deSolve (version 1.2-1)

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 step 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). 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-06, atol = 1e-06, 
  tcrit = NULL, verbose = FALSE, hmin = 0, hmax = NULL, hini = hmax, 
  method = rkMethod("rk45dp7", ...), maxsteps = 5000, ...)

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

an R-function that computes the values of the derivatives in the ODE system (the model definition) at time t. The R-function func must be defined as: yprime = func(t, y, parms, ...). t

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. The solver routines may overshoot their targets (times points in the vector times), and interpolates values for the desired time po

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 largest difference in times, to avoid that the simulation possibly ignores short-term events. If 0, no maximal

hini

initial step size to be attempted; if 0, the initial step size is determined by the solver.

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

maxsteps

maximal number of steps during one call to the solver.

...

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

Value

• A matrix 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 solver returns with an unrecoverable error. If y has a names attribute, it will be used to label the columns of the output value. The output will have the attributes istate, and rstate, two vectors with several useful elements, whose interpretation is compatible with lsoda:
• el 1:
  0 for normal return, -2 means excess accuracy requested (tolerances too small).
• el 12:
  The number of steps taken for the problem so far.
• el 13:
  The number of function evaluations for the problem so far.
• el 15:
  The order of the method.

Details

The Runge-Kutta solvers are primarily provided for didactic reasons. For most practical cases, solvers of the Livermore family (lsoda, lsode, lsodes, lsodar, vode, daspk) are superior because of higher efficiency and faster implementation (FORTRAN and C). In addition to this, some of the Livermore solvers are also suitable for stiff ODEs, differential algebraic equations (DAEs), or partial differential equations (PDEs). Function rk is a generalized implementation that can be used to evaluate different solvers of the Runge-Kutta family. 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. An example is given below: model<-function(t,Y,parameters) { with(as.list(parameters),{ dy1 = -k1*Y[1] + k2*Y[2]*Y[3] dy3 = k3*Y[2]*Y[2] dy2 = -dy1 - dy3 list(c(dy1,dy2,dy3)) }) }

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

See Also

rkMethod, ode, lsoda, lsode, lsodes, lsodar, vode, daspk

Examples

#########################################
  ## Example: Lotka-volterra model
  #########################################
  
  ## Note: 
  ## parameters are a list, names accessible via "with" statement
  ## (see also ode and lsoda examples)

  lvmodel <- function(t, x, parms) {
    S <- x[1] # substrate
    P <- x[2] # producer
    K <- x[3] # consumer
    
    with(parms,{
      import <- approx(signal$times, signal$import, t)$y
      dS <- import - b * S * P + g * K
      dP <- c * S * P  - d * K * P
      dK <- e * P * K  - f * K
      res<-c(dS, dP, dK)
      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, K=1)
  
  ## Euler method
  out1  <- as.data.frame(rk(xstart, times, lvmodel, parms, 
                            hini = 0.1, method="euler"))
  
  ## classical Runge-Kutta 4th order
  out2 <- as.data.frame(rk(xstart, times, lvmodel, parms, 
                           hini = 1, method="rk4"))
  
  ## Dormand-Prince method of order 5(4)
  out3 <- as.data.frame(rk(xstart, times, lvmodel, 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$time, out1$K, type="l",    ylab="Consumer")
  lines(out2$time, out2$K, col="red",   lty="dotted", lwd=2)
  lines(out3$time, out3$K, col="green", lty="dotted")
  
  plot (out1$P, out1$K, type="l", xlab="Producer",ylab="Consumer")
  lines(out2$P, out2$K, col="red",   lty="dotted", lwd=2)
  lines(out3$P, out3$K, 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)