vode: Solver for Ordinary Differential Equations (ODE)
In deSolve: Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')
vode R Documentation
Solver for Ordinary Differential Equations (ODE)
Description
Solves the initial value problem for stiff or nonstiff systems of
ordinary differential equations (ODE) in the form:
dy/dt = f(t,y)
The R function vode provides an interface to the FORTRAN ODE
solver of the same name, written by Peter N. Brown, Alan C. Hindmarsh
and George D. Byrne.
The system of ODE's is written as an R function or be defined in
compiled code that has been dynamically loaded.
In contrast to lsoda, the user has to specify whether or
not the problem is stiff and choose the appropriate solution method.
vode is very similar to lsode, but uses a
variable-coefficient method rather than the fixed-step-interpolate
methods in lsode. In addition, in vode it is possible
to choose whether or not a copy of the Jacobian is saved for reuse in
the corrector iteration algorithm; In lsode, a copy is not
kept.
Usage
vode(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
jacfunc = NULL, jactype = "fullint", mf = NULL, verbose = FALSE,
tcrit = NULL, hmin = 0, hmax = NULL, hini = 0, ynames = TRUE,
maxord = NULL, 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 ODE 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
either an R-function that computes the values of the
derivatives in the ODE system (the model definition) at time
t, or a character string giving the name of a compiled function in a
dynamically loaded shared library.
If func is an R-function, it 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.
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
must be specified in the same order as the state variables y.
If func is
a string, then dllname must give the name of the shared
library (without extension) which must be loaded before
vode() is called. See package vignette "compiledCode"
for more details.
parms
vector or list of parameters used in func or
jacfunc.
rtol
relative error tolerance, either a scalar or an array as
long as y. See details.
atol
absolute error tolerance, either a scalar or an array as
long as y. See details.
jacfunc
if not NULL, an R function that computes the
Jacobian of the system of differential equations
\partial\dot{y}_i/\partial y_j, or
a string giving the name of a function or subroutine in
‘dllname’ that computes the Jacobian (see vignette
"compiledCode" for more about this option).
In some circumstances, supplying
jacfunc can speed up the computations, if the system is
stiff. 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.
jactype
the structure of the Jacobian, one of
"fullint", "fullusr", "bandusr" or
"bandint" - either full or banded and estimated internally or
by user; overruled if mf is not NULL.
mf
the "method flag" passed to function vode - overrules
jactype - provides more options than jactype - see
details.
verbose
if TRUE: full output to the screen, e.g. will
print the diagnostiscs of the integration - see details.
tcrit
if not NULL, then vode cannot integrate
past tcrit. The FORTRAN routine dvode 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 multi-D models.
maxord
the maximum order to be allowed. NULL uses the default,
i.e. order 12 if implicit Adams method (meth = 1), order 5 if BDF
method (meth = 2). Reduce maxord to save storage space.
bandup
number of non-zero bands above the diagonal, in case
the Jacobian is banded.
banddown
number of non-zero bands below the diagonal, in case
the Jacobian is banded.
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 refered to in func and jacfunc.
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 func and jacfunc.
ipar
only when ‘dllname’ is specified: a vector with
integer values passed to the dll-functions whose names are specified
by func and jacfunc.
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 func, 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 func, 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.
forcings or package 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 and
jacfunc allowing this to be a generic function.
Details
Before using the integrator vode, the user has to decide
whether or not the problem is stiff.
If the problem is nonstiff, use method flag mf = 10, which
selects a nonstiff (Adams) method, no Jacobian used.
If the problem is stiff, there are four standard choices which can be
specified with jactype or mf.
The options for jactype are
- jac = "fullint":
a full Jacobian, calculated internally by
vode, corresponds to mf = 22,
- jac = "fullusr":
a full Jacobian, specified by user function
jacfunc, corresponds to mf = 21,
- jac = "bandusr":
a banded Jacobian, specified by user
function jacfunc; the size of the bands specified by
bandup and banddown, corresponds to mf = 24,
- jac = "bandint":
a banded Jacobian, calculated by vode; the
size of the bands specified by bandup and banddown,
corresponds to mf = 25.
More options are available when specifying mf directly.
