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

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" (alias "ode23"), "rk45dp7" (alias "ode45") or "rk78f".
...: 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 currently 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:

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)
"rk78f"	\|	Runge-Kutta-Fehlberg, order 7(8)
"rk78dp"	\|	Dormand-Prince, order 7(8)

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.

Methods "rk45dp7" (alias "ode45") and "rk45ck" contain specific and efficient built-in interpolation schemes (dense output).

As an alternative, Neville-Aitken polynomials can be used to interpolate between time steps. This is available for all RK methods and may be useful to speed up computation if no dense-output formula is available. Note however, that this can introduce considerable local error; it is disabled by default (see nknots below).

Note

Adaptive stepsize Runge-Kuttas are preferred if the solution contains parts when the states change fast, and parts when not much happens. They will take small steps over bumpy ground and long steps over uninteresting terrain.
As a suggestion, one may use "rk23" (alias "ode23") for simple problems and "rk45dp7" (alias "ode45") for rough problems. The default solver is "rk45dp7" (alias "ode45"), because of its relatively high order (4), re-use of the last intermediate steps (FSAL = first same as last) and built-in polynomial interpolation (dense output).
Solver "rk23bs", that supports also FSAL, may be useful for slightly stiff systems if demands on precision are relatively low.
Another good choice, assuring medium accuracy, is the Cash-Karp Runge-Kutta method, "rk45ck".
Classical "rk4" is traditionally used in cases where an adequate stepsize is known a-priori or if external forcing data are provided for fixed time steps only and frequent interpolation of external data needs to be avoided.
Method "rk45dp7" (alias "ode45") contains an efficient built-in interpolation scheme (dense output) based on intermediate function evaluations.

Starting with version 1.8 implicit Runge-Kutta (irk) methods are also supported by the general rk interface, however their implementation is still experimental. Instead of this you may consider radau for a specific full implementation of an implicit Runge-Kutta method.

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

coefficients of the lower order Runge-Kutta pair.

b2

coefficients of the higher order Runge-Kutta pair (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).

densetype

optional integer value specifying the dense output formula; currently only densetype = 1 for rk45dp7 (Dormand-Prince) and densetype = 2 for rk45ck (Cash-Karp) are supported. Undefined values (e.g., densetype = NULL) disable dense output.

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. If nknots < 2 (the default) then internal interpolation is switched off and integration is performed step by step between external time steps.

If nknots is between 3 and 8, Neville-Aitken polynomials are used, which need at least nknots + 1 internal time steps. Interpolation may speed up integration but can lead to local errors higher than the tolerance, especially if external and internal time steps are very different.

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 \(0.4/Qerr\).

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. doi:10.1145/79505.79507

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

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. https://octave.sourceforge.io

Prince, P. J. and Dormand, J. R. (1981) High order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7(1), 67–75. doi:10.1016/0771-050X(81)90010-3

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

MATLAB (R) is a registed property of The Mathworks Inc.

Author

Thomas Petzoldt thomas.petzoldt@tu-dresden.de

Examples

if (FALSE) { # \dontrun{
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)

## rk using ode45 as the default method
out <- rk(x, times, func, parms)

## all methods can be called also from 'ode' by using rkMethod
out <- ode(x, times, func, parms, method = rkMethod("rk4"))

## 'ode' has aliases for the most common RK methods
out <- ode(x, times, func, parms, method = "ode45")

##===========================================================================
## Comparison of local error from different interpolation methods
##===========================================================================

## lsoda with lower tolerances (1e-10) used as reference
o0 <- ode(x, times, func, parms, method = "lsoda", atol = 1e-10, rtol = 1e-10)

## rk45dp7 with hmax = 10 > delta_t  = 2
o1 <- ode(x, times, func, parms, method = rkMethod("rk45dp7"), hmax = 10)

## disable dense-output interpolation
## and use only Neville-Aitken polynomials instead
o2 <- ode(x, times, func, parms,
  method = rkMethod("rk45dp7", densetype = NULL, nknots = 5), hmax = 10)

## stop and go: disable interpolation completely
## and integrate explicitly between external time steps
o3 <- ode(x, times, func, parms,
  method = rkMethod("rk45dp7", densetype = NULL, nknots = 0, hmax=10))

## compare different interpolation methods with lsoda
mf <- par("mfrow" = c(4, 1))
matplot(o1[,1], o1[,-1], type = "l", xlab = "Time", main = "State Variables",
  ylab = "P, C")
matplot(o0[,1], o0[,-1] - o1[,-1], type = "l", xlab = "Time", ylab = "Diff.",
  main="Difference between lsoda and ode45 with dense output")
abline(h = 0, col = "grey")
matplot(o0[,1], o0[,-1] - o2[,-1], type = "l", xlab = "Time", ylab = "Diff.",
  main="Difference between lsoda and ode45 with Neville-Aitken")
abline(h = 0, col = "grey")
matplot(o0[,1], o0[,-1] - o3[,-1], type = "l", xlab = "Time", ylab = "Diff.",
  main="Difference between lsoda and ode45 in 'stop and go' mode")
abline(h = 0, col = "grey")
par(mf)

##===========================================================================
## rkMethod allows to define user-specified Runge-Kutta methods
##===========================================================================
out <- 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
          )
        )
plot(out)

## compare method diagnostics
times  <- seq(0, 200, length = 10)
o1 <- ode(x, times, func, parms, method = rkMethod("rk45ck"))
o2 <- ode(x, times, func, parms, method = rkMethod("rk78dp"))
diagnostics(o1)
diagnostics(o2)
} # }