TNL::Solvers::ODE::StaticEuler< Real > Class Template Reference

Solver of ODEs with the first order of accuracy. More...

#include <TNL/Solvers/ODE/StaticEuler.h>

Inheritance diagram for TNL::Solvers::ODE::StaticEuler< Real >:

Collaboration diagram for TNL::Solvers::ODE::StaticEuler< Real >:

Public Types
using	DofVectorType = VectorType
	Alias for type of unknown variable \( x \).
using	IndexType = int
	Type for indexing.
using	RealType = Real
	Type of floating-point arithemtics.
using	VectorType = Real
	Type of unknown variable \( x \).
Public Types inherited from TNL::Solvers::ODE::StaticExplicitSolver< Real, int >
using	IndexType = int
	Indexing type.
using	RealType = Real
	Type of the floating-point arithmetics or static vector.

Public Member Functions
__cuda_callable__	StaticEuler ()=default
	Default constructor.
__cuda_callable__ const RealType &	getCourantNumber () const
	Getter for the Courant number.
__cuda_callable__ void	setCourantNumber (const RealType &c)
	This method sets the Courant number in the CFL condition.
bool	setup (const Config::ParameterContainer &parameters, const std::string &prefix="")
	Method for setup of the explicit solver based on configuration parameters.
template<typename RHSFunction , typename... Args>
__cuda_callable__ bool	solve (VectorType &u, RHSFunction &&f, Args... args)
	Solve ODE given by a lambda function.
Public Member Functions inherited from TNL::Solvers::ODE::StaticExplicitSolver< Real, int >
__cuda_callable__	StaticExplicitSolver ()=default
	Default constructor.
bool __cuda_callable__	checkNextIteration ()
	Checks if the solver is allowed to do the next iteration.
__cuda_callable__ const RealType &	getMaxTau () const
	Getter of maximal value of the time step.
__cuda_callable__ const RealType &	getStopTime () const
	Getter of the time where the evolution computation shall by stopped.
__cuda_callable__ const RealType &	getTau () const
	Getter of the time step used for the computation.
__cuda_callable__ const RealType &	getTime () const
	Getter of the current time of the evolution computed by the solver.
__cuda_callable__ void	setMaxTau (const RealType &maxTau)
	Setter of maximal value of the time step.
__cuda_callable__ void	setStopTime (const RealType &stopTime)
	Setter of the time where the evolution computation shall by stopped.
__cuda_callable__ void	setTau (const RealType &tau)
	Setter of the time step used for the computation.
__cuda_callable__ void	setTestingMode (bool testingMode)
__cuda_callable__ void	setTime (const RealType &t)
	Settter of the current time of the evolution computed by the solver.
bool	setup (const Config::ParameterContainer &parameters, const std::string &prefix="")
	Method for setup of the iterative solver based on configuration parameters.
Public Member Functions inherited from TNL::Solvers::StaticIterativeSolver< Real, Index >
__cuda_callable__	StaticIterativeSolver ()=default
	Default constructor.
__cuda_callable__ bool	checkConvergence ()
	Checks whether the convergence occurred already.
__cuda_callable__ bool	checkNextIteration ()
	Checks if the solver is allowed to do the next iteration.
__cuda_callable__ const Real &	getConvergenceResidue () const
	Gets the the convergence threshold.
__cuda_callable__ const Real &	getDivergenceResidue () const
	Gets the limit for the divergence criterion.
__cuda_callable__ const Index &	getIterations () const
	Gets the number of iterations performed by the solver so far.
__cuda_callable__ const Index &	getMaxIterations () const
	Gets the maximal number of iterations the solver is allowed to perform.
__cuda_callable__ const Index &	getMinIterations () const
	Gets the minimal number of iterations the solver is supposed to do.
__cuda_callable__ const Real &	getResidue () const
	Gets the residue reached at the current iteration.
__cuda_callable__ bool	nextIteration ()
	Proceeds to the next iteration.
__cuda_callable__ void	resetIterations ()
	Sets the the number of the current iterations to zero.
__cuda_callable__ void	setConvergenceResidue (const Real &convergenceResidue)
	Sets the threshold for the convergence.
__cuda_callable__ void	setDivergenceResidue (const Real &divergenceResidue)
	Sets the residue limit for the divergence criterion.
__cuda_callable__ void	setMaxIterations (const Index &maxIterations)
	Sets the maximal number of iterations the solver is allowed to perform.
__cuda_callable__ void	setMinIterations (const Index &minIterations)
	Sets the minimal number of iterations the solver is supposed to do.
__cuda_callable__ void	setResidue (const Real &residue)
	Sets the residue reached at the current iteration.
bool	setup (const Config::ParameterContainer &parameters, const std::string &prefix="")
	Method for setup of the iterative solver based on configuration parameters.

