/usr/include/deal.II/base/time_stepping.templates.h is in libdeal.ii-dev 8.4.2-2+b1.
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//
// Copyright (C) 2014 - 2015 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__time_stepping_templates_h
#define dealii__time_stepping_templates_h
#include <deal.II/base/std_cxx11/bind.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/time_stepping.h>
DEAL_II_NAMESPACE_OPEN
namespace TimeStepping
{
// ----------------------------------------------------------------------
// RungeKutta
// ----------------------------------------------------------------------
template <typename VectorType>
double RungeKutta<VectorType>::evolve_one_time_step(
std::vector<std_cxx11::function<VectorType (const double, const VectorType &)> > &F,
std::vector<std_cxx11::function<VectorType (const double, const double, const VectorType &)> > &J_inverse,
double t,
double delta_t,
VectorType &y)
{
AssertThrow(F.size()==0,
ExcMessage("RungeKutta methods cannot handle more that one function to integate."));
AssertThrow(J_inverse.size()==0,
ExcMessage("RungeKutta methods cannot handle more that one function to integate."));
return evolve_one_time_step(F[0],J_inverse[0],t,delta_t,y);
}
// ----------------------------------------------------------------------
// ExplicitRungeKutta
// ----------------------------------------------------------------------
template <typename VectorType>
ExplicitRungeKutta<VectorType>::ExplicitRungeKutta(runge_kutta_method method)
{
initialize(method);
}
template <typename VectorType>
void ExplicitRungeKutta<VectorType>::initialize(runge_kutta_method method)
{
status.method = method;
switch (method)
{
case (FORWARD_EULER) :
{
this->n_stages = 1;
this->b.push_back(1.0);
this->c.push_back(0.0);
break;
}
case (RK_THIRD_ORDER) :
{
this->n_stages = 3;
this->b.reserve(this->n_stages);
this->c.reserve(this->n_stages);
this->b.push_back(1.0/6.0);
this->b.push_back(2.0/3.0);
this->b.push_back(1.0/6.0);
this->c.push_back(0.0);
this->c.push_back(0.5);
this->c.push_back(1.0);
std::vector<double> tmp;
this->a.push_back(tmp);
tmp.resize(1);
tmp[0] = 0.5;
this->a.push_back(tmp);
tmp.resize(2);
tmp[0] = -1.0;
tmp[1] = 2.0;
this->a.push_back(tmp);
break;
}
case (RK_CLASSIC_FOURTH_ORDER) :
{
this->n_stages = 4;
this->b.reserve(this->n_stages);
this->c.reserve(this->n_stages);
std::vector<double> tmp;
this->a.push_back(tmp);
tmp.resize(1);
tmp[0] = 0.5;
this->a.push_back(tmp);
tmp.resize(2);
tmp[0] = 0.0;
tmp[1] = 0.5;
this->a.push_back(tmp);
tmp.resize(3);
tmp[1] = 0.0;
tmp[2] = 1.0;
this->a.push_back(tmp);
this->b.push_back(1.0/6.0);
this->b.push_back(1.0/3.0);
this->b.push_back(1.0/3.0);
this->b.push_back(1.0/6.0);
this->c.push_back(0.0);
this->c.push_back(0.5);
this->c.push_back(0.5);
this->c.push_back(1.0);
break;
}
default :
{
AssertThrow(false,ExcMessage("Unimplemented explicit Runge-Kutta method."));
}
}
}
template <typename VectorType>
double ExplicitRungeKutta<VectorType>::evolve_one_time_step
(std_cxx11::function<VectorType (const double, const VectorType &)> f,
std_cxx11::function<VectorType (const double, const double, const VectorType &)> /*id_minus_tau_J_inverse*/,
double t,
double delta_t,
VectorType &y)
{
return evolve_one_time_step(f,t,delta_t,y);
}
template <typename VectorType>
double ExplicitRungeKutta<VectorType>::evolve_one_time_step
(std_cxx11::function<VectorType (const double, const VectorType &)> f,
double t,
double delta_t,
VectorType &y)
{
std::vector<VectorType> f_stages(this->n_stages,y);
// Compute the different stages needed.
