Description Usage Arguments Details Value Note Author(s) See Also Examples
Solving initial value problems for systems of firstorder ordinary differential equations (ODEs) using Euler's method or the classical RungeKutta 4th order integration.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  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, ...)

y 
the initial (state) values for the ODE system. If 
times 
times at which explicit estimates for 
func 
either an Rfunction 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 The return value of If 
parms 
vector or list of parameters used in 
nspec 
for 1D models only: the number of species (components)
in the model. If 
dimens 
for 1D models only: the number of boxes in the
model. If 
names 
for 1D models only: the names of the components; used for plotting. 
verbose 
a logical value that, when 
ynames 
if 
dllname 
a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions refered to in 
initfunc 
if not 
initpar 
only when ‘dllname’ is specified and an
initialisation function 
rpar 
only when ‘dllname’ is specified: a vector with
double precision values passed to the DLLfunctions whose names are
specified by 
ipar 
only when ‘dllname’ is specified: a vector with
integer values passed to the dllfunctions whose names are specified
by 
nout 
only used if 
outnames 
only used if ‘dllname’ is specified and

forcings 
only used if ‘dllname’ is specified: a list with
the forcing function data sets, each present as a twocolumned matrix,
with (time, value); interpolation outside the interval
[min( See forcings or package vignette 
initforc 
if not 
fcontrol 
A list of control parameters for the forcing functions.
See forcings or vignette 
... 
additional arguments passed to 
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 RungeKutta 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.
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.
For most practical cases, solvers with flexible timestep
(e.g. rk(method = "ode45")
and especially solvers of the
Livermore family (ODEPACK, e.g. lsoda
) are superior.
Thomas Petzoldt [email protected]
rkMethod
for a list of available RungeKutta
parameter sets,
rk
for the more general RungeCode,
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 1D models,
ode.2D
for integrating 2D models,
ode.3D
for integrating 3D models,
dede
for integrating models with delay
differential equations,
diagnostics
to print diagnostic messages.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46  ## =======================================================================
## 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/N01) * 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 underestimation
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"))

Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.