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Excerpt from the official documentation

GSL solvers for Ordinary Differential Equations (ODEs) are documented on www.gnu.org.

This package has two utility functions to select the integration method:

rgsl::integrationMethod("msbdf") == 0
rgsl::nameMethod(0) == "msbdf"

The value of integrationMethod can be used here:

rgsl::r_gsl_odeiv2_outer(...,method=integrationMethod("msbdf"))
rgsl::r_gsl_odeiv2_outer_sens(...,method=integrationMethod("rkf45"))
rgsl::r_gsl_odeiv2_outer_state_only(...,method=integrationMethod("msadams"))

rk2

gsl_odeiv2_step_type *gsl_odeiv2_step_rk2

Explicit embedded Runge-Kutta (2, 3) method.

rk4

gsl_odeiv2_step_type *gsl_odeiv2_step_rk4

Explicit 4th order (classical) Runge-Kutta. Error estimation is carried out by the step doubling method. For more efficient estimate of the error, use the embedded methods described below.

rkf45

gsl_odeiv2_step_type *gsl_odeiv2_step_rkf45

Explicit embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.

rkck

gsl_odeiv2_step_type *gsl_odeiv2_step_rkck

Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.

rk8pd

gsl_odeiv2_step_type *gsl_odeiv2_step_rk8pd

Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.

rk1imp

gsl_odeiv2_step_type *gsl_odeiv2_step_rk1imp

Implicit Gaussian first order Runge-Kutta. Also known as implicit Euler or backward Euler method. Error estimation is carried out by the step doubling method. This algorithm requires the Jacobian and access to the driver object via gsl_odeiv2_step_set_driver().

rk2imp

gsl_odeiv2_step_type *gsl_odeiv2_step_rk2imp

Implicit Gaussian second order Runge-Kutta. Also known as implicit mid-point rule. Error estimation is carried out by the step doubling method. This stepper requires the Jacobian and access to the driver object via gsl_odeiv2_step_set_driver().

rk4imp

gsl_odeiv2_step_type *gsl_odeiv2_step_rk4imp

Implicit Gaussian 4th order Runge-Kutta. Error estimation is carried out by the step doubling method. This algorithm requires the Jacobian and access to the driver object via gsl_odeiv2_step_set_driver().

bsimp

gsl_odeiv2_step_type *gsl_odeiv2_step_bsimp

Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method is generally suitable for stiff problems. This stepper requires the Jacobian.

msadams

gsl_odeiv2_step_type *gsl_odeiv2_step_msadams

A variable-coefficient linear multistep Adams method in Nordsieck form. This stepper uses explicit Adams-Bashforth (predictor) and implicit Adams-Moulton (corrector) methods in $P(EC)^m$ functional iteration mode. Method order varies dynamically between 1 and 12. This stepper requires the access to the driver object via gsl_odeiv2_step_set_driver().

msbdf

gsl_odeiv2_step_type *gsl_odeiv2_step_msbdf

A variable-coefficient linear multistep backward differentiation formula (BDF) method in Nordsieck form. This stepper uses the explicit BDF formula as predictor and implicit BDF formula as corrector. A modified Newton iteration method is used to solve the system of non-linear equations. Method order varies dynamically between 1 and 5. The method is generally suitable for stiff problems. This stepper requires the Jacobian and the access to the driver object via gsl_odeiv2_step_set_driver().

List of all integrators in reverse order

Taken from this file, using coreutils:

printf "{" ; egrep "^gsl_" integrators.md | sed -r 's/^gsl_[^ ]+ [*]//' | tac | sed 's/$/, /' | tr -d '\n'; printf "NULL}\n"
{gsl_odeiv2_step_msbdf, gsl_odeiv2_step_msadams, gsl_odeiv2_step_bsimp, gsl_odeiv2_step_rk4imp, gsl_odeiv2_step_rk2imp, gsl_odeiv2_step_rk1imp, gsl_odeiv2_step_rk8pd, gsl_odeiv2_step_rkck, gsl_odeiv2_step_rkf45, gsl_odeiv2_step_rk4, gsl_odeiv2_step_rk2, NULL}

The list is NULL terminated.