mebdfi: Solver for Differential Algebraic Equations (DAE) up to index...
In deTestSet: Test Set for Differential Equations
mebdfi R Documentation
Solver for Differential Algebraic Equations (DAE) up to index 3
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 the Modified Extended Backward Differentiation formulas for stiff
fully implicit inital value problems
These formulas increase the absolute stability regions of
the classical BDFs.
The orders of the implemented formulae range from 1 to 8.
The R function mebdfi provides an interface to the Fortran DAE
solver of the same name, written by T.J. Abdulla and J.R. Cash.
The system of DE's is written as an R function or can be defined in
compiled code that has been dynamically loaded.
Usage
mebdfi(y, times, func = NULL, parms, dy = NULL, res = NULL,
nind=c(length(y),0,0), rtol = 1e-6, atol = 1e-6, jacfunc = NULL,
jacres = NULL, jactype = "fullint", mass = NULL, verbose = FALSE,
tcrit = NULL, hini = 0, ynames = TRUE, maxord = 7, 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, ...)
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
cannot be used if the model is a DAE system. If an ODE
system, 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 or DAE 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
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 DAE 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
mebdfi() is called (see package vignette "compiledCode"
for more information).
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.
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 defined as: jacres <-
function(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 + cj*dG/dyprime, 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 dy_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 mass dy/dt = f(t,y). The mass-matrix
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
verbose
if TRUE: full output to the screen, e.g. will
print the diagnostiscs of the integration - see details.
tcrit
the Fortran routine mebdfi 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.
hini
initial step size to be attempted; if 0, the initial step
size is set to 1e-6, but it may be better to set it equal to rtol.
The solver is quite sensitive to values of hini; sometimes
if it fails, it helps to decrease/increase hini
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, an integer between 2 and 7.
The default is maxord = 7, but values of 4-5 may be better for
difficult problems; hihger order methods are more efficient but less stable.
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.
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 calculed 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 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 package vignette "compiledCode".
fcontrol
A list of control parameters for the forcing functions.
vignette compiledCode from package deSolve.
...
additional arguments passed to func,
jacfunc, res and jacres, allowing this to be a
generic function.
Details
The mebdfi solver uses modified extended backward differentiation
formulas of orders one through eight (specified with maxord)
to solve either:
an ODE system of the form
y' = f(t,y,...)
for y = Y,
or
a DAE system of the form
F(t,y,y') = 0
for y = Y and y'
= YPRIME.
The recommended value of maxord is eight, unless it is believed
that there are severe stability problems in which case maxord =
4 or 5 should be tried instead.
ODEs are specified in func, DAEs are specified in res.
If a DAE system, Values for Y and YPRIME at the initial time
must be given as input. Ideally,these values should be consistent,
that is, if T, Y, YPRIME are the given initial values, they should
satisfy F(T,Y,YPRIME) = 0.
The form of the Jacobian can be specified by
jactype. This is one of:
- jactype = "fullint":
a full Jacobian, calculated internally
by mebdfi, 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
mebdfi; 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, mebdfi will return
an error code.
res and jacres may be defined in compiled C or Fortran code, as
well as in an R-function. See deSolve's 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.
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 ‘mebdfi’ returns with an unrecoverable error. If
y has a names attribute, it will be used to label the columns
of the output value.
Author(s)
Karline Soetaert <karline.soetaert@nioz.nl>
Jeff Cash
References
J. R. Cash, The integration of stiff initial value problems
in O.D.E.S using modified extended backward differentiation
formulae, Comp. and Maths. with applics., 9, 645-657, (1983).
J.R. Cash and S. Considine, an MEBDF code for stiff
initial value problems, ACM Trans Math Software, 142-158,
(1992).
J.R. Cash, Stable recursions with applications to the
numerical solution of stiff systems, Academic Press,(1979).
