schemes marked with both have equal elements only for i>1 (named ESDIRK).

Template Parameters

real_type type of the coefficients

See also: RungeKutta, ERKStep, ARKStep

Member Typedef Documentation

◆ value_type

template<class real_type >

using dg::ButcherTableau< real_type >::value_type = real_type

Constructor & Destructor Documentation

◆ ButcherTableau() [1/4]

template<class real_type >

dg::ButcherTableau< real_type >::ButcherTableau ( )

default

No memory allocation.

◆ ButcherTableau() [2/4]

template<class real_type >

dg::ButcherTableau< real_type >::ButcherTableau	(	unsigned	s,
		unsigned	order,
		const real_type *	a,
		const real_type *	b,
		const real_type *	c
	)

inline

Construct a classic non-embedded tableau.

Parameters

s	number of stages
order	(global) order of the resulting method
a	pointer to ss real numbers interpreted as a_{ij}=a[is+j]
b	pointer to s real numbers interpreted as b_j=b[j]
c	pointer to s real numbers interpreted as c_i=c[i]

Note: This constructor initializes the embedded coefficients bt=b which makes the embedded method equal to the actual method

◆ ButcherTableau() [3/4]

template<class real_type >

dg::ButcherTableau< real_type >::ButcherTableau	(	unsigned	s,
		unsigned	embedded_order,
		unsigned	order,
		const real_type *	a,
		const real_type *	b,
		const real_type *	bt,
		const real_type *	c
	)

inline

Construct an embedded tableau.

Parameters

s	number of stages
embedded_order	(global) order of the embedded method (corresponding to `bt`)
order	(global) order of the method (corresponding to `b`)
a	pointer to ss real numbers interpreted as a_{ij}=a[is+j]
b	pointer to s real numbers interpreted as b_j=b[j]
bt	pointer to s real numbers interpreted as bt_j=bt[j]
c	pointer to s real numbers interpreted as c_i=c[i]

◆ ButcherTableau() [4/4]

template<class real_type >

dg::ButcherTableau< real_type >::ButcherTableau	(	unsigned	s,
		real_type *	data
	)

inline

Construct from ARKode standard format.

Construct embedded method from single array

Parameters

s	number of stages
data	Array of (s+1)(s+2) real numbers, interpreted as: c[i]=data[i(s+1)], a_ij=data[i(s+1)+j+1], order=data[s(s+1)], b_j=data[s(s+1)+j+1], embedded_order = data[(s+1)(s+1)], bt_j=data[(s+1)*(s+1)+j+1]

Member Function Documentation

◆ a()

template<class real_type >

real_type dg::ButcherTableau< real_type >::a	(	unsigned	i,
		unsigned	j
	)		const

inline

Read the a_ij coefficients.

Parameters

i	row number 0<=i<s, i>=s results in undefined behaviour
j	col number 0<=j<s, j>=s results in undefined behaviour

Returns: a_ij

◆ b()

template<class real_type >

real_type dg::ButcherTableau< real_type >::b ( unsigned j ) const

inline

Read the b_j coefficients.

Parameters

j	col number 0<=j<s, j>=s results in undefined behaviour

Returns: b_j

◆ bt()

template<class real_type >

real_type dg::ButcherTableau< real_type >::bt ( unsigned j ) const

inline

Read the embedded bt_j coefficients.

Parameters

j	col number 0<=j<s, j>=s results in undefined behaviour

Returns: bt_j

◆ c()

template<class real_type >

real_type dg::ButcherTableau< real_type >::c ( unsigned i ) const

inline

Read the c_i coefficients.

Parameters

i	row number 0<=i<s, i>=s results in undefined behaviour

Returns: c_i

◆ d()

template<class real_type >

real_type dg::ButcherTableau< real_type >::d ( unsigned j ) const

inline

Return the coefficients for the error estimate Equivalent to b(j)-bt(j)

Parameters

j	col number 0<=j<s, j>=s results in undefined behaviour

Returns: b(j)-bt(j)

◆ embedded_order()

template<class real_type >

unsigned dg::ButcherTableau< real_type >::embedded_order ( ) const

inline

global order of accuracy for the embedded method represented by bt

◆ isEmbedded()

template<class real_type >

bool dg::ButcherTableau< real_type >::isEmbedded ( ) const

inline

True if the method has an embedding.

◆ isFsal()

template<class real_type >

bool dg::ButcherTableau< real_type >::isFsal ( ) const

inline

True if the method has the "First Same As Last" property: the last stage is evaluated at the same point as the first stage of the next step.

◆ isImplicit()

template<class real_type >

bool dg::ButcherTableau< real_type >::isImplicit ( ) const

inline

True if an element on or above the diagonal in a is non-zero.

◆ num_stages()

template<class real_type >

unsigned dg::ButcherTableau< real_type >::num_stages ( ) const

inline

The number of stages s.

◆ order()

template<class real_type >

unsigned dg::ButcherTableau< real_type >::order ( ) const

inline

global order of accuracy for the method represented by b

The documentation for this struct was generated from the following file:

tableau.h

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ value_type

Constructor & Destructor Documentation

◆ ButcherTableau() [1/4]

◆ ButcherTableau() [2/4]

◆ ButcherTableau() [3/4]

◆ ButcherTableau() [4/4]

Member Function Documentation

◆ a()

◆ b()

◆ bt()

◆ c()

◆ d()

◆ embedded_order()

◆ isEmbedded()

◆ isFsal()

◆ isImplicit()

◆ num_stages()

◆ order()