/* ode-initval/rk4imp.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* Runge-Kutta 4, Gaussian implicit */
/* Author: G. Jungman
*/
/* Error estimation by step doubling, see eg. Ascher, U.M., Petzold,
L.R., Computer methods for ordinary differential and
differential-algebraic equations, SIAM, Philadelphia, 1998.
Method coefficients can also be found in it.
*/
#include <config.h>
#include <stdlib.h>
#include <string.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_odeiv.h>
#include "odeiv_util.h"
typedef struct
{
double *k1nu;
double *k2nu;
double *ytmp1;
double *ytmp2;
double *y0;
double *y0_orig;
double *y_onestep;
}
rk4imp_state_t;
static void *
rk4imp_alloc (size_t dim)
{
rk4imp_state_t *state = (rk4imp_state_t *) malloc (sizeof (rk4imp_state_t));
if (state == 0)
{
GSL_ERROR_NULL ("failed to allocate space for rk4imp_state",
GSL_ENOMEM);
}
state->k1nu = (double *) malloc (dim * sizeof (double));
if (state->k1nu == 0)
{
free (state);
GSL_ERROR_NULL ("failed to allocate space for k1nu", GSL_ENOMEM);
}
state->k2nu = (double *) malloc (dim * sizeof (double));
if (state->k2nu == 0)
{
free (state->k1nu);
free (state);
GSL_ERROR_NULL ("failed to allocate space for k2nu", GSL_ENOMEM);
}
state->ytmp1 = (double *) malloc (dim * sizeof (double));
if (state->ytmp1 == 0)
{
free (state->k2nu);
free (state->k1nu);
free (state);
GSL_ERROR_NULL ("failed to allocate space for ytmp1", GSL_ENOMEM);
}
state->ytmp2 = (double *) malloc (dim * sizeof (double));
if (state->ytmp2 == 0)
{
free (state->ytmp1);
free (state->k2nu);
free (state->k1nu);
free (state);
GSL_ERROR_NULL ("failed to allocate space for ytmp2", GSL_ENOMEM);
}
state->y0 = (double *) malloc (dim * sizeof (double));
if (state->y0 == 0)
{
free (state->ytmp2);
free (state->ytmp1);
free (state->k2nu);
free (state->k1nu);
free (state);
GSL_ERROR_NULL ("failed to allocate space for y0", GSL_ENOMEM);
}
state->y0_orig = (double *) malloc (dim * sizeof (double));
if (state->y0_orig == 0)
{
free (state->y0);
free (state->ytmp2);
free (state->ytmp1);
free (state->k2nu);
free (state->k1nu);
free (state);
GSL_ERROR_NULL ("failed to allocate space for y0_orig", GSL_ENOMEM);
}
state->y_onestep = (double *) malloc (dim * sizeof (double));
if (state->y_onestep == 0)
{
free (state->y0_orig);
free (state->y0);
free (state->ytmp2);
free (state->ytmp1);
free (state->k2nu);
free (state->k1nu);
free (state);
GSL_ERROR_NULL ("failed to allocate space for y_onestep", GSL_ENOMEM);
}
return state;
}
static int
rk4imp_step (double *y, rk4imp_state_t *state,
const double h, const double t,
const size_t dim, const gsl_odeiv_system *sys)
{
/* Makes a Runge-Kutta 4th order implicit advance with step size h.
y0 is initial values of variables y.
The implicit matrix equations to solve are:
Y1 = y0 + h * a11 * f(t + h * c1, Y1) + h * a12 * f(t + h * c2, Y2)
Y2 = y0 + h * a21 * f(t + h * c1, Y1) + h * a22 * f(t + h * c2, Y2)
y = y0 + h * b1 * f(t + h * c1, Y1) + h * b2 * f(t + h * c2, Y2)
with constant coefficients a, b and c. For this method
they are: b=[0.5 0.5] c=[(3-sqrt(3))/6 (3+sqrt(3))/6]
a11=1/4, a12=(3-2*sqrt(3))/12, a21=(3+2*sqrt(3))/12 and a22=1/4
*/
const double ir3 = 1.0 / M_SQRT3;
const int iter_steps = 3;
int nu;
size_t i;
double *const k1nu = state->k1nu;
double *const k2nu = state->k2nu;
double *const ytmp1 = state->ytmp1;
double *const ytmp2 = state->ytmp2;
/* iterative solution of Y1 and Y2.
Note: This method does not check for convergence of the
iterative solution!