compute_stages(f,t,delta_t,y,f_stages);
// Linear combinations of the stages.
for (unsigned int i=0; i<this->n_stages; ++i)
y.sadd(1.,delta_t *this->b[i],f_stages[i]);
return (t+delta_t);
}
template <typename VectorType>
const typename ExplicitRungeKutta<VectorType>::Status &ExplicitRungeKutta<VectorType>::get_status() const
{
return status;
}
template <typename VectorType>
void ExplicitRungeKutta<VectorType>::compute_stages
(std_cxx11::function<VectorType (const double, const VectorType &)> f,
const double t,
const double delta_t,
const VectorType &y,
std::vector<VectorType> &f_stages) const
{
for (unsigned int i=0; i<this->n_stages; ++i)
{
VectorType Y(y);
for (unsigned int j=0; j<i; ++j)
Y.sadd(1.,delta_t *this->a[i][j],f_stages[j]);
// Evaluate the function f at the point (t+c[i]*delta_t,Y).
f_stages[i] = f(t+this->c[i]*delta_t,Y);
}
}
// ----------------------------------------------------------------------
// ImplicitRungeKutta
// ----------------------------------------------------------------------
template <typename VectorType>
ImplicitRungeKutta<VectorType>::ImplicitRungeKutta(runge_kutta_method method,
unsigned int max_it,
double tolerance)
:
RungeKutta<VectorType> (),
skip_linear_combi(false),
max_it(max_it),
tolerance(tolerance)
{
initialize(method);
}
template <typename VectorType>
void ImplicitRungeKutta<VectorType>::initialize(runge_kutta_method method)
{
status.method = method;
switch (method)
{
case (BACKWARD_EULER) :
{
this->n_stages = 1;
this->a.push_back(std::vector<double>(1, 1.0));
this->b.push_back(1.0);
this->c.push_back(1.0);
break;
}
case (IMPLICIT_MIDPOINT) :
{
this->a.push_back(std::vector<double>(1, 0.5));
this->b.push_back(1.0);
this->c.push_back(0.5);
this->n_stages = 1;
break;
}
case (CRANK_NICOLSON) :
{
this->n_stages = 2;
this->b.reserve(this->n_stages);
this->c.reserve(this->n_stages);
this->a.push_back(std::vector<double>(1, 0.0));
this->a.push_back(std::vector<double>(2, 0.5));
this->b.push_back(0.5);
this->b.push_back(0.5);
this->c.push_back(0.0);
this->c.push_back(1.0);
break;
}
case (SDIRK_TWO_STAGES) :
{
this->n_stages = 2;
this->b.reserve(this->n_stages);
this->c.reserve(this->n_stages);
double const gamma = 1.0 - 1.0 / std::sqrt(2.0);
this->b.push_back(1.0 - gamma);
this->b.push_back(gamma);
this->a.push_back(std::vector<double>(1, gamma));
this->a.push_back(this->b);
this->c.push_back(gamma);
this->c.push_back(1.0);
break;
}
default :
{
AssertThrow(false,ExcMessage("Unimplemented implicit Runge-Kutta method."));
}
}
}
template <typename VectorType>
double ImplicitRungeKutta<VectorType>::evolve_one_time_step
(std_cxx11::function<VectorType (const double, const VectorType &)> f,
std_cxx11::function<VectorType (const double, const double, const VectorType &)> id_minus_tau_J_inverse,
double t,
double delta_t,
VectorType &y)
{
VectorType old_y(y);
std::vector<VectorType> f_stages(this->n_stages,y);
// Compute the different stages needed.
compute_stages(f,id_minus_tau_J_inverse,t,delta_t,y,f_stages);
// If necessary, compute the linear combinations of the stages.
