R: Solver for Differential Algebraic Equations (DAE)

daspk {deSolve} | R Documentation

## Solver for Differential Algebraic Equations (DAE)

### Description

Solves either:

- a system of ordinary differential equations (ODE) of the form

  y' = f(t, y, ...)

  or

- a system of differential algebraic equations (DAE) of the form

  F(t,y,y') = 0

  or

- a system of linearly implicit DAES in the form

  M y' = f(t, y)

using a combination of backward differentiation formula (BDF) and a direct linear system solution method (dense or banded).

The R function `daspk` provides an interface to the FORTRAN DAE solver of the same name, written by Linda R. Petzold, Peter N. Brown, Alan C. Hindmarsh and Clement W. Ulrich.

The system of DE'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.

### Usage

    daspk(y, times, func = NULL, parms, nind = c(length(y), 0, 0),
      dy = NULL, res = NULL, nalg = 0,
      rtol = 1e-6, atol = 1e-6, jacfunc = NULL,
      jacres = NULL, jactype = "fullint", mass = NULL, estini = NULL,
      verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL,
      hini = 0, ynames = TRUE, maxord = 5, bandup = NULL,
      banddown = NULL, maxsteps = 5000, dllname = NULL,
      initfunc = dllname, initpar = parms, rpar = NULL,
      ipar = NULL, nout = 0, outnames = NULL,
      forcings=NULL, initforc = NULL, fcontrol=NULL,
      events = NULL, lags = NULL, ...)

