#include <AdamsBashforth.hpp>

Classes
struct	Order

Public Types
using	options = implementation defined

Public Types inherited from LtsTimeStepper
using	provided_time_stepper_interfaces = implementation defined

template<typename LocalVars , typename RemoteVars , typename Coupling >
using	BoundaryHistoryType = TimeSteppers::BoundaryHistory< LocalVars, RemoteVars, std::invoke_result_t< const Coupling &, LocalVars, RemoteVars > >

template<typename LocalVars , typename RemoteVars , typename Coupling >
using	BoundaryReturn = db::unprefix_variables< std::invoke_result_t< const Coupling &, LocalVars, RemoteVars > >
	Return type of boundary-related functions. The coupling returns the derivative of the variables, but this is multiplied by the time step so the return type should not have `dt` prefixes.

Public Types inherited from TimeStepper
using	provided_time_stepper_interfaces = implementation defined

Public Member Functions
	AdamsBashforth (size_t order)

	AdamsBashforth (const AdamsBashforth &)=default

AdamsBashforth &	operator= (const AdamsBashforth &)=default

	AdamsBashforth (AdamsBashforth &&)=default

AdamsBashforth &	operator= (AdamsBashforth &&)=default

size_t	order () const override
	The convergence order of the stepper. More...

uint64_t	number_of_substeps () const override
	Number of substeps in this TimeStepper. More...

uint64_t	number_of_substeps_for_error () const override
	Number of substeps in this TimeStepper when providing an error measure for adaptive time-stepping. More...

size_t	number_of_past_steps () const override
	Number of past time steps needed for multi-step method. More...

double	stable_step () const override
	Rough estimate of the maximum step size this method can take stably as a multiple of the step for Euler's method. More...

bool	monotonic () const override
	Whether computational and temporal orderings of operations match. More...

TimeStepId	next_time_id (const TimeStepId &current_id, const TimeDelta &time_step) const override
	The TimeStepId after the current substep. More...

TimeStepId	next_time_id_for_error (const TimeStepId &current_id, const TimeDelta &time_step) const override
	The TimeStepId after the current substep when providing an error measure for adaptive time-stepping. More...

bool	neighbor_data_required (const TimeStepId &next_substep_id, const TimeStepId &neighbor_data_id) const override
	Check whether a neighbor record is needed for boundary output. More...

bool	neighbor_data_required (double dense_output_time, const TimeStepId &neighbor_data_id) const override
	Check whether a neighbor record is needed for boundary output. More...

	WRAPPED_PUPable_decl_template (AdamsBashforth)

	AdamsBashforth (CkMigrateMessage *)

void	pup (PUP::er &p) override

Public Member Functions inherited from LtsTimeStepper
	WRAPPED_PUPable_abstract (LtsTimeStepper)

template<typename LocalVars , typename RemoteVars , typename Coupling >
void	add_boundary_delta (const gsl::not_null< BoundaryReturn< LocalVars, RemoteVars, Coupling > * > result, const BoundaryHistoryType< LocalVars, RemoteVars, Coupling > &history, const TimeDelta &time_step, const Coupling &coupling) const
	Compute the change in a boundary quantity due to the coupling on the interface. More...

template<typename LocalVars , typename RemoteVars , typename CouplingResult >
void	clean_boundary_history (const gsl::not_null< TimeSteppers::BoundaryHistory< LocalVars, RemoteVars, CouplingResult > * > history) const
	Remove old entries from the history. More...

template<typename LocalVars , typename RemoteVars , typename Coupling >
void	boundary_dense_output (const gsl::not_null< BoundaryReturn< LocalVars, RemoteVars, Coupling > * > result, const BoundaryHistoryType< LocalVars, RemoteVars, Coupling > &history, const double time, const Coupling &coupling) const
	Derived classes must implement this as a function with signature. More...