if (skip_linear_combi==false)
{
y = old_y;
for (unsigned int i=0; i<this->n_stages; ++i)
y.sadd(1.,delta_t *this->b[i],f_stages[i]);
}
return (t+delta_t);
}
template <typename VectorType>
void ImplicitRungeKutta<VectorType>::set_newton_solver_parameters(unsigned int max_it_, double tolerance_)
{
max_it = max_it_;
tolerance = tolerance_;
}
template <typename VectorType>
const typename ImplicitRungeKutta<VectorType>::Status &ImplicitRungeKutta<VectorType>::get_status() const
{
return status;
}
template <typename VectorType>
void ImplicitRungeKutta<VectorType>::compute_stages(
std_cxx11::function<VectorType (const double, const VectorType &)> f,
std_cxx11::function<VectorType (const double, const double, const VectorType &)> id_minus_tau_J_inverse,
double t,
double delta_t,
VectorType &y,
std::vector<VectorType> &f_stages)
{
VectorType z(y);
for (unsigned int i=0; i<this->n_stages; ++i)
{
VectorType old_y(z);
for (unsigned int j=0; j<i; ++j)
old_y.sadd(1.,delta_t *this->a[i][j],f_stages[j]);
// Solve the nonlinear system using Newton's method
const double new_t = t+this->c[i]*delta_t;
const double new_delta_t = this->a[i][i]*delta_t;
newton_solve(std_cxx11::bind(&ImplicitRungeKutta<VectorType>::compute_residual,this,f,new_t,new_delta_t,
std_cxx11::cref(old_y),std_cxx11::_1,std_cxx11::ref(f_stages[i]),std_cxx11::_2),
std_cxx11::bind(id_minus_tau_J_inverse,new_t,new_delta_t,std_cxx11::_1),y);
}
}
template <typename VectorType>
void ImplicitRungeKutta<VectorType>::newton_solve(
std_cxx11::function<void (const VectorType &,VectorType &)> get_residual,
std_cxx11::function<VectorType (const VectorType &)> id_minus_tau_J_inverse,
VectorType &y)
{
VectorType residual(y);
get_residual(y,residual);
unsigned int i=0;
const double initial_residual_norm = residual.l2_norm();
double norm_residual = initial_residual_norm;
while (i<max_it)
{
y.sadd(1.0,-1.0,id_minus_tau_J_inverse(residual));
get_residual(y,residual);
norm_residual = residual.l2_norm();
if (norm_residual < tolerance)
break;
++i;
}
status.n_iterations = i+1;
status.norm_residual = norm_residual;
}
template <typename VectorType>
void ImplicitRungeKutta<VectorType>::compute_residual
(std_cxx11::function<VectorType (const double, const VectorType &)> f,
double t,
double delta_t,
const VectorType &old_y,
const VectorType &y,
VectorType &tendency,
VectorType &residual) const
{
// The tendency is stored to save one evaluation of f.