### Arguments

`y` | the initial (state) values for the DE 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` | to be used if the model is an ODE, or a DAE written in linearly implicit form (M y' = f(t, y)). `func` should be an R-function that computes the values of the derivatives in the ODE system (the model definition) at time t. `func` 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`, unless `ynames` is FALSE. `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 should be specified in the same order as the specification of the state variables `y`. Note that it is not possible to define `func` as a compiled function in a dynamically loaded shared library. Use `res` instead.
`parms` | vector or list of parameters used in `func`, `jacfunc`, or `res`
`nind` | if a DAE system: a three-valued vector with the number of variables of index 1, 2, 3 respectively. The equations must be defined such that the index 1 variables precede the index 2 variables which in turn precede the index 3 variables. The sum of the variables of different index should equal N, the total number of variables. Note that this has been added for consistency with radau. If used, then the variables are weighed differently than in the original daspk code, i.e. index 2 variables are scaled with 1/h, index 3 variables are scaled with 1/h^2. In some cases this allows daspk to solve index 2 or index 3 problems.
`dy` | the initial derivatives of the state variables of the DE system. Ignored if an ODE.
`res` | if a DAE system: either an R-function that computes the residual function F(t,y,y') of the DAE system (the model defininition) at time `t`, or a character string giving the name of a compiled function in a dynamically loaded shared library. If `res` is a user-supplied R-function, it must be defined as: `res <- function(t, y, dy, parms, ...)`. Here `t` is the current time point in the integration, `y` is the current estimate of the variables in the ODE system, `dy` are the corresponding derivatives. If the initial `y` or `dy` have a `names` attribute, the names will be available inside `res`, unless `ynames` is `FALSE`. `parms` is a vector of parameters. The return value of `res` should be a list, whose first element is a vector containing the residuals of the DAE system, i.e. delta = F(t,y,y'), and whose next elements contain output variables that are required at each point in `times`. If `res` is a string, then `dllname` must give the name of the shared library (without extension) which must be loaded before `daspk()` is called (see package vignette `"compiledCode"` for more information).
`nalg` | if a DAE system: the number of algebraic equations (equations not involving derivatives). Algebraic equations should always be the last, i.e. preceeded by the differential equations. Only used if `estini` = 1.
`rtol` | relative error tolerance, either a scalar or a vector, one value for each y,
`atol` | absolute error tolerance, either a scalar or a vector, one value for each y.
`jacfunc` | if not `NULL`, an R function that computes the Jacobian of the system of differential equations. Only used in case the system is an ODE (y' = f(t, y)), specified by `func`. The R calling sequence for `jacfunc` is identical to that of `func`. If the Jacobian is a full matrix, `jacfunc` should return a matrix dydot/dy, where the ith row contains the derivative of dy_i/dt with respect to y_j, or a vector containing the matrix elements by columns (the way R and FORTRAN store matrices). If the Jacobian is banded, `jacfunc` should return a matrix containing only the nonzero bands of the Jacobian, rotated row-wise. See first example of lsode.
`jacres` | `jacres` and not `jacfunc` should be used if the system is specified by the residual function F(t, y, y'), i.e. `jacres` is used in conjunction with `res`. If `jacres` is an R-function, the calling sequence for `jacres` is identical to that of `res`, but with extra parameter `cj`. Thus it should be called as: `jacres = func(t, y, dy, parms, cj, ...)`. Here `t` is the current time point in the integration, `y` is the current estimate of the variables in the ODE system, y' are the corresponding derivatives and `cj` is a scalar, which is normally proportional to the inverse of the stepsize. If the initial `y` or `dy` have a `names` attribute, the names will be available inside `jacres`, unless `ynames` is `FALSE`. `parms` is a vector of parameters (which may have a names attribute). If the Jacobian is a full matrix, `jacres` should return the matrix dG/d y + cj*dG/d y', where the ith row is the sum of the derivatives of G_i with respect to y_j and the scaled derivatives of G_i with respect to y'_j. If the Jacobian is banded, `jacres` should return only the nonzero bands of the Jacobian, rotated rowwise. See details for the calling sequence when `jacres` is a string.
`jactype` | the structure of the Jacobian, one of `"fullint"`, `"fullusr"`, `"bandusr"` or `"bandint"` - either full or banded and estimated internally or by the user.
`mass` | the mass matrix. If not `NULL`, the problem is a linearly implicit DAE and defined as M dy/dt = f(t,y). The mass-matrix M should be of dimension n*n where n is the number of y-values. If `mass=NULL` then the model is either an ODE or a DAE, specified with `res`
`estini` | only if a DAE system, and if initial values of `y` and `dy` are not consistent (i.e. F(t, y, dy) != 0), setting `estini` = 1 or 2, will solve for them. If `estini` = 1: dy and the algebraic variables are estimated from `y`; in this case, the number of algebraic equations must be given (`nalg`). If `estini` = 2: `y` will be estimated from `dy`.
`verbose` | if TRUE: full output to the screen, e.g. will print the `diagnostiscs` of the integration - see details.
`tcrit` | the FORTRAN routine `daspk` overshoots its targets (times points in the vector `times`), and interpolates values for the desired time points. If there is a time beyond which integration should not proceed (perhaps because of a singularity), that should be provided in `tcrit`.
`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 size is specified.
`hini` | initial step size to be attempted; if 0, the initial step size is determined by the solver
`ynames` | logical, if `FALSE`, names of state variables are not passed to function `func`; this may speed up the simulation especially for large models.
`maxord` | the maximum order to be allowed. Reduce `maxord` to save storage space ( <= 5)
`bandup` | number of non-zero bands above the diagonal, in case the Jacobian is banded (and `jactype` one of "bandint", "bandusr")
`banddown` | number of non-zero bands below the diagonal, in case the Jacobian is banded (and `jactype` one of "bandint", "bandusr")
`maxsteps` | maximal number of steps per output interval taken by the solver; will be recalculated to be at least 500 and a multiple of 500; if `verbose` is `TRUE` the solver will give a warning if more than 500 steps are taken, but it will continue till `maxsteps` steps. (Note this warning was always given in deSolve versions < 1.10.3).
`dllname` | a string giving the name of the shared library (without extension) that contains all the compiled function or subroutine definitions referred to in `res` and `jacres`. 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 `res` and `jacres`.
`ipar` | only when ‘dllname’ is specified: a vector with integer values passed to the dll-functions whose names are specified by `res` and `jacres`.
`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 `res`, 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 `res`, 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 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.
`lags` | A list that specifies timelags, i.e. the number of steps that has to be kept. To be used for delay differential equations. See timelags, dede for more information.
`...` | additional arguments passed to `func`, `jacfunc`, `res` and `jacres`, allowing this to be a generic function.

### Details

The daspk solver uses the backward differentiation formulas of orders one through five (specified with `maxord`) to solve either:

- an ODE system of the form

  y' = f(t,y,...)

  or

- a DAE system of the form

  y' = M f(t,y,...)

  or

- a DAE system of the form

  F(t,y,y') = 0

  . The index of the DAE should be preferable <= 1.

ODEs are specified using argument `func`, DAEs are specified using argument `res`.

If a DAE system, Values for y and y' (argument `dy`) at the initial time must be given as input. Ideally, these values should be consistent, that is, if t, y, y' are the given initial values, they should satisfy F(t,y,y') = 0.
However, if consistent values are not known, in many cases daspk can solve for them: when `estini` = 1, y' and algebraic variables (their number specified with `nalg`) will be estimated, when `estini` = 2, y will be estimated.

