R: Explicit One-Step Solvers for Ordinary Differential Equations...

rk {deSolve} | R Documentation

## 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 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

    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 matrix or data frame 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

    ## =======================================================================
    ## 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)

[Package deSolve version 1.32 Index]