dopri5: Dormand-Prince or CashCarp Runge-Kutta of Order (4)5

Description

Solves the initial value problem for systems of ordinary differential equations (ODE) in the form: $$dy/dt = f(t,y)$$ The Rfunction dopri5 provides an interface to the Fortran ODE solver DOPRI5, written by E. Hairer and G. Wanner. It implements the explicit Runge-Kutta method of order 4(5) due to Dormand & Prince with stepsize contral and dense output The Rfunction cashkarp provides an interface to the Fortran ODE solver CASHCARP, written by J. Cash and F. Mazzia. It implements the explicit Runge-Kutta method of order 4(5) due to Cash-Carp, with stepsize contral and dense output The system of ODE's is written as an Rfunction or can be defined in compiled code that has been dynamically loaded.

Usage

dopri5   (y, times, func, parms, rtol = 1e-6, atol = 1e-6,
  verbose = FALSE, hmax = NULL, hini = hmax, ynames = TRUE, 
  maxsteps = 10000, dllname = NULL, initfunc = dllname, 
  initpar=parms, rpar = NULL, ipar = NULL, nout = 0, 
  outnames = NULL, forcings = NULL, initforc = NULL, fcontrol = NULL, ...)
cashkarp (y, times, func, parms, rtol = 1e-6, atol = 1e-6,
  verbose = FALSE, hmax = NULL, hini = hmax, ynames = TRUE, 
  maxsteps = 10000, dllname = NULL, initfunc = dllname, initpar = parms, 
  rpar = NULL, ipar = NULL, nout = 0, outnames = NULL, forcings = NULL,
  initforc = NULL, fcontrol = NULL, stiffness = 0, ...)

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

time sequence for which output is wanted; the first value of times must be the initial time; if only one step is to be taken; set times = NULL.

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

parms

vector or list of parameters used in func or jacfunc.

rtol

relative error tolerance, either a scalar or an array as long as y. See details.

atol

absolute error tolerance, either a scalar or an array as long as y. See details.

verbose

if TRUE: full output to the screen, e.g. will print the diagnostiscs of the integration - if the method becomes stiff it will rpint a message.

hmax

an optional maximum value of the integration stepsize. If not specified, hmax is set to the largest difference in times.

hini

initial step size to be attempted.

ynames

logical, if FALSE names of state variables are not passed to function func; this may speed up the simulation especially for multi-D models.

maxsteps

maximal number of steps taken by the solver, for the entire integration. This is different from the settings of this argument in the solvers from package deSolve!

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 vignette "compiledCode" from pa

initfunc

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

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. These names will be used to label the output matrix.

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 forcin

fcontrol

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

stiffness

How the stiffness of the solution should be estimated. Default = No stiffness estimate; when = stiffness = 1 or -1: all stiffness estimates calculated ; when = stiffness = 2 or -2: stiffness based

...

additional arguments passed to func and jacfunc 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 FORTRAN routine `lsoda' returns with an unrecoverable error. If y has a names attribute, it will be used to label the columns of the output value.

Details

The work is done by the FORTRAN subroutine dop853, whose documentation should be consulted for details. The implementation is based on the Fortran 77 version fromOctober 11, 2009. The input parameters rtol, and atol determine the error control performed by the solver, which roughly keeps the local error of y(i) below rtol(i)*abs(y(i))+atol(i). The diagnostics of the integration can be printed to screen by calling diagnostics. If verbose = TRUE, the diagnostics will written to the screen at the end of the integration.

See vignette("deSolve") from the deSolve package for an explanation of each element in the vectors containing the diagnostic properties and how to directly access them.

Models may be defined in compiled C or FORTRAN code, as well as in an R-function. See package vignette "compiledCode" from package deSolve for details.

Information about linking forcing functions to compiled code is in forcings (from package deSolve).

References

E. Hairer, S.P. Norsett AND G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series In Computational Mathematics, SPRINGER-VERLAG (1993)

Examples

Run this code

## =======================================================================
## Example :
##   The Arenstorff orbit model
## =======================================================================

Arenstorff <- function(t, y, parms) {

  D1 <- ((y[1]+mu)^2+y[2]^2)^(3/2)
  D2 <- ((y[1]-(1-mu))^2+y[2]^2)^(3/2)

  dy1 <- y[3]
  dy2 <- y[4]
  dy3 <- y[1] + 2*y[4]-(1-mu)*(y[1]+mu)/D1 -mu*(y[1]-(1-mu))/D2
  dy4 <- y[2] - 2*y[3]-(1-mu)*y[2]/D1 - mu*y[2]/D2

  list(c(dy1,dy2,dy3,dy4))
}

#-----------------------------
# parameters, initial values and times
#-----------------------------
mu    <- 0.012277471
yini  <- c(x = 0.994, y = 0, dx = 0, 
  dy = -2.00158510637908252240537862224)
times <- seq(0, 18, 0.01)

#-----------------------------
# solve the model
#-----------------------------

#out <- dopri5 (times=times, y=yini, func = Arenstorff, parms=NULL )
out  <- cashkarp (times = times, y = yini, func = Arenstorff, parms = NULL )
plot(out[,c("x", "y")], type = "l", lwd = 2, main = "Arenstorff")

#-----------------------------
# First and last value should be the same
#-----------------------------

times <- c(0, 17.0652165601579625588917206249)

Test  <- dopri5 (times = times, y = yini, func = Arenstorff, parms = NULL)

diagnostics(Test)

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