Public Member Functions inherited from TimeStepper
	WRAPPED_PUPable_abstract (TimeStepper)

template<typename Vars >
void	update_u (const gsl::not_null< Vars * > u, const TimeSteppers::History< Vars > &history, const TimeDelta &time_step) const
	Set `u` to the value at the end of the current substep. More...

template<typename Vars >
std::optional< StepperErrorEstimate >	update_u (const gsl::not_null< Vars * > u, const TimeSteppers::History< Vars > &history, const TimeDelta &time_step, const std::optional< StepperErrorTolerances > &tolerances) const
	Set `u` to the value at the end of the current substep; report the error measure when available. More...

template<typename Vars >
void	clean_history (const gsl::not_null< TimeSteppers::History< Vars > * > history) const
	Remove old entries from the history. More...

template<typename Vars >
bool	dense_update_u (const gsl::not_null< Vars * > u, const TimeSteppers::History< Vars > &history, const double time) const
	Compute the solution value at a time between steps. To evaluate at a time within a given step, call this method at the start of the step containing the time. The function returns true on success, otherwise the call should be retried after the next substep. More...

virtual size_t	order () const =0
	The convergence order of the stepper. More...

virtual uint64_t	number_of_substeps () const =0
	Number of substeps in this TimeStepper. More...

virtual uint64_t	number_of_substeps_for_error () const =0
	Number of substeps in this TimeStepper when providing an error measure for adaptive time-stepping. More...

virtual size_t	number_of_past_steps () const =0
	Number of past time steps needed for multi-step method. More...

virtual double	stable_step () const =0
	Rough estimate of the maximum step size this method can take stably as a multiple of the step for Euler's method. More...

virtual bool	monotonic () const =0
	Whether computational and temporal orderings of operations match. More...

virtual TimeStepId	next_time_id (const TimeStepId &current_id, const TimeDelta &time_step) const =0
	The TimeStepId after the current substep. More...

virtual TimeStepId	next_time_id_for_error (const TimeStepId &current_id, const TimeDelta &time_step) const =0
	The TimeStepId after the current substep when providing an error measure for adaptive time-stepping. More...

template<typename Vars >
bool	can_change_step_size (const TimeStepId &time_id, const TimeSteppers::History< Vars > &history) const
	Whether a change in the step size is allowed before taking a step. Step sizes can never be changed on a substep. More...

Static Public Attributes
static constexpr const size_t	minimum_order = 1

static constexpr const size_t	maximum_order = 8

static constexpr Options::String	help

Static Public Attributes inherited from LtsTimeStepper
static constexpr bool	local_time_stepping = true

Static Public Attributes inherited from TimeStepper
static constexpr bool	local_time_stepping = false

static constexpr bool	imex = false

Friends
bool	operator== (const AdamsBashforth &lhs, const AdamsBashforth &rhs)

Detailed Description

An Nth order Adams-Bashforth time stepper.

The stable step size factors for different orders are given by:

Order	CFL Factor
1	1
2	1 / 2
3	3 / 11
4	3 / 20
5	45 / 551
6	5 / 114
7	945 / 40663
8	945 / 77432

Local time stepping calculation

Suppose the local and remote sides of the interface are evaluated at times $\dots, t_{- 1}^{L}, t_{0}^{L}, t_{1}^{L}, \dots$ and $\dots, t_{- 1}^{R}, t_{0}^{R}, t_{1}^{R}, \dots$ , respectively, with the starting location of the numbering arbitrary in each case. Let the step we wish to calculate the effect of be the step from $t_{m_{S}}^{L}$ to $t_{m_{S} + 1}^{L}$ . We call the sequence produced from the union of the local and remote time sequences $\dots, \tilde{t}_{- 1}, \tilde{t}_{0}, \tilde{t}_{1}, \dots$ . For example, one possible sequence of times is:

$\begin{matrix} (1) & \begin{aligned} Local side: \\ Union times: \\ Remote side: \end{aligned} \dots \begin{matrix} \tilde{t}_{1} \\ t_{5}^{R} \end{matrix} \leftarrow Δ \tilde{t}_{1} \to \begin{matrix} t_{4}^{L} \\ \tilde{t}_{2} \end{matrix} \leftarrow Δ \tilde{t}_{2} \to \begin{matrix} \tilde{t}_{3} \\ t_{6}^{R} \end{matrix} \leftarrow Δ \tilde{t}_{3} \to \begin{matrix} \tilde{t}_{4} \\ t_{7}^{R} \end{matrix} \leftarrow Δ \tilde{t}_{4} \to \begin{matrix} t_{5}^{L} \\ \tilde{t}_{5} \end{matrix} \dots \end{matrix}$

