rkMethod {deSolve}R Documentation

Collection of Parameter Sets (Butcher Arrays) for the Runge-Kutta Family of ODE Solvers

Description

This function returns a list specifying coefficients and properties of ODE solver methods from the Runge-Kutta family.

Usage

rkMethod(method = NULL, ...)

Arguments

method a string constant naming one of the pre-defined methods of the Runge-Kutta family of solvers. The most common methods are the fixed-step methods "euler", "rk2", "rk4" or the variable step methods "rk23bs", "rk34f", "rk45f" or "rk45dp7".
... specification of a user-defined solver, see Value and example below.

Details

This function supplies method settings for rk or ode. If called without arguments, the names of all implemented solvers of the Runge-Kutta family is returned.

The following comparison gives an idea how the algorithms of deSolve are related to similar algorithms of other simulation languages:

rkMethod | Description
"euler" | Euler's Method
"rk2" | 2nd order Runge-Kutta, fixed time step (Heun's method)
"rk4" | classical 4th order Runge-Kutta, fixed time step
"rk23" | Runge-Kutta, order 2(3), Octave: ode23
"rk23bs", "ode23" | Bogacki-Shampine, order 2(3), Matlab: ode23
"rk34f" | Runge-Kutta-Fehlberg, order 3(4)
"rk45f" | Runge-Kutta-Fehlberg, order 4(5), Octave: ode45, pair=1
"rk45e" | Runge-Kutta-England, order 4(5)
"rk45dp6" | Dormand-Prince, order 4(5), local order 6
"rk45dp7", "ode45" | Dormand-Prince 4(5), local order 7
| (also known as dopri5, MATLAB: ode45, Octave: ode45, pair=0)

Note that this table is based on the Runge-Kutta coefficients only, but the algorithms differ also in their implementation, in their stepsize adaption strategy and interpolation methods.

Value

A list with the following elements:

ID name of the method (character)
varstep boolean value specifying if the method allows for variable time step (TRUE) or not (FALSE).
FSAL (first same as last) boolean value specifying if the method allows re-use of the last function evaluation (TRUE) or not (FALSE or NULL).
A coefficient matrix of the method. As link{rk} supports only explicit methods, this matrix must be lower triangular. A can also be a vector if only the subdiagonal values are different from zero.
b1 weighting coefficients for averaging the function evaluations of method 1.
b2 weighting coefficients for averaging the function evaluations of method 2 (optional, for embedded methods that allow variable time step).
c coefficients for calculating the intermediate time steps.
d coefficients for polynomial interpolation of the outputs from internal steps (dense output), currently only available for method rk45dp7 (Dormand-Prince).
stage number of function evaluations needed (corresponds to number of rows in A).
Qerr global error order of the method, important for automatic time-step adjustment.

Note

Author(s)

Thomas Petzoldt thomas.petzoldt@tu-dresden.de

References

Bogacki, P. and Shampine L.F. (1989) A 3(2) pair of Runge-Kutta formulas, Appl. Math. Lett. 2, 1–9.

Butcher, J. C. (1987) The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods, Wiley, Chichester and New York.

Dormand, J. R. and Prince, P. J. (1980) A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6(1), 19–26.

Dormand, J. R. and Prince, P. J. (1981) High order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7(1), 67–75.

Engeln-Muellges, G. and Reutter, F. (1996) Numerik Algorithmen: Entscheidungshilfe zur Auswahl und Nutzung. VDI Verlag, Duesseldorf.

Fehlberg, E. (1967) Klassische Runge-Kutta-Formeln fuenfter and siebenter Ordnung mit Schrittweiten-Kontrolle, Computing (Arch. Elektron. Rechnen) 4, 93–106.

Kutta, W. (1901) Beitrag zur naeherungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys. 46, 435–453.

Octave-Forge - Extra Packages for GNU Octave, Package OdePkg. http://octave.sourceforge.net/doc/odepkg.html

Runge, C. (1895) Ueber die numerische Aufloesung von Differentialgleichungen, Math. Ann. 46, 167–178.

MATLAB (R) is a registed property of The Mathworks Inc. http://www.mathworks.com/

See Also

rk, ode

Examples

rkMethod()          # returns the names of all available methods
rkMethod("rk45dp7") # parameters of the Dormand-Prince 5(4) method
rkMethod("ode45")   # an alias for the same method

func <- function(t, x, parms) {
  with(as.list(c(parms, x)),{
    dP  <- a * P      - b * K * P
    dK  <- b * P * K  - c * K
    res <- c(dP, dK)
    list(res)
  })
}
times  <- seq(0, 20, length = 21)
parms  <- c(a = 0.1, b = 0.1, c = 0.1)
x <- c(P = 2, K = 1)

ode(x, times, func, parms, method = rkMethod("rk4"))

ode(x, times, func, parms, method = "ode45")

## disable polynomial interpolation (dense output)
## and fall back to linear approximation
ode(x, times, func, parms, method = rkMethod("rk45dp7", d = NULL))

## define and use a new rk method
ode(x, times, func, parms, 
  method = rkMethod(ID = "midpoint",
    varstep = FALSE,
    #A       = matrix(c(0, 0, 1/2, 0), nrow = 2, byrow = TRUE), 
    # or simply, because this A is nonzero only in the subdiagonal
    A      = c(0, 1/2),
    b1      = c(0, 1),
    c       = c(0, 1/2),
    stage   = 2,
    Qerr    = 1
  )
)


[Package deSolve version 1.2-3 Index]