Sweeper for the van der Pol oscillator. More...

#include <vdp_sweeper.hpp>

Inheritance diagram for pfasst::examples::vdp::VdpSweeper< time >:

Collaboration diagram for pfasst::examples::vdp::VdpSweeper< time >:

Public Member Functions
	VdpSweeper (double nu, double x0, double y0)
	generic constructor; initialize all function call counters with zero. More...

virtual	~VdpSweeper ()
	upon destruction, report final error and number of function calls and close output file. More...

void	echo_error (time t)
	computes and prints out error, which is meaningful only for \( \nu=0 \). More...

double	get_errors ()
	returns error, but does not update it! More...

void	post_predict () override
	post prediction step, update error. More...

void	post_sweep () override
	post sweep, update error. More...

void	exact (real_vector_type &q, time t)
	if \( \nu=0 \), this computes the exact solution at time `t`. More...

void	exact (shared_ptr< encap_type > q_encap, time t)

void	f_impl_eval (shared_ptr< encap_type > f_encap, shared_ptr< encap_type > q_encap, time t) override
	Evaluate the implicit part of the right hand side. More...

void	impl_solve (shared_ptr< encap_type > f_encap, shared_ptr< encap_type > q_encap, time t, time dt, shared_ptr< encap_type > rhs_encap) override
	for given \( b \), solve \( \left( u - \Delta t f(u) \right) u = b \) for \( u \) and set `f_encap` to \( f(u) \). More...

Public Member Functions inherited from pfasst::encap::ImplicitSweeper< time >
	ImplicitSweeper ()=default

virtual	~ImplicitSweeper ()=default

virtual void	setup (bool coarse) override
	Setup (allocate etc) the sweeper. More...

virtual void	predict (bool initial) override
	Compute low-order provisional solution. More...

virtual void	sweep () override
	Perform one SDC sweep/iteration. More...

virtual void	advance () override
	Advance the end solution to start solution. More...

virtual void	reevaluate (bool initial_only) override

virtual void	integrate (time dt, vector< shared_ptr< Encapsulation< time >>> dst) const override
	Integrates values of right hand side at all time nodes \( t \in [0,M-1] \) simultaneously. More...

virtual void	f_impl_eval (shared_ptr< Encapsulation< time >> f_impl_encap, shared_ptr< Encapsulation< time >> u_encap, time t)
	Evaluate the implicit part of the ODE. More...

virtual void	impl_solve (shared_ptr< Encapsulation< time >> f_encap, shared_ptr< Encapsulation< time >> u_encap, time t, time dt, shared_ptr< Encapsulation< time >> rhs_encap)
	Solve \( U - \Delta t F_{\rm impl}(U) = RHS \) for \( U \). More...

Public Member Functions inherited from pfasst::encap::EncapSweeper< time >
	EncapSweeper ()

virtual shared_ptr< Encapsulation< time > >	get_state (size_t m) const
	Retrieve solution values of current iteration at time node index `m`. More...

virtual shared_ptr< Encapsulation< time > >	get_tau (size_t m) const
	Retrieve FAS correction of current iteration at time node index `m`. More...

virtual shared_ptr< Encapsulation< time > >	get_saved_state (size_t m) const
	Retrieve solution values of previous iteration at time node index `m`. More...

virtual void	set_options () override
	Set options from command line etc. More...

virtual void	spread () override
	Initialize solution values at all time nodes with meaningful values. More...

virtual void	save (bool initial_only) override
	Save states (and/or function values) at all nodes. More...

virtual void	set_quadrature (shared_ptr< IQuadrature< time >> quadrature)

virtual shared_ptr< const IQuadrature< time > >	get_quadrature () const

virtual const vector< time >	get_nodes () const

virtual void	set_factory (shared_ptr< EncapFactory< time >> factory)

virtual shared_ptr< EncapFactory< time > >	get_factory () const

virtual shared_ptr< Encapsulation< time > >	get_start_state () const

virtual shared_ptr< Encapsulation< time > >	get_end_state ()

void	set_residual_tolerances (time abs_residual_tol, time rel_residual_tol, int order=0)
	Set residual tolerances for convergence checking. More...