See Also
-
gamd and bimd two other DAE solvers,
-
daspk another DAE solver from package deSolve,
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))
})
}
times <- seq(0, 100, by = 1)
## Initial conc; D is in equilibrium with A,B
y <- c(A = 2, B = 3, D = 2*3/K)
## ODE model solved with mebdfi
ODE <- as.data.frame(mebdfi(y = y, times = times, func = Fun_ODE,
parms = pars, atol = 1e-8, rtol = 1e-8))
## Initial rate of change
dy <- c(dA = 0, dB = 0, dD = 0)
## DAE model solved with mebdfi
DAE <- as.data.frame(mebdfi(y = y, dy = dy, times = times,
res = Res_DAE, parms = pars, atol = 1e-8, rtol = 1e-8))
## =======================================================================
## 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(nr=3, nc=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 <- as.data.frame(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 <- as.data.frame(daspk(y = y, dy = dy, times = times,
res = Res_DAE, parms = pars, atol = 1e-10, rtol = 1e-10))
MASS<- mebdfi(y = y, times = times, func = Mass_FUN,
parms = pars, mass = Mass)
## ================
## plotting output
## ================
opa <- par(mfrow = c(2, 2))
for (i in 2:5)
{
plot(ODE$time, ODE[, i], xlab = "time",
ylab = "conc", main = names(ODE)[i], type = "l")
points(DAE$time, DAE[,i], col = "red")
}
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))
par(mfrow = opa)
## =============================================================================
##
## Example 3: higher index DAE
##
## Car axis problem, index 3 DAE, 8 differential, 2 algebraic equations
## from
## F. Mazzia and C. Magherini. Test Set for Initial Value Problem Solvers,
## release 2.4. Department
## of Mathematics, University of Bari and INdAM, Research Unit of Bari,
## February 2008.
## Available at http://www.dm.uniba.it/~testset.
## =============================================================================
# car returns the residuals of the implicit DAE
car <- function(t, y, dy, pars){
with(as.list(c(pars, y)), {
f <- rep(0, 10)
yb <- r*sin(w*t)
xb <- sqrt(L*L - yb*yb)
Ll <- sqrt(xl^2 + yl^2)
Lr <- sqrt((xr-xb)^2 + (yr-yb)^2)
f[1:4] <- y[5:8]
k <- M*eps*eps/2
f[5] <- (L0-Ll)*xl/Ll + lam1*xb+2*lam2*(xl-xr)
f[6] <- (L0-Ll)*yl/Ll + lam1*yb+2*lam2*(yl-yr)-k*g
f[7] <- (L0-Lr)*(xr-xb)/Lr - 2*lam2*(xl-xr)
f[8] <- (L0-Lr)*(yr-yb)/Lr - 2*lam2*(yl-yr)-k*g
f[9] <- xb*xl+yb*yl
f[10] <- (xl-xr)^2+(yl-yr)^2-L*L
delt <- dy-f
delt[5:8] <- k*dy[5:8]-f[5:8]
delt[9:10] <- -f[9:10]
list(delt=delt,f=f)
})
}
# parameters
pars <- c(eps = 1e-2, M = 10, L = 1, L0 = 0.5,
r = 0.1, w = 10, g = 1)
# initial conditions: state variables
yini <- with (as.list(pars),
c(xl = 0, yl = L0, xr = L, yr = L0, xla = -L0/L,
yla = 0, xra = -L0/L, yra = 0, lam1 = 0, lam2 = 0)
)
# initial conditions: derivates
dyini <- rep(0, 10)
FF <- car(0, yini, dyini, pars)
dyini[1:4] <- yini[5:8]
dyini[5:8] <- 2/pars["M"]/(pars["eps"])^2*FF$f[5:8]
# check consistency of initial condition: delt should be = 0.
car(0, yini, dyini, pars)
# running the model
times <- seq(0, 3, by = 0.01)
nind <- c(4, 4, 2) # index 1, 2 and 3 variables
out <- mebdfi(y = yini, dy = dyini, times, res = car, parms = pars,
nind = nind, rtol = 1e-5, atol = 1e-5)
plot(out, which = 1:4, type = "l", lwd=2)
mtext(outer = TRUE, side = 3, line = -0.5, cex = 1.5, "car axis")
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| mebdfi | R Documentation |
Solver for Differential Algebraic Equations (DAE) up to index 3
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 the Modified Extended Backward Differentiation formulas for stiff fully implicit inital value problems
These formulas increase the absolute stability regions of the classical BDFs.