We call the indices of the step's start and end times in the union time sequence $n_{S}$ and $n_{E}$ , respectively. We define $n_{m}^{L}$ to be the union-time index corresponding to $t_{m}^{L}$ and $m_{n}^{L}$ to be the index of the last local time not later than $\tilde{t}_{n}$ and similarly for the remote side. So for the above example, $n_{4}^{L} = 2$ and $m_{2}^{R} = 5$ , and if we wish to compute the step from $t_{4}^{L}$ to $t_{5}^{L}$ we would have $m_{S} = 4$ , $n_{S} = 2$ , and $n_{E} = 5$ .

If we wish to evaluate the change over this step to $k$ th order, we can write the change in the value as a linear combination of the values of the coupling between the elements at unequal times:

$\begin{matrix} (2) & F_{m_{S}} = \sum_{q^{L} = m_{S} - (k - 1)}^{m_{S}} \sum_{q^{R} = m_{n_{S}}^{R} - (k - 1)}^{m_{n_{E} - 1}^{R}} D_{q^{L} q^{R}} I_{q^{L} q^{R}}, \end{matrix}$

where $D_{q^{L} q^{R}}$ is the coupling function evaluated between data from $t_{q^{L}}^{L}$ and $t_{q^{R}}^{R}$ . The coefficients can be written as the sum of three terms,

$\begin{matrix} (3) & I_{q^{L} q^{R}} = I_{q^{L} q^{R}}^{E} + I_{q^{L} q^{R}}^{R} + I_{q^{L} q^{R}}^{L}, \end{matrix}$

which can be interpreted as a contribution from equal-time evaluations and contributions related to the remote and local evaluation times. These are given by

$\begin{aligned} (4) & I_{q^{L} q^{R}}^{E} & = \sum_{n = n_{S}}^{min {n_{E}, n^{L} + k} - 1} {\tilde{α}}_{n, n - n^{L}} Δ \tilde{t}_{n} & if t_{q^{L}}^{L} = t_{q^{R}}^{R}, otherwise 0 \\ (5) & I_{q^{L} q^{R}}^{R} & = ℓ_{q^{L} - m_{S} + k} (\tilde{t}_{n^{R}}; t_{m_{S} - (k - 1)}^{L}, \dots, t_{m_{S}}^{L}) \sum_{n = max {n_{S}, n^{R}}}^{min {n_{E}, n^{R} + k} - 1} {\tilde{α}}_{n, n - n^{R}} Δ \tilde{t}_{n} & if t_{q^{R}}^{R} is not in {t_{\dots}^{L}}, otherwise 0 \\ (6) & I_{q^{L} q^{R}}^{L} & = \sum_{n = max {n_{S}, n^{R}}}^{min {n_{E}, n^{L} + k, n_{q^{R} + k}^{R}} - 1} ℓ_{q^{R} - m_{n}^{R} + k} (\tilde{t}_{n^{L}}; t_{m_{n}^{R} - (k - 1)}^{R}, \dots, t_{m_{n}^{R}}^{R}) {\tilde{α}}_{n, n - n^{L}} Δ \tilde{t}_{n} & if t_{q^{L}}^{L} is not in {t_{\dots}^{R}}, otherwise 0, \end{aligned}$

where for brevity we write $n^{L} = n_{q^{L}}^{L}$ and $n^{R} = n_{q^{R}}^{R}$ , and where $ℓ_{a} (t; x_{1}, \dots, x_{k})$ a Lagrange interpolating polynomial and ${\tilde{α}}_{n j}$ is the $j$ th coefficient for an Adams-Bashforth step over the union times from step $n$ to step $n + 1$ .