virtual void	residual (time dt, vector< shared_ptr< Encapsulation< time >>> dst) const
	Compute residual at each SDC node (including FAS corrections). More...

virtual bool	converged () override
	Return convergence status. More...

virtual void	post (ICommunicator *comm, int tag) override

virtual void	send (ICommunicator *comm, int tag, bool blocking) override

virtual void	recv (ICommunicator *comm, int tag, bool blocking) override

virtual void	broadcast (ICommunicator *comm) override

Public Member Functions inherited from pfasst::ISweeper< time >
	ISweeper ()

virtual	~ISweeper ()

virtual void	set_controller (Controller< time > *ctrl)
	Set the sweepers controller. More...

virtual Controller< time > *	get_controller ()
	Accessor to the controller managing this sweeper. More...

virtual void	post_step ()
	Hook automatically run after each completed time step. More...

Private Types
typedef encap::Encapsulation< time >	encap_type

typedef encap::VectorEncapsulation< double >	real_vector_type
	Define a type for a complex PFASST vector encapsulation. More...

Private Attributes
double	nu
	Parameter \( \nu \) in the van-der-Pol oscillator. More...

double	x0
	Starting values. More...

double	y0

size_t	newton_maxit
	Maximum number of iterations for the nonlinear Newton solver. More...

double	newton_tol
	Tolerance for the nonlinear Newton iteration. More...

size_t	n_f_impl_eval
	Counters for how often `f_impl_eval` and `impl_solve` are called. More...

size_t	n_impl_solve

size_t	n_newton_iter

fstream	output_file
	Output file. More...

double	error
	error: For \( \nu=0 \), vdP reduces to linear oscillator, otherwise no analytic solution is available More...

Additional Inherited Members
Protected Member Functions inherited from pfasst::encap::ImplicitSweeper< time >
void	set_end_state ()

Protected Attributes inherited from pfasst::encap::ImplicitSweeper< time >
Matrix< time >	q_tilde

vector< shared_ptr< Encapsulation< time > > >	s_integrals
	Node-to-node integrals of \( F(t,u) \) at all time nodes of the current iteration. More...

vector< shared_ptr< Encapsulation< time > > >	fs_impl
	Values of the implicit part of the right hand side \( F_{impl}(t,u) \) at all time nodes of the current iteration. More...

Protected Attributes inherited from pfasst::encap::EncapSweeper< time >
string	FORMAT_STR

shared_ptr< IQuadrature< time > >	quadrature

shared_ptr< EncapFactory< time > >	factory
	Encapsulation data structure factory. More...

shared_ptr< Encapsulation< time > >	start_state
	Separate start state, i.e. initial condition for the sweeper's current time step. More...

shared_ptr< Encapsulation< time > >	end_state
	Current solution at \( T_{end} \). More...

vector< shared_ptr< Encapsulation< time > > >	residuals
	Place for the residuals at the different time nodes. More...

vector< shared_ptr< Encapsulation< time > > >	state
	Solution values \( U \) at all time nodes of the current iteration. More...

vector< shared_ptr< Encapsulation< time > > >	saved_state
	Solution values \( U \) at all time nodes of the previous iteration. More...

vector< shared_ptr< Encapsulation< time > > >	fas_corrections
	FAS corrections \( \tau \) at all time nodes of the current iteration. More...

int	residual_norm_order

time	abs_residual_tol
	Tolerance for absolute residual. More...

time	rel_residual_tol
	Tolerance for relative residual. More...

Protected Attributes inherited from pfasst::ISweeper< time >
Controller< time > *	controller
	Backreference to the controller managing the sweeper instance. More...

