Acts::ButcherTableau Class Reference

Coefficients of an explicit Runge-Kutta method. More...

#include <Acts/Propagator/ButcherTableau.hpp>

Public Member Functions
	ButcherTableau (std::string name, unsigned order, unsigned embeddedOrder, std::vector< double > c, std::vector< std::vector< double > > a, std::vector< double > b, std::vector< double > bEmbedded)
	Construct and validate a tableau.
double	a (std::size_t i, std::size_t j) const
double	b (std::size_t i) const
double	bEmbedded (std::size_t i) const
double	c (std::size_t i) const
unsigned	embeddedOrder () const
bool	firstSameAsLast () const
bool	hasEmbedded () const
const std::string &	name () const
unsigned	order () const
std::size_t	stages () const

Static Public Member Functions
static std::shared_ptr< const ButcherTableau >	classicalRk4 ()
	The classical fourth-order method without an embedded solution (W.
static std::shared_ptr< const ButcherTableau >	dormandPrince54 ()
	The Dormand-Prince 5(4) pair RK5(4)7M (J.
static std::shared_ptr< const ButcherTableau >	verner98 ()
	The "most efficient" Verner 9(8) pair.

Detailed Description

Coefficients of an explicit Runge-Kutta method.

The tableau defines the stages

Y_i = y + h * sum_{j < i} a(i, j) * K_j, K_i = f(Y_i),

the solution y_1 = y + h * sum_i b(i) * K_i and optionally an embedded solution with the weights bEmbedded for an error estimate.

The equation of motion is a second-order equation for the position. A tableau with the coefficients A and b, applied to the first-order system of the free parameters, is the same as the general Runge-Kutta-Nyström method with the position coefficients A^2 and b A (Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I, 2nd ed., Springer 1993, Section II.14).

Note

The Runge-Kutta-Nyström step of EigenStepper and of the ATLAS stepper is a different fourth-order method than classicalRk4 on the first-order system, so the two do not give the same result.

Constructor & Destructor Documentation

ButcherTableau()

Acts::ButcherTableau::ButcherTableau	(	std::string	name,
		unsigned	order,
		unsigned	embeddedOrder,
		std::vector< double >	c,
		std::vector< std::vector< double > >	a,
		std::vector< double >	b,
		std::vector< double >	bEmbedded )

Construct and validate a tableau.

Parameters

name	Name of the method
order	Order of the solution
embeddedOrder	Order of the embedded solution, ignored without embedded weights
c	Stage nodes
a	Stage coefficients, one row per stage, each row with the coefficients of the previous stages
b	Weights of the solution
bEmbedded	Weights of the embedded solution, or empty

Exceptions

std::invalid_argument

if the sizes do not match, if the first node is not zero, or if a row of a does not sum to its node

Member Function Documentation

a()

double Acts::ButcherTableau::a	(	std::size_t	i,
		std::size_t	j ) const

Parameters

i	Stage index
j	Index of a previous stage, j < i

Returns

the coefficient of stage j in stage i

b()

double Acts::ButcherTableau::b

(

std::size_t

i

)

const

Parameters

i	Stage index

Returns

the weight of stage i in the solution

bEmbedded()

double Acts::ButcherTableau::bEmbedded

(

std::size_t

i

)

const

Parameters

i	Stage index

Returns

the weight of stage i in the embedded solution

c()

double Acts::ButcherTableau::c

(

std::size_t

i

)

const

Parameters

i	Stage index

Returns

the node of stage i

classicalRk4()

std::shared_ptr< const ButcherTableau > Acts::ButcherTableau::classicalRk4 ( )

static

The classical fourth-order method without an embedded solution (W.

Kutta, Z. Math. Phys. 46 (1901) 435-453; Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I, Section II.1)

Returns

the shared tableau

dormandPrince54()

std::shared_ptr< const ButcherTableau > Acts::ButcherTableau::dormandPrince54 ( )

static

The Dormand-Prince 5(4) pair RK5(4)7M (J.

R. Dormand, P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19-26, Table 2; Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I, Section II.5, DOPRI5). The stepper propagates the fifth-order solution and uses the fourth-order solution for the error estimate.

Returns

the shared tableau

embeddedOrder()

unsigned Acts::ButcherTableau::embeddedOrder

(

)

const

Returns

the order of the embedded solution

firstSameAsLast()

bool Acts::ButcherTableau::firstSameAsLast

(

)

const

Returns

true if the last stage is at the end of the step (FSAL), so its field is the field at the end

hasEmbedded()

bool Acts::ButcherTableau::hasEmbedded

(

)

const

Returns

true if the tableau has embedded weights

name()

const std::string & Acts::ButcherTableau::name

(

)

const

Returns

the name of the method

order()

unsigned Acts::ButcherTableau::order

(

)

const

Returns

the order of the solution

stages()

std::size_t Acts::ButcherTableau::stages

(

)

const

Returns

the number of stages

verner98()

std::shared_ptr< const ButcherTableau > Acts::ButcherTableau::verner98 ( )

static

The "most efficient" Verner 9(8) pair.

The method is described in J. H. Verner, Numerically optimal Runge-Kutta pairs with interpolants, Numerical Algorithms 53 (2010) 383-396, doi:10.1007/s11075-009-9290-3. The coefficients are from J. H. Verner, https://www.sfu.ca/~jverner/, file RKV98.IIa.Efficient.000000349.081210.CoeffsOnlyRADandFLOATS. The stepper propagates the ninth-order solution and uses the eighth-order solution for the error estimate.

Returns

the shared tableau