tendency = f(t,y);
residual = tendency;
residual.sadd(delta_t,1.0,old_y);
residual.sadd(-1.0,1.,y);
}
// ----------------------------------------------------------------------
// EmbeddedExplicitRungeKutta
// ----------------------------------------------------------------------
template <typename VectorType>
EmbeddedExplicitRungeKutta<VectorType>::EmbeddedExplicitRungeKutta
(runge_kutta_method method,
double coarsen_param,
double refine_param,
double min_delta,
double max_delta,
double refine_tol,
double coarsen_tol)
:
coarsen_param(coarsen_param),
refine_param(refine_param),
min_delta_t(min_delta),
max_delta_t(max_delta),
refine_tol(refine_tol),
coarsen_tol(coarsen_tol),
last_same_as_first(false),
last_stage(NULL)
{
initialize(method);
}
template <typename VectorType>
void EmbeddedExplicitRungeKutta<VectorType>::initialize(runge_kutta_method method)
{
status.method = method;
switch (method)
{
case (HEUN_EULER) :
{
this->n_stages = 2;
this->a.push_back(std::vector<double>());
this->a.push_back(std::vector<double>(1, 1.0));
this->c.push_back(0.0);
this->c.push_back(1.0);
b1.push_back(0.5);
b1.push_back(0.5);
b2.push_back(1.0);
b2.push_back(0.0);
break;
}
case (BOGACKI_SHAMPINE) :
{
last_same_as_first = true;
this->n_stages = 4;
this->c.reserve(this->n_stages);
this->b1.reserve(this->n_stages);
this->b2.reserve(this->n_stages);
std::vector<double> tmp;
this->a.push_back(tmp);
tmp.resize(1);
tmp[0] = 0.5;
this->a.push_back(tmp);
tmp.resize(2);
tmp[0] = 0.0;
tmp[1] = 0.75;
this->a.push_back(tmp);
tmp.resize(3);
tmp[0] = 2.0/9.0;
tmp[1] = 1.0/3.0;
tmp[2] = 4.0/9.0;
this->a.push_back(tmp);
this->c.push_back(0.0);
this->c.push_back(0.5);
this->c.push_back(0.75);
this->c.push_back(1.0);
this->b1.push_back(2.0/9.0);
this->b1.push_back(1.0/3.0);
this->b1.push_back(4.0/9.0);
this->b1.push_back(0.0);
this->b2.push_back(7.0/24.0);
this->b2.push_back(0.25);
this->b2.push_back(1.0/3.0);
this->b2.push_back(0.125);
break;
}
case (DOPRI) :
{
last_same_as_first = true;
this->n_stages = 7;
this->c.reserve(this->n_stages);
this->b1.reserve(this->n_stages);
this->b2.reserve(this->n_stages);
std::vector<double> tmp;
this->a.push_back(tmp);
tmp.resize(1);
tmp[0] = 1./5.;
this->a.push_back(tmp);
tmp.resize(2);
tmp[0] = 3./40.;
tmp[1] = 9./40.;
this->a.push_back(tmp);
tmp.resize(3);
tmp[0] = 44./45.;
tmp[1] = -56./15.;
tmp[2] = 32./9.;
this->a.push_back(tmp);
tmp.resize(4);
tmp[0] = 19372./6561.;
tmp[1] = -25360./2187.;
tmp[2] = 64448./6561.;
tmp[3] = -212./729.;
this->a.push_back(tmp);
tmp.resize(5);
tmp[0] = 9017./3168.;
tmp[1] = -355./33.;
tmp[2] = 46732./5247.;
tmp[3] = 49./176.;
tmp[4] = -5103./18656;
this->a.push_back(tmp);
tmp.resize(6);
tmp[0] = 35./384.;
tmp[1] = 0.;
tmp[2] = 500./1113.;
tmp[3] = 125./192.;
tmp[4] = -2187./6784.;
tmp[5] = 11./84.;
this->a.push_back(tmp);
this->c.push_back(0.);
this->c.push_back(1./5.);
this->c.push_back(3./10.);
this->c.push_back(4./5.);
this->c.push_back(8./9.);
this->c.push_back(1.);
this->c.push_back(1.);
this->b1.push_back(35./384.);
this->b1.push_back(0.);
this->b1.push_back(500./1113.);
this->b1.push_back(125./192.);
this->b1.push_back(-2187./6784.);
this->b1.push_back(11./84.);
this->b1.push_back(0.);
this->b2.push_back(5179./57600.);
this->b2.push_back(0.);
this->b2.push_back(7571./16695.);
this->b2.push_back(393./640.);
this->b2.push_back(-92097./339200.);
this->b2.push_back(187./2100.);
this->b2.push_back(1./40.);
break;
}
case (FEHLBERG) :
{
this->n_stages = 6;