Detailed Description

template<typename time = pfasst::time_precision>
class pfasst::examples::vdp::VdpSweeper< time >

Sweeper for the van der Pol oscillator.

\[ x' = y\\ y' = \nu*(1 - x^2)*y - x \]

Note that an analytical solution is available only for \( \nu=0 \), where the vdP simplifies to the standard linear oscillator. Hence, actual errors are computed only for \( \nu=0 \).

Definition at line 53 of file vdp_sweeper.hpp.

Member Typedef Documentation

template<typename time = pfasst::time_precision>

typedef encap::Encapsulation<time> pfasst::examples::vdp::VdpSweeper< time >::encap_type

private

Definition at line 57 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

typedef encap::VectorEncapsulation<double> pfasst::examples::vdp::VdpSweeper< time >::real_vector_type

private

Define a type for a complex PFASST vector encapsulation.

Definition at line 60 of file vdp_sweeper.hpp.

Constructor & Destructor Documentation

template<typename time = pfasst::time_precision>

pfasst::examples::vdp::VdpSweeper< time >::VdpSweeper	(	double	nu,
		double	x0,
		double	y0
	)

inline

generic constructor; initialize all function call counters with zero.

Also open file to write solution.

Parameters

[in] coefficient nu of van-der-Pol oscillator

Definition at line 91 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

virtual pfasst::examples::vdp::VdpSweeper< time >::~VdpSweeper ( )

inlinevirtual

upon destruction, report final error and number of function calls and close output file.

Definition at line 110 of file vdp_sweeper.hpp.

References ML_LOG.

Member Function Documentation

template<typename time = pfasst::time_precision>

void pfasst::examples::vdp::VdpSweeper< time >::echo_error ( time t )

inline

computes and prints out error, which is meaningful only for \( \nu=0 \).

Also writes the current solution into an ascii file

Parameters

[in] t Time

Definition at line 124 of file vdp_sweeper.hpp.

References pfasst::examples::boris::max(), and ML_LOG.

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template<typename time = pfasst::time_precision>

void pfasst::examples::vdp::VdpSweeper< time >::exact	(	real_vector_type &	q,
		time	t
	)

inline

if \( \nu=0 \), this computes the exact solution at time t.

For other values of \( \nu \), it just returns the inital value, because there is no analytical solution to evaluate.

Parameters

[in] t Time

For \( \nu=0 \) and given initial value \( x0 \), \( y0 \), the analytic solution reads

\[ x(t) = y0*sin(t) + x0*cos(t)\\ y(t) = -x0*sin(t) + y0*cos(t) \]

Note that for \( t=0 \), we recover \( x(0) = x0 \), \( y(0) = y0 \).

Definition at line 178 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

void pfasst::examples::vdp::VdpSweeper< time >::exact	(	shared_ptr< encap_type >	q_encap,
		time	t
	)

inline

Definition at line 205 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

void pfasst::examples::vdp::VdpSweeper< time >::f_impl_eval	(	shared_ptr< encap_type >	f_encap,
		shared_ptr< encap_type >	q_encap,
		time	t
	)

inlineoverride

Evaluate the implicit part of the right hand side.

Definition at line 214 of file vdp_sweeper.hpp.

References UNUSED.

template<typename time = pfasst::time_precision>

double pfasst::examples::vdp::VdpSweeper< time >::get_errors ( )

inline

returns error, but does not update it!

Definition at line 144 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

void pfasst::examples::vdp::VdpSweeper< time >::impl_solve	(	shared_ptr< encap_type >	f_encap,
		shared_ptr< encap_type >	q_encap,
		time	t,
		time	dt,
		shared_ptr< encap_type >	rhs_encap
	)

inlineoverride

for given \( b \), solve \( \left( u - \Delta t f(u) \right) u = b \) for \( u \) and set f_encap to \( f(u) \).

