Sandals::Tableau< Real, S > Struct Template Reference

Struct container for the Butcher tableau of a Runge-Kutta method. More...

#include <Tableau.hh>

Public Types
using	Type = enum class type : Integer {ERK = 0, IRK = 1, DIRK = 2}
using	Vector = Eigen::Vector<Real, S>
using	Matrix = Eigen::Matrix<Real, S, S>

Public Member Functions
bool	check (bool verbose=false) const

Public Attributes
const Real	SQRT_EPSILON {std::sqrt(EPSILON)}
std::string	name
Type	type
Integer	order
Integer	order_e {-1}
Matrix	A
Vector	b
Vector	b_e {Vector::Zero()}
Vector	c
bool	is_embedded {false}

Private Member Functions
Integer	compute_order (Matrix const &A, Vector const &b, Vector const &c, bool verbose=false) const

Detailed Description

template<typename Real, Integer S>
struct Sandals::Tableau< Real, S >

Butcher tableau

Runge-Kutta methods are a family of numerical techniques for solving ODEs and DAEs. These methods approximate the solution by combining weighted evaluations of the system's functions at specific points called nodes within the integration domain.

A Runge-Kutta method is characterized by its Butcher tableau, a tabular representation of the method's parameters. For a method of order \(p\) and \(s\) stages, the Butcher tableau is defined as

\[ \begin{array}{c|ccc} c_1 & a_{11} & a_{12} & \dots & a_{1s} \\ c_2 & a_{21} & a_{22} & \dots & a_{2s} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c_s & a_{s1} & a_{s2} & \dots & a_{ss} \\ \hline & b_1 & b_2 & \dots & b_s \end{array} \text{,} \]

where \(\mathbf{A}\) is the Runge-Kutta coefficient matrix, \(\mathbf{b}\) is the Runge-Kutta weights vector, \(\hat{\mathbf{b}}\) is the (optional) embedded Runge-Kutta weights vector, and \(\mathbf{c}\) is the Runge-Kutta nodes vector. These parameters define the method's structure and determine its accuracy and stability.

Template Parameters

S	The size of the Butcher tableau.

Member Typedef Documentation

Matrix

template<typename Real, Integer S>

using Sandals::Tableau< Real, S >::Matrix = Eigen::Matrix<Real, S, S>

Templetized matrix type.

Type

template<typename Real, Integer S>

using Sandals::Tableau< Real, S >::Type = enum class type : Integer {ERK = 0, IRK = 1, DIRK = 2}

Runge-Kutta type enumeration.

Vector

template<typename Real, Integer S>

using Sandals::Tableau< Real, S >::Vector = Eigen::Vector<Real, S>

Templetized vector type.

Member Function Documentation

check()

template<typename Real, Integer S>

bool Sandals::Tableau< Real, S >::check ( bool verbose = false ) const

inline

Check the Butcher tableau consistency for a generic Runge-Kutta method.

Parameters

[in]

verbose

Verbosity flag.

Returns

True if the Butcher tableau is consistent and its order is verified, false otherwise.

compute_order()

template<typename Real, Integer S>

Integer Sandals::Tableau< Real, S >::compute_order ( Matrix const & A, Vector const & b, Vector const & c, bool verbose = false ) const

inlineprivate

Check the order of a Runge-Kutta method according to the conditions taken from: A family of embedded Runge-Kutta formulae, J. R. Dormand and P. J. Prince, Journal of Computational and Applied Mathematics, volume 6(1), 1980. DOI 10.1016/0771-0509(80)90013-3.

Parameters

[in]	A	Matrix \( \mathbf{A} \).
[in]	b	Weights vector \( \mathbf{b} \) or embedded weights vector \( \hat{\mathbf{b}} \).
[in]	c	Nodes vector \( \mathbf{c} \).
[in]	verbose	Verbosity flag.

Returns

The calculated order of the Runge-Kutta method.

Member Data Documentation

A

template<typename Real, Integer S>

Matrix Sandals::Tableau< Real, S >::A

Matrix \( \mathbf{A} \).

b

template<typename Real, Integer S>

Vector Sandals::Tableau< Real, S >::b

Weights vector \( \mathbf{b} \).

b_e

template<typename Real, Integer S>

Vector Sandals::Tableau< Real, S >::b_e {Vector::Zero()}

Embedded weights vector \( \hat{\mathbf{b}} \).

c

template<typename Real, Integer S>

Vector Sandals::Tableau< Real, S >::c

Nodes vector \( \mathbf{c} \).

is_embedded

template<typename Real, Integer S>

bool Sandals::Tableau< Real, S >::is_embedded {false}

Embedded method boolean.

name

template<typename Real, Integer S>

std::string Sandals::Tableau< Real, S >::name

Name of the Runge-Kutta method.

order

template<typename Real, Integer S>

Integer Sandals::Tableau< Real, S >::order

Order of the Runge-Kutta method.

order_e

template<typename Real, Integer S>

Integer Sandals::Tableau< Real, S >::order_e {-1}

Order of the Runge-Kutta embedded method.

SQRT_EPSILON

template<typename Real, Integer S>

const Real Sandals::Tableau< Real, S >::SQRT_EPSILON {std::sqrt(EPSILON)}

< Basic constants. Square root of machine epsilon epsilon static constant value.

type

template<typename Real, Integer S>

Type Sandals::Tableau< Real, S >::type

Runge-Kutta type.

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

• include/Sandals/Tableau.hh