deSolve (version 1.5-1)

rkMethod: 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 "

...

specification of a user-defined solver, see Value and example below.

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) optional 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 must be a vector for fixed step methods where 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
  optional coefficients for built-in 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.
• nknots
  integer value specifying the order of interpolation polynomials for methods without dense output (default = 5th order). If nknots < 2 internal interpolation is switched off. Note that this works only if time steps match exactly. This may not work as expected for all cases because of floating point rounding errors (see RFAQ about ``Why doesn't Rthink these numbers are equal?'').
• alpha
  optional tuning parameter for stepsize adjustment. If alpha is omitted, it is set to $1/Qerr - 0.75 beta$. The default value is $1/Qerr$ (for beta = 0).
• beta
  optional tuning parameter for stepsize adjustment. Typical values are $0$ (default) or $0.4/Qerr$

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 are returned. The following comparison gives an idea how the algorithms of deSolve are related to similar algorithms of other simulation languages: lll{ 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) "rk45ck" | Runge-Kutta Cash-Karp, order 4(5) "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. The table reflects the state at time of writing and it is of course possible that implementations change.

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.

Cash, J. R. and Karp A. H., 1990. A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Transactions on Mathematical Software 16, 201--222.

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 * C * P
    dC  <- b * P * C  - c * C
    res <- c(dP, dC)
    list(res)
  })
}
times  <- seq(0, 200, length = 101)
parms  <- c(a = 0.1, b = 0.1, c = 0.1)
x <- c(P = 2, C = 1)

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

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

o0 <- ode(x, times, func, parms, method = "lsoda")
o1 <- ode(x, times, func, parms, method = rkMethod("rk45dp7"))
## disable dense-output interpolation
## and fall back to Neville-Aitken polynomials
o2 <- ode(x, times, func, parms, method = rkMethod("rk45dp7", d = NULL))

## show differences between methods
par(mfrow=c(3,1))
matplot(o1[,1], o1[,-1], type="l", xlab="Time", main="State Variables")
matplot(o1[,1], o2[,-1], type="p", pch=16, add=TRUE)
matplot(o0[,1], o1[,-1]-o0[,-1], type="l", lty="solid", xlab="Time",
  main="Difference: rk45dp7 - lsoda")
matplot(o1[,1], o2[,-1]-o0[,-1], type="l", lty="dashed", xlab="Time",
  main="difference: Neville-Aitken - Dense Output")
abline(h=0, col="grey")

## define and use a new rk method
ode(x, times, func, parms, 
  method = rkMethod(ID = "midpoint",
    varstep = FALSE,
    A      = c(0, 1/2),
    b1      = c(0, 1),
    c       = c(0, 1/2),
    stage   = 2,
    Qerr    = 1
  )
)