Solves the nonlinear equation

\[ P(q) := q - dt*f(q) - rhs = 0 \]

using Newton's method. For the vdP oscillator, it is

\[ P(q) = \left( x - dt*y - rhs_0 ; y - dt*\left( \nu*(1-x^2)*y -x \right) - rhs_1 \right) \]

The Newton update reads

\[ q_new = q + inv(P'(q))*(-P(q)) \]

Note that \( P'(q) = Id - dt*f'(q) \).

The inverse of \( P' \) has been computed symbolically, so that \( P'(q)*x \) can be evaluated directly.

The Jacobian of the right hand side of the van der Pol oscillator for \( q=[x;y] \) is

\[ Df(q) = \left( 0 1 ; -2*\nu*x*y - 1 \nu*(1-x^2) \right) \]

Because here we need to invert \( Id - dt*f(q) = b \), the corresponding Jacobian is

\[ J(q) = \left( 1 -dt ; dt*2*\nu*x*y+1 1 + dt*\nu(x^2-1) \right) \]

which can be inverted analytically

\[ inv(J(q)) = (1/c)*\left( dt*x^2-dt+1 dt ; -2*dt*\nu*x*y-dt 1\right) =: [a dt; b 1.0] \]

with \( c:=2*\nu*x*y*dt^2 + dt^2 + dt*x^2 - dt + 1 \)

Definition at line 234 of file vdp_sweeper.hpp.

References plot::dt, and UNUSED.

template<typename time = pfasst::time_precision>

void pfasst::examples::vdp::VdpSweeper< time >::post_predict ( )

inlineoverridevirtual

post prediction step, update error.

Reimplemented from pfasst::ISweeper< time >.

Definition at line 152 of file vdp_sweeper.hpp.

References plot::dt, and plot::t.

template<typename time = pfasst::time_precision>

void pfasst::examples::vdp::VdpSweeper< time >::post_sweep ( )

inlineoverridevirtual

post sweep, update error.

Reimplemented from pfasst::ISweeper< time >.

Definition at line 162 of file vdp_sweeper.hpp.

References plot::dt, and plot::t.

Member Data Documentation

template<typename time = pfasst::time_precision>

double pfasst::examples::vdp::VdpSweeper< time >::error

private

error: For \( \nu=0 \), vdP reduces to linear oscillator, otherwise no analytic solution is available

Definition at line 81 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

size_t pfasst::examples::vdp::VdpSweeper< time >::n_f_impl_eval

private

Counters for how often f_impl_eval and impl_solve are called.

Definition at line 75 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

size_t pfasst::examples::vdp::VdpSweeper< time >::n_impl_solve

private

Definition at line 75 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

size_t pfasst::examples::vdp::VdpSweeper< time >::n_newton_iter

private

Definition at line 75 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

size_t pfasst::examples::vdp::VdpSweeper< time >::newton_maxit

private

Maximum number of iterations for the nonlinear Newton solver.

Definition at line 69 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

double pfasst::examples::vdp::VdpSweeper< time >::newton_tol

private

Tolerance for the nonlinear Newton iteration.

Definition at line 72 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

double pfasst::examples::vdp::VdpSweeper< time >::nu

private

Parameter \( \nu \) in the van-der-Pol oscillator.

Definition at line 63 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

fstream pfasst::examples::vdp::VdpSweeper< time >::output_file

private

Output file.

Definition at line 78 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

double pfasst::examples::vdp::VdpSweeper< time >::x0

private

Starting values.

Definition at line 66 of file vdp_sweeper.hpp.

template<typename time = pfasst::time_precision>

double pfasst::examples::vdp::VdpSweeper< time >::y0

private

Definition at line 66 of file vdp_sweeper.hpp.

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

examples/vanderpol/vdp_sweeper.hpp

Public Member Functions

Private Types

Private Attributes

Additional Inherited Members

Detailed Description

template<typename time = pfasst::time_precision> class pfasst::examples::vdp::VdpSweeper< time >

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename time = pfasst::time_precision>
class pfasst::examples::vdp::VdpSweeper< time >