deSolve (version 1.10-9)

rk4: Solve System of ODE (Ordinary Differential Equation)s by Euler's Method or Classical Runge-Kutta 4th Order Integration.

Description

Solving initial value problems for systems of first-order ordinary differential equations (ODEs) using Euler's method or the classical Runge-Kutta 4th order integration.

Usage

euler(y, times, func, parms, verbose = FALSE, ynames = TRUE,
  dllname = NULL, initfunc = dllname, initpar = parms,
  rpar = NULL, ipar = NULL, nout = 0, outnames = NULL,
  forcings = NULL, initforc = NULL, fcontrol = NULL, ...)
rk4(y, times, func, parms, verbose = FALSE, ynames = TRUE,
  dllname = NULL, initfunc = dllname, initpar = parms,
  rpar = NULL, ipar = NULL, nout = 0, outnames = NULL,
  forcings = NULL, initforc = NULL, fcontrol = NULL, ...)
euler.1D(y, times, func, parms, nspec = NULL, dimens = NULL,
  names = NULL, verbose = FALSE, ynames = TRUE,
  dllname = NULL, initfunc = dllname, initpar = parms,
  rpar = NULL,  ipar = NULL, nout = 0, outnames = NULL,
  forcings = NULL, initforc = NULL, fcontrol = 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 fu

parms

vector or list of parameters used in func.

nspec

for 1D models only: the number of species (components) in the model. If NULL, then dimens should be specified.

dimens

for 1D models only: the number of boxes in the model. If NULL, then nspec should be specified.

names

for 1D models only: the names of the components; used for plotting.

verbose

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

ynames

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

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. 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 don

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.

...

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

rk4 and euler are special versions of the two fixed step solvers with less overhead and less functionality (e.g. no interpolation and no events) compared to the generic Runge-Kutta codes called by ode resp. rk. If you need different internal and external time steps or want to use events, please use: rk(y, times, func, parms, method = "rk4") or rk(y, times, func, parms, method = "euler").

See help pages of rk and rkMethod for details.

Function euler.1D essentially calls functioneuler but contains additional code to support plotting of 1D models, see ode.1D and plot.1D for details.

See Also

• rkMethodfor a list of available Runge-Kutta parameter sets,
• rkfor the more general Runge-Code,
• lsoda,lsode,lsodes,lsodar,vode,daspkfor solvers of the Livermore family,
• odefor a general interface to most of the ODE solvers,
• ode.bandfor solving models with a banded Jacobian,
• ode.1Dfor integrating 1-D models,
• ode.2Dfor integrating 2-D models,
• ode.3Dfor integrating 3-D models,
• dedefor integrating models with delay differential equations,

diagnostics to print diagnostic messages.

Examples

## =======================================================================
## Example: Analytical and numerical solutions of logistic growth
## =======================================================================

## the derivative of the logistic
logist <- function(t, x, parms) {
  with(as.list(parms), {
    dx <- r * x[1] * (1 - x[1]/K)
    list(dx)
  })
}

time  <- 0:100
N0    <- 0.1; r <- 0.5; K <- 100
parms <- c(r = r, K = K)
x <- c(N = N0)

## analytical solution
plot(time, K/(1 + (K/N0-1) * exp(-r*time)), ylim = c(0, 120),
  type = "l", col = "red", lwd = 2)

## reasonable numerical solution with rk4
time <- seq(0, 100, 2)
out <- as.data.frame(rk4(x, time, logist, parms))
points(out$time, out$N, pch = 16, col = "blue", cex = 0.5)

## same time step with euler, systematic under-estimation
time <- seq(0, 100, 2)
out <- as.data.frame(euler(x, time, logist, parms))
points(out$time, out$N, pch = 1)

## unstable result
time <- seq(0, 100, 4)
out <- as.data.frame(euler(x, time, logist, parms))
points(out$time, out$N, pch = 8, cex = 0.5)

## method with automatic time step
out <- as.data.frame(lsoda(x, time, logist, parms))
points(out$time, out$N, pch = 1, col = "green")

legend("bottomright",
  c("analytical","rk4, h=2", "euler, h=2",
    "euler, h=4", "lsoda"),
  lty = c(1, NA, NA, NA, NA), lwd = c(2, 1, 1, 1, 1),
  pch = c(NA, 16, 1, 8, 1),
  col = c("red", "blue", "black", "black", "green"))