Member Function Documentation

◆ monotonic()

bool TimeSteppers::AdamsBashforth::monotonic ( ) const

overridevirtual

Whether computational and temporal orderings of operations match.

If this method returns true, then, for two time-stepper operations occurring at different simulation times, the temporally earlier operation will be performed first. These operations include RHS evaluation, dense output, and neighbor communication. In particular, dense output never requires performing a RHS evaluation later than the output time, so control systems measurements cannot cause deadlocks.

Warning: This guarantee only holds if the time steps themselves are monotonic, which can be violated during initialization.

Implements TimeStepper.

◆ neighbor_data_required() [1/2]

bool TimeSteppers::AdamsBashforth::neighbor_data_required	(	const TimeStepId &	next_substep_id,
		const TimeStepId &	neighbor_data_id
	)		const

overridevirtual

Check whether a neighbor record is needed for boundary output.

In order to perform boundary output, all records from the neighbor with TimeStepIds for which this method returns true should have been added to the history. Versions are provided for a substep and for dense output.

Implements LtsTimeStepper.

◆ neighbor_data_required() [2/2]

bool TimeSteppers::AdamsBashforth::neighbor_data_required	(	double	dense_output_time,
		const TimeStepId &	neighbor_data_id
	)		const

overridevirtual

Check whether a neighbor record is needed for boundary output.

In order to perform boundary output, all records from the neighbor with TimeStepIds for which this method returns true should have been added to the history. Versions are provided for a substep and for dense output.

Implements LtsTimeStepper.

◆ next_time_id()

TimeStepId TimeSteppers::AdamsBashforth::next_time_id	(	const TimeStepId &	current_id,
		const TimeDelta &	time_step
	)		const

overridevirtual

The TimeStepId after the current substep.

Implements TimeStepper.

◆ next_time_id_for_error()

TimeStepId TimeSteppers::AdamsBashforth::next_time_id_for_error	(	const TimeStepId &	current_id,
		const TimeDelta &	time_step
	)		const

overridevirtual

The TimeStepId after the current substep when providing an error measure for adaptive time-stepping.

Certain substep methods (e.g. embedded RK4(3)) require additional steps when providing an error measure of the integration.

Implements TimeStepper.

◆ number_of_past_steps()

size_t TimeSteppers::AdamsBashforth::number_of_past_steps ( ) const

overridevirtual

Number of past time steps needed for multi-step method.

Implements TimeStepper.

◆ number_of_substeps()

uint64_t TimeSteppers::AdamsBashforth::number_of_substeps ( ) const

overridevirtual

Number of substeps in this TimeStepper.

Implements TimeStepper.

◆ number_of_substeps_for_error()

uint64_t TimeSteppers::AdamsBashforth::number_of_substeps_for_error ( ) const

overridevirtual

Number of substeps in this TimeStepper when providing an error measure for adaptive time-stepping.

Details

Certain substep methods (e.g. embedded RK4(3)) require additional steps when providing an error measure of the integration.

Implements TimeStepper.

◆ order()

size_t TimeSteppers::AdamsBashforth::order ( ) const

overridevirtual

The convergence order of the stepper.

Implements TimeStepper.

◆ stable_step()

double TimeSteppers::AdamsBashforth::stable_step ( ) const

overridevirtual

Rough estimate of the maximum step size this method can take stably as a multiple of the step for Euler's method.

Implements TimeStepper.

Member Data Documentation

◆ help

constexpr Options::String TimeSteppers::AdamsBashforth::help

staticconstexpr

Initial value:

= {

"An Adams-Bashforth Nth order time-stepper."}

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

src/Time/TimeSteppers/AdamsBashforth.hpp

Classes

Public Types

Public Member Functions

Static Public Attributes

Friends

Detailed Description

Local time stepping calculation

Member Function Documentation

◆ monotonic()

◆ neighbor_data_required() [1/2]

◆ neighbor_data_required() [2/2]

◆ next_time_id()

◆ next_time_id_for_error()

◆ number_of_past_steps()

◆ number_of_substeps()

◆ number_of_substeps_for_error()

Details

◆ order()

◆ stable_step()

Member Data Documentation

◆ help