rk: Explicit One-Step Solvers for Ordinary Differential Equations (ODE)

Description

Solving initial value problems for non-stiff systems of first-order ordinary differential equations (ODEs).

The Rfunction rk is a top-level function that provides interfaces to a collection of common explicit one-step solvers of the Runge-Kutta family with fixed or variable time steps.

The system of ODE's is written as an Rfunction (which may, of course, use .C, .Fortran, .Call, etc., to call foreign code) or be defined in compiled code that has been dynamically loaded. A vector of parameters is passed to the ODEs, so the solver may be used as part of a modeling package for ODEs, or for parameter estimation using any appropriate modeling tool for non-linear models in Rsuch as optim, nls, nlm or nlme

Usage

rk(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
  verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL,
  hini = hmax, ynames = TRUE, method = rkMethod("rk45dp7", ... ),
  maxsteps = 5000, dllname = NULL, initfunc = dllname,
  initpar = parms, rpar = NULL, ipar = NULL,
  nout = 0, outnames = NULL, forcings = NULL,
  initforc = NULL, fcontrol = NULL, events = NULL, ...)

Arguments

the initial (state) values for the ODE system. If y has a name attribute, the names will be used to label the output matrix.

times

times at which explicit estimates for y are desired. The first value in times must be the initial time.

func

either an R-function that computes the values of the derivatives in the ODE system (the model definition) at time t, or a character string giving the name of a compiled function in a dynamically loaded shared library.

If fu

parms

vector or list of parameters used in func.

rtol

relative error tolerance, either a scalar or an array as long as y. Only applicable to methods with variable time step, see details.

atol

absolute error tolerance, either a scalar or an array as long as y. Only applicable to methods with variable time step, see details.

tcrit

if not NULL, then rk cannot integrate past tcrit. This parameter is for compatibility with other solvers.

verbose

a logical value that, when TRUE, triggers more verbose output from the ODE solver.

hmin

an optional minimum value of the integration stepsize. In special situations this parameter may speed up computations with the cost of precision. Don't use hmin if you don't know why!

hmax

an optional maximum value of the integration stepsize. If not specified, hmax is set to the maximum of hini and the largest difference in times, to avoid that the simulation possibly ignores short-term ev

hini

initial step size to be attempted; if 0, the initial step size is determined automatically by solvers with flexible time step. Setting hini = 0 for fixed step methods forces setting of internal time steps identically to external t

ynames

if FALSE: names of state variables are not passed to function func ; this may speed up the simulation especially for large models.

method

the integrator to use. This can either be a string constant naming one of the pre-defined methods or a call to function rkMethod specifying a user-defined method. The most common methods are t

maxsteps

average maximal number of steps per output interval taken by the solver. This argument is defined such as to ensure compatibility with the Livermore-solvers. rk only accepts the maximal number of steps for the entire integration,

dllname

a string giving the name of the shared library (without extension) that contains all the compiled function or subroutine definitions refered to in func and jacfunc. See package vignette "compiledCode".

initfunc

if not NULL, the name of the initialisation function (which initialises values of parameters), as provided in dllname. See package vignette "compiledCode".

initpar

only when dllname is specified and an initialisation function initfunc is in the dll: the parameters passed to the initialiser, to initialise the common blocks (FORTRAN) or global variables (C, C++).

rpar

only when dllname is specified: a vector with double precision values passed to the dll-functions whose names are specified by func and jacfunc.

ipar

only when dllname is specified: a vector with integer values passed to the dll-functions whose names are specified by func and jacfunc.

nout

only used if dllname is specified and the model is defined in compiled code: the number of output variables calculated in the compiled function func, present in the shared library. Note: it is not automatically checke

outnames

only used if dllname is specified and nout > 0: the names of output variables calculated in the compiled function func, present in the shared library.

forcings

only used if dllname is specified: a list with the forcing function data sets, each present as a two-columned matrix, with (time,value); interpolation outside the interval [min(times), max(times)] is done

initforc

if not NULL, the name of the forcing function initialisation function, as provided in dllname. It MUST be present if forcings has been given a value. See forcings

fcontrol

A list of control parameters for the forcing functions. See forcings or vignette compiledCode.

events

A list that specifies events, i.e. when the value of a state variable is suddenly changed. See events for more information. Not also that if events are specified, then polynomial interpolation is switched of

...

additional arguments passed to func allowing this to be a generic function.