The form of the Jacobian can be specified by `jactype`. This is one of:

jactype = "fullint":

  a full Jacobian, calculated internally by `daspk`, the default,

jactype = "fullusr":

  a full Jacobian, specified by user function `jacfunc` or `jacres`,

jactype = "bandusr":

  a banded Jacobian, specified by user function `jacfunc` or `jacres`; the size of the bands specified by `bandup` and `banddown`,

jactype = "bandint":

  a banded Jacobian, calculated by `daspk`; the size of the bands specified by `bandup` and `banddown`.

If `jactype` = "fullusr" or "bandusr" then the user must supply a subroutine `jacfunc`.

If jactype = "fullusr" or "bandusr" then the user must supply a subroutine `jacfunc` or `jacres`.

The input parameters `rtol`, and `atol` determine the error control performed by the solver. If the request for precision exceeds the capabilities of the machine, `daspk` will return an error code. See `lsoda` for details.

When the index of the variables is specified (argument `nind`), and higher index variables are present, then the equations are scaled such that equations corresponding to index 2 variables are multiplied with 1/h, for index 3 they are multiplied with 1/h^2, where h is the time step. This is not in the standard DASPK code, but has been added for consistency with solver radau. Because of this, daspk can solve certain index 2 or index 3 problems.

res and jacres may be defined in compiled C or FORTRAN code, as well as in an R-function. See package vignette `"compiledCode"` for details. Examples in FORTRAN are in the ‘dynload’ subdirectory of the `deSolve` package directory.

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") 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"` for details.

More information about models defined in compiled code is in the package vignette ("compiledCode"); information about linking forcing functions to compiled code is in forcings.

Examples in both C and FORTRAN are in the ‘dynload’ subdirectory of the `deSolve` package directory.

### 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` or `res`, plus an additional column (the first) for the time value. There will be one row for each element in `times` unless the FORTRAN routine ‘daspk’ returns with an unrecoverable error. If `y` has a names attribute, it will be used to label the columns of the output value.

### Note

In this version, the Krylov method is not (yet) supported.

From `deSolve` version 1.10.4 and above, the following changes were made

1. the argument list to `daspk` now also includes `nind`, the index of each variable. This is used to scale the variables, such that `daspk` in R can also solve certain index 2 or index 3 problems, which the original Fortran version may not be able to solve.

2. the default of `atol` was changed from 1e-8 to 1e-6, to be consistent with the other solvers.

3. the multiple warnings from daspk when the number of steps exceed 500 were toggled off unless `verbose` is `TRUE`

### Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

### References

L. R. Petzold, A Description of DASSL: A Differential/Algebraic System Solver, in Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pp. 65-68.

K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier, New York, 1989.

P. N. Brown and A. C. Hindmarsh, Reduced Storage Matrix Methods in Stiff ODE Systems, J. Applied Mathematics and Computation, 31 (1989), pp. 40-91.

P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems, SIAM J. Sci. Comp., 15 (1994), pp. 1467-1488.

P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Consistent Initial Condition Calculation for Differential-Algebraic Systems, LLNL Report UCRL-JC-122175, August 1995; submitted to SIAM J. Sci. Comp.

Netlib: http://www.netlib.org

### See Also

- `radau` for integrating DAEs up to index 3,

- `rk`,

- `rk4` and `euler` for Runge-Kutta integrators.

- `lsoda`, `lsode`, `lsodes`, `lsodar`, `vode`, for other 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

    ## =======================================================================
    ## Coupled chemical reactions including an equilibrium
    ## modeled as (1) an ODE and (2) as a DAE
    ##
    ## The model describes three chemical species A,B,D:
    ## subjected to equilibrium reaction D <- > A + B
    ## D is produced at a constant rate, prod
    ## B is consumed at 1s-t order rate, r
    ## Chemical problem formulation 1: ODE
    ## =======================================================================

    ## Dissociation constant
    K <- 1

    ## parameters
    pars <- c(
            ka   = 1e6,     # forward rate
            r    = 1,
            prod = 0.1)

    Fun_ODE <- function (t, y, pars)
    {
      with (as.list(c(y, pars)), {
        ra  <- ka*D        # forward rate
        rb  <- ka/K *A*B   # backward rate

        ## rates of changes
        dD  <- -ra + rb + prod
        dA  <-  ra - rb
        dB  <-  ra - rb - r*B
        return(list(dy = c(dA, dB, dD),
                    CONC = A+B+D))
      })
    }

    ## =======================================================================
    ## Chemical problem formulation 2: DAE
    ## 1. get rid of the fast reactions ra and rb by taking
    ## linear combinations   : dD+dA = prod (res1) and
    ##                         dB-dA = -r*B (res2)
    ## 2. In addition, the equilibrium condition (eq) reads:
    ## as ra = rb : ka*D = ka/K*A*B = >      K*D = A*B
    ## =======================================================================