The orders of the implemented formulae range from 1 to 8.
The R function mebdfi provides an interface to the Fortran DAE
solver of the same name, written by T.J. Abdulla and J.R. Cash.
The system of DE's is written as an R function or can be defined in compiled code that has been dynamically loaded.
Usage
mebdfi(y, times, func = NULL, parms, dy = NULL, res = NULL,
nind=c(length(y),0,0), rtol = 1e-6, atol = 1e-6, jacfunc = NULL,
jacres = NULL, jactype = "fullint", mass = NULL, verbose = FALSE,
tcrit = NULL, hini = 0, ynames = TRUE, maxord = 7, 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, ...)
Arguments
y |
the initial (state) values for the DE system. If |
times |
time sequence for which output is wanted; the first
value of |
func |
cannot be used if the model is a DAE system. If an ODE
system,
The return value of Note that it is not possible to define |
parms |
vector or list of parameters used in |
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 |
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. |
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 |
verbose |
if TRUE: full output to the screen, e.g. will
print the |
tcrit |
the Fortran routine |
hini |
initial step size to be attempted; if 0, the initial step
size is set to 1e-6, but it may be better to set it equal to |
ynames |
logical, if |
maxord |
the maximum order to be allowed, an integer between 2 and 7.
The default is |
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. |
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 package vignette |
initforc |
if not |
fcontrol |
A list of control parameters for the forcing functions.
vignette |
... |
additional arguments passed to |
Details
The mebdfi solver uses modified extended backward differentiation
formulas of orders one through eight (specified with maxord)
to solve either:
an ODE system of the form
y' = f(t,y,...)for y = Y, or
a DAE system of the form
F(t,y,y') = 0for y = Y and y' = YPRIME.
The recommended value of maxord is eight, unless it is believed
that there are severe stability problems in which case maxord =
4 or 5 should be tried instead.
ODEs are specified in func, DAEs are specified in res.
If a DAE system, Values for Y and YPRIME at the initial time must be given as input. Ideally,these values should be consistent, that is, if T, Y, YPRIME are the given initial values, they should satisfy F(T,Y,YPRIME) = 0.
The form of the Jacobian can be specified by
jactype. This is one of:
- jactype = "fullint":
a full Jacobian, calculated internally by
mebdfi, 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
mebdfi; 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, mebdfi will return
an error code.
res and jacres may be defined in compiled C or Fortran code, as
well as in an R-function. See deSolve's 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.
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 ‘mebdfi’ returns with an unrecoverable error. If
y has a names attribute, it will be used to label the columns
of the output value.
Author(s)
Karline Soetaert <karline.soetaert@nioz.nl>
Jeff Cash
References
J. R. Cash, The integration of stiff initial value problems in O.D.E.S using modified extended backward differentiation formulae, Comp. and Maths. with applics., 9, 645-657, (1983).
J.R. Cash and S. Considine, an MEBDF code for stiff initial value problems, ACM Trans Math Software, 142-158, (1992).
J.R. Cash, Stable recursions with applications to the numerical solution of stiff systems, Academic Press,(1979).