Value

A matrix of class deSolve with up to as many rows as elements in times and as many columns as elements in y plus the number of "global" values returned in the next elements of the return from func, plus and additional column for the time value. There will be a row for each element in times unless the integration routine returns with an unrecoverable error. If y has a names attribute, it will be used to label the columns of the output value.

Details

Function rk is a generalized implementation that can be used to evaluate different solvers of the Runge-Kutta family of explicit ODE solvers. A pre-defined set of common method parameters is in function rkMethod which also allows to supply user-defined Butcher tables.

The input parameters rtol, and atol determine the error control performed by the solver. The solver will control the vector of estimated local errors in y, according to an inequality of the form max-norm of ( e/ewt ) $\leq$ 1, where ewt is a vector of positive error weights. The values of rtol and atol should all be non-negative. The form of ewt is:

$$\mathbf{rtol} \times \mathrm{abs}(\mathbf{y}) + \mathbf{atol}$$

where multiplication of two vectors is element-by-element.

Models can be defined in Ras a user-supplied R-function, that must be called as: yprime = func(t, y, parms). t is the current time point in the integration, y is the current estimate of the variables in the ODE system.

The return value of func should be a list, whose first element is a vector containing the derivatives of y with respect to time, and whose second element contains output variables that are required at each point in time. Examples are given below.

References

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

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

Hindmarsh, Alan C. (1983) ODEPACK, A Systematized Collection of ODE Solvers; in p.55--64 of Stepleman, R.W. et al.[ed.] (1983) Scientific Computing, North-Holland, Amsterdam.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2007) Numerical Recipes in C. Cambridge University Press.

Examples

Run this code

## =======================================================================
## Example: Resource-producer-consumer Lotka-Volterra model
## =======================================================================

## Note:
## parameters are a list, names accessible via "with" function
## (see also ode and lsoda examples)

SPCmod <- function(t, x, parms) {
  S <- x[1] # substrate
  P <- x[2] # producer
  C <- x[3] # consumer

  with(parms, {
    import <- approx(signal$times, signal$import, t)$y
    dS <- import - b * S * P + g * C
    dP <- c * S * P  - d * C * P
    dC <- e * P * C  - f * C
    res <- c(dS, dP, dC)
    list(res)
  })
}

## vector of timesteps
times  <- seq(0, 100, length = 101)

## external signal with rectangle impulse
signal <- as.data.frame(list(times = times,
                            import = rep(0,length(times))))

signal$import[signal$times >= 10 & signal$times <= 11] <- 0.2

## Parameters for steady state conditions
parms <- list(b = 0.0, c = 0.1, d = 0.1, e = 0.1, f = 0.1, g = 0.0)

## Start values for steady state
y <- xstart <- c(S = 1, P = 1, C = 1)

system.time({
## Euler method
out1  <- as.data.frame(rk(xstart, times, SPCmod, parms,
                          hini = 0.1, method = "euler"))

## classical Runge-Kutta 4th order
out2 <- as.data.frame(rk(xstart, times, SPCmod, parms,
                         hini = 1, method = "rk4"))

## Dormand-Prince method of order 5(4)
out3 <- as.data.frame(rk(xstart, times, SPCmod, parms,
                         hmax = 1, method = "rk45dp7"))
})

mf <- par(mfrow = c(2,2))
plot (out1$time, out1$S, type = "l", ylab = "Substrate")
lines(out2$time, out2$S, col = "red", lty = "dotted", lwd = 2)
lines(out3$time, out3$S, col = "green", lty = "dotted")

plot (out1$time, out1$P, type = "l", ylab = "Producer")
lines(out2$time, out2$P, col = "red", lty = "dotted")
lines(out3$time, out3$P, col = "green", lty = "dotted")

plot (out1$time, out1$C, type = "l", ylab = "Consumer")
lines(out2$time, out2$C, col = "red", lty = "dotted", lwd = 2)
lines(out3$time, out3$C, col = "green", lty = "dotted")

plot (out1$P, out1$C, type = "l", xlab = "Producer", ylab = "Consumer")
lines(out2$P, out2$C, col = "red",   lty = "dotted", lwd = 2)
lines(out3$P, out3$C, col = "green", lty = "dotted")

legend("center", legend = c("euler", "rk4", "rk45dp7"),
  lty = c(1, 3, 3), lwd = c(1, 2, 1),
  col = c("black", "red", "green"))
par(mfrow = mf)

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