this->c.reserve(this->n_stages);
this->b1.reserve(this->n_stages);
this->b2.reserve(this->n_stages);
std::vector<double> tmp;
this->a.push_back(tmp);
tmp.resize(1);
tmp[0] = 0.25;
this->a.push_back(tmp);
tmp.resize(2);
tmp[0] = 0.09375;
tmp[1] = 0.28125;
this->a.push_back(tmp);
tmp.resize(3);
tmp[0] = 1932.0/2197.0;
tmp[1] = -7200.0/2197.0;
tmp[2] = 7296.0/2197.0;
this->a.push_back(tmp);
tmp.resize(4);
tmp[0] = 439.0/216.0;
tmp[1] = -8.0;
tmp[2] = 3680.0/513.0;
tmp[3] = -845.0/4104.0;
this->a.push_back(tmp);
tmp.resize(5);
tmp[0] = -8.0/27.0;
tmp[1] = 2.0;
tmp[2] = -3544.0/2565.0;
tmp[3] = 1859.0/4104.0;
tmp[4] = -0.275;
this->a.push_back(tmp);
this->c.push_back(0.0);
this->c.push_back(0.25);
this->c.push_back(0.375);
this->c.push_back(12.0/13.0);
this->c.push_back(1.0);
this->c.push_back(0.5);
this->b1.push_back(16.0/135.0);
this->b1.push_back(0.0);
this->b1.push_back(6656.0/12825.0);
this->b1.push_back(28561.0/56430.0);
this->b1.push_back(-0.18);
this->b1.push_back(2.0/55.0);
this->b2.push_back(25.0/216.0);
this->b2.push_back(0.0);
this->b2.push_back(1408.0/2565.0);
this->b2.push_back(2197.0/4104.0);
this->b2.push_back(-0.2);
this->b2.push_back(0.0);
break;
}
case (CASH_KARP) :
{
this->n_stages = 6;
this->c.reserve(this->n_stages);
this->b1.reserve(this->n_stages);
this->b2.reserve(this->n_stages);
std::vector<double> tmp;
this->a.push_back(tmp);
tmp.resize(1);
tmp[0] = 0.2;
this->a.push_back(tmp);
tmp.resize(2);
tmp[0] = 0.075;
tmp[1] = 0.225;
this->a.push_back(tmp);
tmp.resize(3);
tmp[0] = 0.3;
tmp[1] = -0.9;
tmp[2] = 1.2;
this->a.push_back(tmp);
tmp.resize(4);
tmp[0] = -11.0/54.0;
tmp[1] = 2.5;
tmp[2] = -70.0/27.0;
tmp[3] = 35.0/27.0;
this->a.push_back(tmp);
tmp.resize(5);
tmp[0] = 1631.0/55296.0;
tmp[1] = 175.0/512.0;
tmp[2] = 575.0/13824.0;
tmp[3] = 44275.0/110592.0;
tmp[4] = 253.0/4096.0;
this->a.push_back(tmp);
this->c.push_back(0.0);
this->c.push_back(0.2);
this->c.push_back(0.3);
this->c.push_back(0.6);
this->c.push_back(1.0);
this->c.push_back(0.875);
this->b1.push_back(37.0/378.0);
this->b1.push_back(0.0);
this->b1.push_back(250.0/621.0);
this->b1.push_back(125.0/594.0);
this->b1.push_back(0.0);
this->b1.push_back(512.0/1771.0);
this->b2.push_back(2825.0/27648.0);
this->b2.push_back(0.0);
this->b2.push_back(18575.0/48384.0);
this->b2.push_back(13525.0/55296.0);
this->b2.push_back(277.0/14336.0);
this->b2.push_back(0.25);
break;
}
default :
{
AssertThrow(false,ExcMessage("Unimplemented Embedded Runge-Kutta method."));
}
}
}
template <typename VectorType>
void EmbeddedExplicitRungeKutta<VectorType>::free_memory()
{
if (last_stage!=NULL)
delete last_stage;
last_stage = NULL;
}
template <typename VectorType>
double EmbeddedExplicitRungeKutta<VectorType>::evolve_one_time_step(
std_cxx11::function<VectorType (const double, const VectorType &)> f,
std_cxx11::function<VectorType (const double, const double, const VectorType &)> /*id_minus_tau_J_inverse*/,
double t,
double delta_t,
VectorType &y)
{
return evolve_one_time_step(f,t,delta_t,y);
}
template <typename VectorType>
double EmbeddedExplicitRungeKutta<VectorType>::evolve_one_time_step(
std_cxx11::function<VectorType (const double, const VectorType &)> f,
double t, double delta_t, VectorType &y)
{
bool done = false;
unsigned int count = 0;
double error_norm = 0.;
VectorType old_y(y);
VectorType error(y);
std::vector<VectorType> f_stages(this->n_stages,y);
while (!done)
{
error = 0.;
y = old_y;
// Compute the different stages needed.