See Also
-
gamdandbimdtwo other DAE solvers, -
daspkanother DAE solver from packagedeSolve,
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))
})
}
times <- seq(0, 100, by = 1)
## Initial conc; D is in equilibrium with A,B
y <- c(A = 2, B = 3, D = 2*3/K)
## ODE model solved with mebdfi
ODE <- as.data.frame(mebdfi(y = y, times = times, func = Fun_ODE,
parms = pars, atol = 1e-8, rtol = 1e-8))
## Initial rate of change
dy <- c(dA = 0, dB = 0, dD = 0)
## DAE model solved with mebdfi
DAE <- as.data.frame(mebdfi(y = y, dy = dy, times = times,
res = Res_DAE, parms = pars, atol = 1e-8, rtol = 1e-8))
## =======================================================================
## 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(nr=3, nc=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 <- as.data.frame(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 <- as.data.frame(daspk(y = y, dy = dy, times = times,
res = Res_DAE, parms = pars, atol = 1e-10, rtol = 1e-10))
MASS<- mebdfi(y = y, times = times, func = Mass_FUN,
parms = pars, mass = Mass)
## ================
## plotting output
## ================
opa <- par(mfrow = c(2, 2))
for (i in 2:5)
{
plot(ODE$time, ODE[, i], xlab = "time",
ylab = "conc", main = names(ODE)[i], type = "l")
points(DAE$time, DAE[,i], col = "red")
}
legend("bottomright",lty = c(1,NA),pch = c(NA,1),
col = c("black","red"),legend = c("ODE","DAE"))
# difference between both implementations:
max(abs(ODE-DAE))
par(mfrow = opa)
## =============================================================================
##
## Example 3: higher index DAE
##
## Car axis problem, index 3 DAE, 8 differential, 2 algebraic equations
## from
## F. Mazzia and C. Magherini. Test Set for Initial Value Problem Solvers,
## release 2.4. Department
## of Mathematics, University of Bari and INdAM, Research Unit of Bari,
## February 2008.
## Available at http://www.dm.uniba.it/~testset.
## =============================================================================
# car returns the residuals of the implicit DAE
car <- function(t, y, dy, pars){
with(as.list(c(pars, y)), {
f <- rep(0, 10)
yb <- r*sin(w*t)
xb <- sqrt(L*L - yb*yb)
Ll <- sqrt(xl^2 + yl^2)
Lr <- sqrt((xr-xb)^2 + (yr-yb)^2)
f[1:4] <- y[5:8]
k <- M*eps*eps/2
f[5] <- (L0-Ll)*xl/Ll + lam1*xb+2*lam2*(xl-xr)
f[6] <- (L0-Ll)*yl/Ll + lam1*yb+2*lam2*(yl-yr)-k*g
f[7] <- (L0-Lr)*(xr-xb)/Lr - 2*lam2*(xl-xr)
f[8] <- (L0-Lr)*(yr-yb)/Lr - 2*lam2*(yl-yr)-k*g
f[9] <- xb*xl+yb*yl
f[10] <- (xl-xr)^2+(yl-yr)^2-L*L
delt <- dy-f
delt[5:8] <- k*dy[5:8]-f[5:8]
delt[9:10] <- -f[9:10]
list(delt=delt,f=f)
})
}
# parameters
pars <- c(eps = 1e-2, M = 10, L = 1, L0 = 0.5,
r = 0.1, w = 10, g = 1)
# initial conditions: state variables
yini <- with (as.list(pars),
c(xl = 0, yl = L0, xr = L, yr = L0, xla = -L0/L,
yla = 0, xra = -L0/L, yra = 0, lam1 = 0, lam2 = 0)
)
# initial conditions: derivates
dyini <- rep(0, 10)
FF <- car(0, yini, dyini, pars)
dyini[1:4] <- yini[5:8]
dyini[5:8] <- 2/pars["M"]/(pars["eps"])^2*FF$f[5:8]
# check consistency of initial condition: delt should be = 0.
car(0, yini, dyini, pars)
# running the model
times <- seq(0, 3, by = 0.01)
nind <- c(4, 4, 2) # index 1, 2 and 3 variables
out <- mebdfi(y = yini, dy = dyini, times, res = car, parms = pars,
nind = nind, rtol = 1e-5, atol = 1e-5)
plot(out, which = 1:4, type = "l", lwd=2)
mtext(outer = TRUE, side = 3, line = -0.5, cex = 1.5, "car axis")
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