*/
for (nu = 0; nu < iter_steps; nu++)
{
for (i = 0; i < dim; i++)
{
ytmp1[i] =
y[i] + h * (0.25 * k1nu[i] + 0.5 * (0.5 - ir3) * k2nu[i]);
ytmp2[i] =
y[i] + h * (0.25 * k2nu[i] + 0.5 * (0.5 + ir3) * k1nu[i]);
}
{
int s =
GSL_ODEIV_FN_EVAL (sys, t + 0.5 * h * (1.0 - ir3), ytmp1, k1nu);
if (s != GSL_SUCCESS)
{
return s;
}
}
{
int s =
GSL_ODEIV_FN_EVAL (sys, t + 0.5 * h * (1.0 + ir3), ytmp2, k2nu);
if (s != GSL_SUCCESS)
{
return s;
}
}
}
/* assignment */
for (i = 0; i < dim; i++)
{
const double d_i = 0.5 * (k1nu[i] + k2nu[i]);
y[i] += h * d_i;
}
return GSL_SUCCESS;
}
static int
rk4imp_apply (void *vstate,
size_t dim,
double t,
double h,
double y[],
double yerr[],
const double dydt_in[],
double dydt_out[],
const gsl_odeiv_system * sys)
{
rk4imp_state_t *state = (rk4imp_state_t *) vstate;
size_t i;
double *y0 = state->y0;
double *y0_orig = state->y0_orig;
double *y_onestep = state->y_onestep;
double *k1nu = state->k1nu;
double *k2nu = state->k2nu;
/* Initialization step */
DBL_MEMCPY (y0, y, dim);
/* Save initial values in case of failure */
DBL_MEMCPY (y0_orig, y, dim);
if (dydt_in != 0)
{
DBL_MEMCPY (k1nu, dydt_in, dim);
}
else
{
int s = GSL_ODEIV_FN_EVAL (sys, t, y, k1nu);
if (s != GSL_SUCCESS)
{
return s;
}
}
DBL_MEMCPY (k2nu, k1nu, dim);
/* First traverse h with one step (save to y_onestep) */
DBL_MEMCPY (y_onestep, y, dim);
{
int s = rk4imp_step (y_onestep, state, h, t, dim, sys);
if (s != GSL_SUCCESS)
{
return s;
}
}
/* Then with two steps with half step length (save to y) */
{
int s = rk4imp_step (y, state, h/2.0, t, dim, sys);
if (s != GSL_SUCCESS)
{
/* Restore original y vector */
DBL_MEMCPY (y, y0_orig, dim);
return s;
}
}
DBL_MEMCPY (y0, y, dim);
{
int s = GSL_ODEIV_FN_EVAL (sys, t + h/2.0, y, k1nu);
if (s != GSL_SUCCESS)
{
/* Restore original y vector */
DBL_MEMCPY (y, y0_orig, dim);
return s;
}
}
DBL_MEMCPY (k2nu, k1nu, dim);
{
int s = rk4imp_step (y, state, h/2.0, t + h/2.0, dim, sys);
if (s != GSL_SUCCESS)
{
/* Restore original y vector */
DBL_MEMCPY (y, y0_orig, dim);
return s;
}
}
/* Derivatives at output */
if (dydt_out != NULL)
{
int s = GSL_ODEIV_FN_EVAL (sys, t + h, y, dydt_out);
if (s != GSL_SUCCESS) {
/* Restore original y vector */
DBL_MEMCPY (y, y0_orig, dim);
return s;
}
}
/* Error estimation */
/* Denominator in step doubling error equation
* yerr = 0.5 * | y(onestep) - y(twosteps) | / (2^order - 1)
*/
for (i = 0; i < dim; i++)
{
yerr[i] = 8.0 * 0.5 * (y[i] - y_onestep[i]) / 15.0;
}
return GSL_SUCCESS;
}
static int
rk4imp_reset (void *vstate, size_t dim)
{
rk4imp_state_t *state = (rk4imp_state_t *) vstate;
DBL_ZERO_MEMSET (state->y_onestep, dim);
DBL_ZERO_MEMSET (state->y0_orig, dim);
DBL_ZERO_MEMSET (state->y0, dim);
DBL_ZERO_MEMSET (state->k1nu, dim);
DBL_ZERO_MEMSET (state->k2nu, dim);
DBL_ZERO_MEMSET (state->ytmp1, dim);
DBL_ZERO_MEMSET (state->ytmp2, dim);
return GSL_SUCCESS;
}
static unsigned int
rk4imp_order (void *vstate)
{
rk4imp_state_t *state = (rk4imp_state_t *) vstate;
state = 0; /* prevent warnings about unused parameters */
return 4;
}
static void
rk4imp_free (void *vstate)
{
rk4imp_state_t *state = (rk4imp_state_t *) vstate;
free (state->y_onestep);
free (state->y0_orig);
free (state->y0);
free (state->k1nu);
free (state->k2nu);
free (state->ytmp1);
free (state->ytmp2);
free (state);
}
static const gsl_odeiv_step_type rk4imp_type = { "rk4imp", /* name */
1, /* can use dydt_in? */
1, /* gives exact dydt_out? */
&rk4imp_alloc,
&rk4imp_apply,
&rk4imp_reset,
&rk4imp_order,
&rk4imp_free
};
const gsl_odeiv_step_type *gsl_odeiv_step_rk4imp = &rk4imp_type;