dg::ShuOsherTableau< real_type > Struct Template Reference

Manage coefficients in Shu-Osher form. More...

Public Types
using	value_type = real_type

Public Member Functions
	ShuOsherTableau ()=default
	No memory allocation.
	ShuOsherTableau (unsigned stages, unsigned order, const std::vector< real_type > &alpha_v, const std::vector< real_type > &beta_v)
	Construct a non-embedded explicit tableau.
	operator ButcherTableau< real_type > () const
	A Shu-Osher Tableau can be converted to a Butcher table.
real_type	alpha (unsigned i, unsigned j)
	Read the alpha_ij coefficients.
real_type	beta (unsigned i, unsigned j)
	Read the beta_ij coefficients.
unsigned	num_stages () const
	The number of stages s.
unsigned	order () const
	global order of accuracy for the method

Detailed Description

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

Manage coefficients in Shu-Osher form.

Currently only explicit tables that are marked with "Shu-Osher-Form" are available in this form

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 (but we do not) [Hairer, Noersett, Wanner, Solving ordinary differential Equations I, 1987]
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	The original uses the embedding as the solution (but we do not) 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)

Note

A Shu-Osher Tableau can be uniquely converted to a ButcherTableau but the converse is not true

Template Parameters

real_type

type of the coefficients

See also

ShuOsher

Member Typedef Documentation

value_type

template<class real_type >

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

Constructor & Destructor Documentation

ShuOsherTableau() [1/2]

template<class real_type >

dg::ShuOsherTableau< real_type >::ShuOsherTableau ( )

default

No memory allocation.

ShuOsherTableau() [2/2]

template<class real_type >

dg::ShuOsherTableau< real_type >::ShuOsherTableau ( unsigned stages, unsigned order, const std::vector< real_type > & alpha_v, const std::vector< real_type > & beta_v )

inline

Construct a non-embedded explicit tableau.

Parameters

stages	number of stages
order	(global) order of the resulting method
alpha_v	s*s/2 real numbers interpreted as a linearized triangle
beta_v	s*s/2 real numbers interpreted as a linearized triangle

Member Function Documentation

alpha()

template<class real_type >

real_type dg::ShuOsherTableau< real_type >::alpha ( unsigned i, unsigned j )

inline

Read the alpha_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

alpha_ij

beta()

template<class real_type >

real_type dg::ShuOsherTableau< real_type >::beta ( unsigned i, unsigned j )

inline

Read the beta_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

beta_ij

num_stages()

template<class real_type >

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

inline

The number of stages s.

operator ButcherTableau< real_type >()

template<class real_type >

dg::ShuOsherTableau< real_type >::operator ButcherTableau< real_type > ( ) const

inline

A Shu-Osher Tableau can be converted to a Butcher table.

Returns

the corresponding Butcher Tableau

order()

template<class real_type >

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

inline

global order of accuracy for the method

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

• tableau.h