daspk: Solver for Differential Algebraic Equations (DAE)
In deSolve: Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')
daspk 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 \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 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) \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.
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
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
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.
\Sexpr[results=rd]{tools:::Rd_expr_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.
\Sexpr[results=rd]{tools:::Rd_expr_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
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"))
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| daspk | 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') = 0or
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 |
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 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 |
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
jacfuncorjacres,- jactype = "bandusr":
a banded Jacobian, specified by user function
jacfuncorjacres; the size of the bands specified bybandupandbanddown,- jactype = "bandint":
a banded Jacobian, calculated by
daspk; the size of the bands specified bybandupandbanddown.
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
the argument list to
daspknow also includesnind, the index of each variable. This is used to scale the variables, such thatdaspkin 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
atolwas 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
verboseisTRUE
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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_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
See Also
-
radaufor integrating DAEs up to index 3, -
rk, -
rk4andeulerfor Runge-Kutta integrators. -
lsoda,lsode,lsodes,lsodar,vode, for other solvers of the Livermore family, -
odefor a general interface to most of the ODE solvers, -
ode.bandfor solving models with a banded Jacobian, -
ode.1Dfor integrating 1-D models, -
ode.2Dfor integrating 2-D models, -
ode.3Dfor 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"))
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