Solver for Differential Algebraic Equations (DAE)

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 $\partial\dot{y}/\partial y$, 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 $dG/dy + c_j\cdot dG/dy'$, where the $i$th 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. $F(t,y,dy) \neq 0$), 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.

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.

Author

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. doi:10.1016/0096-3003(89)90110-0

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. doi:10.1137/0915088

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: https://netlib.org

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.

Note

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

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

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.
the default of atol was changed from 1e-8 to 1e-6, to be consistent with the other solvers.
the multiple warnings from daspk when the number of steps exceed 500 were toggled off unless verbose is TRUE

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))
#> [1] 2.731114e-07

## =======================================================================
## 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))
#> [1] 1.725007e-09

## 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"))