Static Public Member Functions
static void	configSetup (Config::ConfigDescription &config, const std::string &prefix="")
	Static method for setup of configuration parameters.
Static Public Member Functions inherited from TNL::Solvers::ODE::StaticExplicitSolver< Real, int >
static void	configSetup (Config::ConfigDescription &config, const std::string &prefix="")
	This method defines configuration entries for setup of the iterative solver.
Static Public Member Functions inherited from TNL::Solvers::StaticIterativeSolver< Real, Index >
static void	configSetup (Config::ConfigDescription &config, const std::string &prefix="")
	This method defines configuration entries for setup of the iterative solver.

Protected Attributes
RealType	courantNumber = 0.0
DofVectorType	k1
Protected Attributes inherited from TNL::Solvers::ODE::StaticExplicitSolver< Real, int >
RealType	maxTau
bool	stopOnSteadyState
RealType	stopTime
RealType	tau
bool	testingMode
RealType	time
Protected Attributes inherited from TNL::Solvers::StaticIterativeSolver< Real, Index >
Real	convergenceResidue = 1e-6
Index	currentIteration = 0
Real	currentResidue = 0
Real	divergenceResidue = std::numeric_limits< Real >::max()
Index	maxIterations = 1000000000
Index	minIterations = 0
Index	refreshRate = 1

Detailed Description

template<typename Real>
class TNL::Solvers::ODE::StaticEuler< Real >

Solver of ODEs with the first order of accuracy.

This solver is based on the Euler method for solving of ordinary differential equations having the following form:

\( \frac{d u}{dt} = f( t, u) \text{ on } (0,T) \)

\( u( 0 ) = u_{ini} \).

It is supposed to be used when the unknown \( x \in R \) is expressed by a scalar, i.e. by a numeric type like double or float.

For problems where \( \vec x\) is represented by TNL::Containers::StaticVector, see TNL::Solvers::ODE::StaticEuler<Containers::StaticVector<Size_,Real>>. For problems where \( \vec x\) is represented by TNL::Containers::Vector, or TNL::Containers::VectorView, see TNL::Solvers::ODE::Euler.

The following example demonstrates the use the solvers:

#include <iostream>
#include <TNL/Solvers/ODE/StaticEuler.h>
 
using Real = double;
 
int main( int argc, char* argv[] )
{
   using ODESolver = TNL::Solvers::ODE::StaticEuler< Real >;
   const Real final_t = 10.0;
   const Real tau = 0.001;
   const Real output_time_step = 0.25;
 
   ODESolver solver;
   solver.setTau(  tau );
   solver.setTime( 0.0 );
   Real u = 0.0;
   while( solver.getTime() < final_t )
   {
      solver.setStopTime( TNL::min( solver.getTime() + output_time_step, final_t ) );
      auto f = [] ( const Real& t, const Real& tau, const Real& u, Real& fu ) {
         fu = t * sin( t );
      };
      solver.solve( u, f );
      std::cout << solver.getTime() << " " << u << std::endl;
   }
}

The result looks as follows:

0.001 0
0.002 1e-09
0.252 0.00526916
0.502 0.0409948
0.752 0.13364
1.002 0.302432
1.252 0.556612
1.502 0.893635
1.752 1.2985
2.002 1.74432
2.252 2.19409
2.502 2.60357
2.752 2.92506
3.002 3.11173
3.252 3.1222
3.502 2.92497
3.752 2.50232
4.002 1.85323
4.252 0.995174
4.502 -0.0355494
4.752 -1.18502
5.002 -2.38443
5.252 -3.55415
5.502 -4.60904
5.752 -5.46451
6.002 -6.04295
6.252 -6.28004
6.502 -6.1306
6.752 -5.57319
7.002 -4.61342
7.252 -3.28541
7.502 -1.65121
7.752 0.201757
8.002 2.16523
8.252 4.11644
8.502 5.92576
8.752 7.46532
9.002 8.61785
9.252 9.28537
9.502 9.3969
9.752 8.9147
10 7.84941

Since this variant of the Euler solver is static, it can be used even inside of GPU kernels and so combined with TNL::Algorithms::parallelFor as demonstrated by the following example:

#include <iostream>
#include <fstream>
#include <TNL/Solvers/ODE/StaticEuler.h>
#include <TNL/Containers/Vector.h>
#include <TNL/Algorithms/parallelFor.h>
 
using Real = double;
 
template< typename Device >
void solveParallelODEs( const char* file_name )
{
   using ODESolver = TNL::Solvers::ODE::StaticEuler< Real >;
   using Vector = TNL::Containers::Vector< Real, Device >;
   const Real final_t = 10.0;
   const Real tau = 0.001;
   const Real output_time_step = 0.1;
   const Real c_min = 1.0;
   const Real c_max = 5.0;
   const int c_vals = 5.0;
   const Real c_step = ( c_max - c_min ) / ( c_vals - 1 );
   const int output_time_steps = ceil( final_t / output_time_step ) + 1;
 
