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

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Description

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

The R function 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 R function (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 R such as optim, nls, nlm or nlme

Usage

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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

y

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 func is an R-function, it must be defined as: func <- function(t, y, parms,...). t is the current time point in the integration, y is the current estimate of the variables in the ODE system. If the initial values y has a names attribute, the names will be available inside func. parms is a vector or list of parameters; ... (optional) are any other arguments passed to the function.

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 next elements are global values that are required at each point in times. The derivatives must be specified in the same order as the state variables y.

If func is a string, then dllname must give the name of the shared library (without extension) which must be loaded before rk is called. See package vignette "compiledCode" for more details.

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 events. If 0, no maximal size is specified. Note that hmin and hmax are ignored by fixed step methods like "rk4" or "euler".

hini

initial step size to be attempted; if 0, the initial step size is determined automatically by solvers with flexible time step. For fixed step methods, setting hini = 0 forces internal time steps identically to external time steps provided by times. Similarly, internal time steps of non-interpolating solvers cannot be bigger than external time steps specified in times.

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 the fixed-step methods "euler", second and fourth-order Runge Kutta ("rk2", "rk4"), or the variable step methods Bogacki-Shampine "rk23bs", Runge-Kutta-Fehlberg "rk34f", the fifth-order Cash-Karp method "rk45ck" or the fifth-order Dormand-Prince method with seven stages "rk45dp7". As a suggestion, one may use "rk23bs" (alias "ode23") for simple problems and "rk45dp7" (alias "ode45") for rough problems.

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. It is calculated as max(length(times) * maxsteps, max(diff(times)/hini + 1).

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 checked whether this is indeed the number of output variables calculated in the dll - you have to perform this check in the code. See package vignette "compiledCode".

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 by taking the value at the closest data extreme.

See forcings or package vignette "compiledCode".

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 or package vignette "compiledCode".

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 off and integration takes place from one external time step to the next, with an internal step size less than or equal the difference of two adjacent points of times.

...

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

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 ) <= 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:

\bold{rtol} * abs(\bold{y}) + \bold{atol}

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

Models can be defined in R as 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.

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.

Note

Arguments rpar and ipar are provided for compatibility with lsoda.

Starting with version 1.8 implicit Runge-Kutta methods are also supported by this 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.

Author(s)

Thomas Petzoldt thomas.petzoldt@tu-dresden.de

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.

See Also

For most practical cases, solvers of the Livermore family (i.e. the ODEPACK solvers, see below) are superior. Some of them are also suitable for stiff ODEs, differential algebraic equations (DAEs), or partial differential equations (PDEs).

  • rkMethod for a list of available Runge-Kutta parameter sets,

  • rk4 and euler for special versions without interpolation (and less overhead),

  • lsoda, lsode, lsodes, lsodar, vode, daspk for solvers of the Livermore family,

  • ode for a general interface to most of the ODE solvers,

  • ode.band for solving models with a banded Jacobian,

  • ode.1D for integrating 1-D models,

  • ode.2D for integrating 2-D models,

  • ode.3D for integrating 3-D models,

  • diagnostics to print diagnostic messages.

Examples

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

## Notes:
## - Parameters are a list, names accessible via "with" function
## - Function sigimp passed as an argument (input) to model
##  (see also ode and lsoda examples)

SPCmod <- function(t, x, parms, input)  {
  with(as.list(c(parms, x)), {
    import <- input(t)
    dS <- import - b*S*P + g*C    # substrate
    dP <- c*S*P  - d*C*P          # producer
    dC <- e*P*C  - f*C            # consumer
    res <- c(dS, dP, dC)
    list(res)
  })
}

## The parameters 
parms <- c(b = 0.001, c = 0.1, d = 0.1, e = 0.1, f = 0.1, g = 0.0)

## vector of timesteps
times <- seq(0, 200, length = 101)

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

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

sigimp <- approxfun(signal$times, signal$import, rule = 2)

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

## Euler method
out1  <- rk(xstart, times, SPCmod, parms, hini = 0.1, 
            input = sigimp, method = "euler")

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

## Dormand-Prince method of order 5(4)
out3 <- rk(xstart, times, SPCmod, parms, hmax = 1, 
           input = sigimp, method = "rk45dp7")

mf <- par("mfrow")
## deSolve plot method for comparing scenarios
plot(out1, out2, out3, which = c("S", "P", "C"), 
     main = c ("Substrate", "Producer", "Consumer"),
     col =c("black", "red", "green"), 
     lty = c("solid", "dotted", "dotted"), lwd = c(1, 2, 1))

## user-specified plot function
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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