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.
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, ...)
y |
the initial (state) values for the DE system. If |
times |
time sequence for which output is wanted; the first
value of |
func |
to be used if the model is an ODE, or a DAE written in linearly
implicit form (M y' = f(t, y)).
The return value of Note that it is not possible to define |
parms |
vector or list of parameters used in |
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 If Here The return value of If |
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 |
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 If the Jacobian is a full matrix, If the Jacobian is banded, |
jacres |
If If the Jacobian is a full matrix, If the Jacobian is banded, |
jactype |
the structure of the Jacobian, one of
|
mass |
the mass matrix.
If not If |
estini |
only if a DAE system, and if initial values of |
verbose |
if TRUE: full output to the screen, e.g. will
print the |
tcrit |
the FORTRAN routine |
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 |
hmax |
an optional maximum value of the integration stepsize. If
not specified, |
hini |
initial step size to be attempted; if 0, the initial step size is determined by the solver |
ynames |
logical, if |
maxord |
the maximum order to be allowed. Reduce |
bandup |
number of non-zero bands above the diagonal, in case
the Jacobian is banded (and |
banddown |
number of non-zero bands below the diagonal, in case
the Jacobian is banded (and |
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 |
dllname |
a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions referred to in |
initfunc |
if not |
initpar |
only when ‘dllname’ is specified and an
initialisation function |
rpar |
only when ‘dllname’ is specified: a vector with
double precision values passed to the dll-functions whose names are
specified by |
ipar |
only when ‘dllname’ is specified: a vector with
integer values passed to the dll-functions whose names are specified
by |
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 |
outnames |
only used if ‘dllname’ is specified and
|
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( See forcings or package vignette |
initforc |
if not |
fcontrol |
A list of control parameters for the forcing functions.
See forcings or vignette |
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 |
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:
a full Jacobian, calculated internally
by daspk, the default,
a full Jacobian, specified by user
function jacfunc or jacres,
a banded Jacobian, specified by user
function jacfunc or jacres; the size of the bands
specified by bandup and banddown,
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.
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.
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
Karline Soetaert <karline.soetaert@nioz.nl>
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
radau for integrating DAEs up to index 3,
rk,
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.
## =======================================================================
## 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"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.