   Vector results( output_time_steps * c_vals, 0.0 );
   auto results_view = results.getView();
   auto f = [=] __cuda_callable__ ( const Real& t, const Real& tau, const Real& u, Real& fu, const Real& c ) {
         fu = t * sin( c * t );
      };
   auto solve = [=] __cuda_callable__ ( int idx ) mutable {
      const Real c = c_min + idx * c_step;
      ODESolver solver;
      solver.setTau(  tau );
      solver.setTime( 0.0 );
      Real u = 0.0;
      int time_step( 1 );
      results_view[ idx ] = u;
      while( time_step < output_time_steps )
      {
         solver.setStopTime( TNL::min( solver.getTime() + output_time_step, final_t ) );
         solver.solve( u, f, c );
         results_view[ time_step++ * c_vals + idx ] = u;
      }
   };
   TNL::Algorithms::parallelFor< Device >( 0, c_vals, solve );
 
   std::fstream file;
   file.open( file_name, std::ios::out );
   for( int i = 0; i < c_vals; i++ )
   {
      file << "# c = " << c_min + i * c_step << std::endl;
      for( int k = 0; k < output_time_steps;k++ )
         file << k * output_time_step << " " << results.getElement( k * c_vals + i ) << std::endl;
      file << std::endl;
   }
}
 
int main( int argc, char* argv[] )
{
   TNL::String file_name( argv[ 1 ] );
   file_name += "/StaticODESolver-SineParallelExample-result.out";
   solveParallelODEs< TNL::Devices::Host >( file_name.getString() );
#ifdef __CUDACC__
   solveParallelODEs< TNL::Devices::Cuda >( file_name.getString() );
#endif
}

Template Parameters

Real	is floating point number type, it is type of \( x \) in this case.

Member Function Documentation

configSetup()

template<typename Real >

void TNL::Solvers::ODE::StaticEuler< Real >::configSetup ( Config::ConfigDescription &  config, const std::string &  prefix = ""  )

static

Static method for setup of configuration parameters.

Parameters

config	is the config description.
prefix	is the prefix of the configuration parameters for this solver.

getCourantNumber()

template<typename Real >

__cuda_callable__ auto TNL::Solvers::ODE::StaticEuler< Real >::getCourantNumber

(

)

const

Getter for the Courant number.

Returns

the Courant number.

setCourantNumber()

template<typename Real >

__cuda_callable__ void TNL::Solvers::ODE::StaticEuler< Real >::setCourantNumber

(

const RealType &

c

)

This method sets the Courant number in the CFL condition.

This method sets the constant \( C \) in the Courant–Friedrichs–Lewy condition. It means that

\[ \Delta t = \frac{C}{\| f( t,x )\|}, \]

.

if \( C > 0\). If \( C = 0 \) the time step stays fixed.

Parameters

c	is the Courant number.

setup()

template<typename Real >

bool TNL::Solvers::ODE::StaticEuler< Real >::setup	(	const Config::ParameterContainer &	parameters,
		const std::string &	prefix = ""
	)

Method for setup of the explicit solver based on configuration parameters.

Parameters

parameters	is the container for configuration parameters.
prefix	is the prefix of the configuration parameters for this solver.

Returns

true if the parameters where parsed successfully.

false if the method did not succeed to read the configuration parameters.

solve()

template<typename Real >

template<typename RHSFunction , typename... Args>

__cuda_callable__ bool TNL::Solvers::ODE::StaticEuler< Real >::solve	(	VectorType &	u,
		RHSFunction &&	f,
		Args...	args
	)

Solve ODE given by a lambda function.

Template Parameters

RHSFunction

is type of a lambda function representing the right-hand side of the ODE system. The definition of the lambda function reads as: auto f = [=] ( const Real& t, const Real& tau, const Real& u, Real& fu, Args... args ) {...} where t is the current time of the evolution, tau is the current time step, u is the solution at the current time, fu is variable/static vector into which the lambda function is suppsed to evaluate the function \( f(t, \vec x) \) at the current time \( t \).

Parameters

u	is a variable/static vector representing the solution of the ODE system at current time.
f	is the lambda function representing the right-hand side of the ODE system.
args	are user define arguments which are passed to the lambda function f.

Returns

true if steady state solution has been reached, false otherwise.

---

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

• src/TNL/Solvers/ODE/StaticEuler.h
• src/TNL/Solvers/ODE/StaticEuler.hpp