    Res_DAE <- function (t, y, yprime, pars)
    {
      with (as.list(c(y, yprime, pars)), {

        ## residuals of lumped rates of changes
        res1 <- -dD - dA + prod
        res2 <- -dB + dA - r*B

        ## and the equilibrium equation
        eq   <- K*D - A*B

        return(list(c(res1, res2, eq),
                    CONC = A+B+D))
      })
    }

    ## =======================================================================
    ## Chemical problem formulation 3: Mass * Func
    ## Based on the DAE formulation
    ## =======================================================================

    Mass_FUN <- function (t, y, pars) {
      with (as.list(c(y, pars)), {

        ## as above, but without the
        f1 <- prod
        f2 <- - r*B

        ## and the equilibrium equation
        f3   <- K*D - A*B

        return(list(c(f1, f2, f3),
                    CONC = A+B+D))
      })
    }
    Mass <- matrix(nrow = 3, ncol = 3, byrow = TRUE,
      data=c(1,  0, 1,         # dA + 0 + dB
            -1,  1, 0,         # -dA + dB +0
             0,  0, 0))        # algebraic

    times <- seq(0, 100, by = 2)

    ## Initial conc; D is in equilibrium with A,B
    y     <- c(A = 2, B = 3, D = 2*3/K)

    ## ODE model solved with daspk
    ODE <- daspk(y = y, times = times, func = Fun_ODE,
                         parms = pars, atol = 1e-10, rtol = 1e-10)

    ## Initial rate of change
    dy  <- c(dA = 0, dB = 0, dD = 0)

    ## DAE model solved with daspk
    DAE <- daspk(y = y, dy = dy, times = times,
             res = Res_DAE, parms = pars, atol = 1e-10, rtol = 1e-10)

    MASS<- daspk(y=y, times=times, func = Mass_FUN, parms = pars, mass = Mass)

    ## ================
    ## plotting output
    ## ================

    plot(ODE, DAE, xlab = "time", ylab = "conc", type = c("l", "p"),
         pch = c(NA, 1))

    legend("bottomright", lty = c(1, NA), pch = c(NA, 1),
      col = c("black", "red"), legend = c("ODE", "DAE"))

    # difference between both implementations:
    max(abs(ODE-DAE))

    ## =======================================================================
    ## same DAE model, now with the Jacobian
    ## =======================================================================
    jacres_DAE <- function (t, y, yprime, pars, cj)
    {
        with (as.list(c(y, yprime, pars)), {
    ##    res1 = -dD - dA + prod
          PD[1,1] <- -1*cj      # d(res1)/d(A)-cj*d(res1)/d(dA)
          PD[1,2] <- 0          # d(res1)/d(B)-cj*d(res1)/d(dB)
          PD[1,3] <- -1*cj      # d(res1)/d(D)-cj*d(res1)/d(dD)
    ##     res2 = -dB + dA - r*B
          PD[2,1] <- 1*cj
          PD[2,2] <- -r -1*cj
          PD[2,3] <- 0
    ##    eq = K*D - A*B
          PD[3,1] <- -B
          PD[3,2] <- -A
          PD[3,3] <- K
          return(PD)
       })
    }

    PD <- matrix(ncol = 3, nrow = 3, 0)

    DAE2 <- daspk(y = y, dy = dy, times = times,
              res = Res_DAE, jacres = jacres_DAE, jactype = "fullusr",
              parms = pars, atol = 1e-10, rtol = 1e-10)

    max(abs(DAE-DAE2))

    ## See \dynload subdirectory for a FORTRAN implementation of this model

    ## =======================================================================
    ## The chemical model as a DLL, with production a forcing function
    ## =======================================================================
    times <- seq(0, 100, by = 2)

    pars <- c(K = 1, ka   = 1e6, r    = 1)

    ## Initial conc; D is in equilibrium with A,B
    y     <- c(A = 2, B = 3, D = as.double(2*3/pars["K"]))

    ## Initial rate of change
    dy  <- c(dA = 0, dB = 0, dD = 0)

    # production increases with time
    prod <- matrix(ncol = 2,
                   data = c(seq(0, 100, by = 10), 0.1*(1+runif(11)*1)))

    ODE_dll <- daspk(y = y, dy = dy, times = times, res = "chemres",
              dllname = "deSolve", initfunc = "initparms",
              initforc = "initforcs", parms = pars, forcings = prod,
              atol = 1e-10, rtol = 1e-10, nout = 2,
              outnames = c("CONC","Prod"))

    plot(ODE_dll, which = c("Prod", "D"), xlab = "time",
         ylab = c("/day", "conc"), main = c("production rate","D"))

[Package deSolve version 1.21 Index]