dg::mat::FunctionalButcherTableau< real_type > Struct Template Reference

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

Public Types
using	value_type = real_type
using	function_type = std::function<value_type(value_type)>

Public Member Functions
	FunctionalButcherTableau ()=default
	No memory allocation.
	FunctionalButcherTableau (unsigned s, unsigned order, const function_type *a, const function_type *b, const real_type *c)
	Construct a classic non-embedded tableau.
	FunctionalButcherTableau (unsigned s, unsigned embedded_order, unsigned order, const function_type *a, const function_type *b, const function_type *bt, const real_type *c)
	Construct an embedded tableau.
function_type	a (unsigned i, unsigned j) const
	Read the a_ij coefficients.
real_type	c (unsigned i) const
	Read the c_i coefficients.
function_type	b (unsigned j) const
	Read the b_j coefficients.
function_type	bt (unsigned j) const
	Read the embedded bt_j coefficients.
unsigned	num_stages () const
	The number of stages s.
unsigned	order () const
	global order of accuracy for the method represented by b
unsigned	embedded_order () const
	global order of accuracy for the embedded method represented by bt
bool	isEmbedded () const
	True if the method has an embedding.
bool	isImplicit () const
	True if an element on or above the diagonal in a is non-zero.

Detailed Description

template<class real_type>
struct dg::mat::FunctionalButcherTableau< real_type >

Manage coefficients of a functional (extended) Butcher tableau.

The goal of this class is to represent a Butcher tableau for the use in Exponential Runge Kutta type time integrators. 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

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::mat::EXPLICIT_EULER_1_1	Hochbruck and Ostermann, Exponential Integrators, Acta Numerica (2010)
Midpoint-2-2	dg::mat::MIDPOINT_2_2	Hochbruck and Ostermann, Exponential Integrators, Acta Numerica (2010)
Runge-Kutta-4-4	dg::mat::CLASSIC_4_4	Cox and Matthews, J. Comput. Phys., 176 (2002)
Hochbruck-3-3-4	dg::mat::HOCHBRUCK_3_3_4	Hochbruck and Ostermann, Exponential Integrators, Acta Numerica (2010) (The exprb43 method)

Note

In exponential Rosenbrock type schemes it is assumed that \( A\) (the matrix) is the Jacobian of the system. If it is not, then the order conditions are different and the order and embedded orders are not what is indicated in our names.

Template Parameters

real_type

type of the coefficients

See also

dg::mat::ExpRungeKutta

Member Typedef Documentation

function_type

template<class real_type >

using dg::mat::FunctionalButcherTableau< real_type >::function_type = std::function<value_type(value_type)>

value_type

template<class real_type >

using dg::mat::FunctionalButcherTableau< real_type >::value_type = real_type

Constructor & Destructor Documentation

FunctionalButcherTableau() [1/3]

template<class real_type >

dg::mat::FunctionalButcherTableau< real_type >::FunctionalButcherTableau ( )

default

No memory allocation.

FunctionalButcherTableau() [2/3]

template<class real_type >

dg::mat::FunctionalButcherTableau< real_type >::FunctionalButcherTableau ( unsigned s, unsigned order, const function_type * a, const function_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 s*s real functions interpreted as a_{ij}=a[i*s+j]
b	pointer to s real functions 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

FunctionalButcherTableau() [3/3]

template<class real_type >

dg::mat::FunctionalButcherTableau< real_type >::FunctionalButcherTableau ( unsigned s, unsigned embedded_order, unsigned order, const function_type * a, const function_type * b, const function_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 s*s real functions interpreted as a_{ij}=a[i*s+j]
b	pointer to s real functions interpreted as b_j=b[j]
bt	pointer to s real functions interpreted as bt_j=bt[j]
c	pointer to s real numbers interpreted as c_i=c[i]

Member Function Documentation

a()

template<class real_type >

function_type dg::mat::FunctionalButcherTableau< 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

Note

The returned function is either strictly positive or zero everywhere, so a test a(i,j)(0) == 0 is valid

b()

template<class real_type >

function_type dg::mat::FunctionalButcherTableau< 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

Note

The returned function is either strictly positive or zero everywhere, so a test b(j)(0) == 0 is valid

bt()

template<class real_type >

function_type dg::mat::FunctionalButcherTableau< 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

Note

The returned function is either strictly positive or zero everywhere, so a test bt(j)(0) == 0 is valid

c()

template<class real_type >

real_type dg::mat::FunctionalButcherTableau< 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

embedded_order()

template<class real_type >

unsigned dg::mat::FunctionalButcherTableau< real_type >::embedded_order ( ) const

inline

global order of accuracy for the embedded method represented by bt

isEmbedded()

template<class real_type >

bool dg::mat::FunctionalButcherTableau< real_type >::isEmbedded ( ) const

inline

True if the method has an embedding.

isImplicit()

template<class real_type >

bool dg::mat::FunctionalButcherTableau< 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::mat::FunctionalButcherTableau< real_type >::num_stages ( ) const

inline

The number of stages s.

order()

template<class real_type >

unsigned dg::mat::FunctionalButcherTableau< 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