compute_stages(f,t,delta_t,y,f_stages);
for (unsigned int i=0; i<this->n_stages; ++i)
{
y.sadd(1.,delta_t *this->b1[i],f_stages[i]);
error.sadd(1.,delta_t *(b2[i]-b1[i]),f_stages[i]);
}
error_norm = error.l2_norm();
// Check if the norm of error is less than the coarsening tolerance
if (error_norm<coarsen_tol)
{
done = true;
// Increase the guessed time step
double new_delta_t = delta_t *coarsen_param;
// Check that the guessed time step is smaller than the maximum time
// step allowed.
if (new_delta_t>max_delta_t)
{
status.exit_delta_t = MAX_DELTA_T;
status.delta_t_guess = max_delta_t;
}
else
{
status.exit_delta_t = DELTA_T;
status.delta_t_guess = delta_t;
}
}
// Check if the norm of error is less than the refining tolerance
else if (error_norm<refine_tol)
{
done = true;
status.exit_delta_t = DELTA_T;
status.delta_t_guess = delta_t;
}
else
{
// If the time step is already the smallest acceptable, exit.
if (delta_t==min_delta_t)
{
done = true;
status.exit_delta_t = MIN_DELTA_T;
status.delta_t_guess = delta_t;
}
// Reduce the time step.
else
{
delta_t *= refine_param;
if (delta_t<min_delta_t)
delta_t = min_delta_t;
}
}
++count;
}
// Save the last stage if necessary
if (last_same_as_first==true)
{
if (last_stage==NULL)
last_stage = new VectorType(f_stages.back());
else
*last_stage = f_stages.back();
}
status.n_iterations = count;
status.error_norm = error_norm;
return (t+delta_t);
}
template <typename VectorType>
void EmbeddedExplicitRungeKutta<VectorType>::set_time_adaptation_parameters(double coarsen_param_,
double refine_param_,
double min_delta_,
double max_delta_,
double refine_tol_,
double coarsen_tol_)
{
coarsen_param = coarsen_param_;
refine_param = refine_param_;
min_delta_t = min_delta_;
max_delta_t = max_delta_;
refine_tol = refine_tol_;
coarsen_tol = coarsen_tol_;
}
template <typename VectorType>
const typename EmbeddedExplicitRungeKutta<VectorType>::Status &EmbeddedExplicitRungeKutta<VectorType>::get_status() const
{
return status;
}
template <typename VectorType>
void EmbeddedExplicitRungeKutta<VectorType>::compute_stages(
std_cxx11::function<VectorType (const double, const VectorType &)> f,
const double t,
const double delta_t,
const VectorType &y,
std::vector<VectorType> &f_stages)
{
VectorType Y(y);
unsigned int i = 0;
// If the last stage is the same as the first, we can skip the evaluation
// of the first stage.
if (last_same_as_first==true)
{
if (last_stage!=NULL)
{
f_stages[0] = *last_stage;
i = 1;
}
}
for (; i<this->n_stages; ++i)
{
Y = y;
for (unsigned int j = 0; j < i; ++j)
Y.sadd(1.0,delta_t *this->a[i][j],f_stages[j]);
f_stages[i] = f(t+this->c[i]*delta_t,Y);
}
}
}
DEAL_II_NAMESPACE_CLOSE
#endif
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