The legal values of mf are 10, 11, 12, 13, 14, 15, 20, 21, 22,
23, 24, 25, -11, -12, -14, -15, -21, -22, -24, -25.
mf is a signed two-digit integer, mf = JSV*(10*METH +
MITER), where
- JSV = SIGN(mf)
indicates the Jacobian-saving strategy: JSV =
1 means a copy of the Jacobian is saved for reuse in the corrector
iteration algorithm. JSV = -1 means a copy of the Jacobian is not
saved.
- METH
indicates the basic linear multistep method: METH = 1
means the implicit Adams method. METH = 2 means the method based
on backward differentiation formulas (BDF-s).
- MITER
indicates the corrector iteration method: MITER = 0
means functional iteration (no Jacobian matrix is involved).
MITER = 1 means chord iteration with a user-supplied full (NEQ by
NEQ) Jacobian.
MITER = 2 means chord iteration with an internally generated
(difference quotient) full Jacobian (using NEQ extra calls to
func per df/dy value).
MITER = 3 means chord iteration with an internally generated
diagonal Jacobian approximation (using 1 extra call to func
per df/dy evaluation).
MITER = 4 means chord iteration with a user-supplied banded
Jacobian.
MITER = 5 means chord iteration with an internally generated
banded Jacobian (using ML+MU+1 extra calls to func per
df/dy evaluation).
If MITER = 1 or 4, the user must supply a subroutine jacfunc.
The example for integrator lsode demonstrates how to
specify both a banded and full Jacobian.
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, vode will return an
error code. See lsoda for details.
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,
plus and additional column for the time value. There will be a row
for each element in times unless the FORTRAN routine ‘vode’
returns with an unrecoverable error. If y has a names
attribute, it will be used to label the columns of the output value.
Note
From version 1.10.4, the default of atol was changed from 1e-8 to 1e-6,
to be consistent with the other solvers.
Author(s)
Karline Soetaert <karline.soetaert@nioz.nl>
References
P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, 1989. VODE: A Variable
Coefficient ODE Solver, SIAM J. Sci. Stat. Comput., 10, pp. 1038-1051.
Also, LLNL Report UCRL-98412, June 1988.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/0910062")}
G. D. Byrne and A. C. Hindmarsh, 1975. A Polyalgorithm for the
Numerical Solution of Ordinary Differential Equations. ACM
Trans. Math. Software, 1, pp. 71-96.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1145/355626.355636")}
A. C. Hindmarsh and G. D. Byrne, 1977. EPISODE: An Effective Package
for the Integration of Systems of Ordinary Differential
Equations. LLNL Report UCID-30112, Rev. 1.
G. D. Byrne and A. C. Hindmarsh, 1976. EPISODEB: An Experimental
Package for the Integration of Systems of Ordinary Differential
Equations with Banded Jacobians. LLNL Report UCID-30132, April 1976.
A. C. Hindmarsh, 1983. ODEPACK, a Systematized Collection of ODE
Solvers. in Scientific Computing, R. S. Stepleman et al., eds.,
North-Holland, Amsterdam, pp. 55-64.
K. R. Jackson and R. Sacks-Davis, 1980. An Alternative Implementation
of Variable Step-Size Multistep Formulas for Stiff ODEs. ACM
Trans. Math. Software, 6, pp. 295-318.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1145/355900.355903")}
Netlib: https://netlib.org
See Also
-
rk,
-
rk4 and euler for
Runge-Kutta integrators.
-
lsoda, lsode,
lsodes, lsodar,
daspk 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
## =======================================================================
## ex. 1
## The famous Lorenz equations: chaos in the earth's atmosphere
## Lorenz 1963. J. Atmos. Sci. 20, 130-141.
## =======================================================================
chaos <- function(t, state, parameters) {
with(as.list(c(state)), {
dx <- -8/3 * x + y * z
dy <- -10 * (y - z)
dz <- -x * y + 28 * y - z
list(c(dx, dy, dz))
})
}
state <- c(x = 1, y = 1, z = 1)
times <- seq(0, 100, 0.01)
out <- vode(state, times, chaos, 0)
plot(out, type = "l") # all versus time
plot(out[,"x"], out[,"y"], type = "l", main = "Lorenz butterfly",
xlab = "x", ylab = "y")
## =======================================================================
## ex. 2
## SCOC model, in FORTRAN - to see the FORTRAN code:
## browseURL(paste(system.file(package="deSolve"),
## "/doc/examples/dynload/scoc.f",sep=""))
## example from Soetaert and Herman, 2009, chapter 3. (simplified)
## =======================================================================
## Forcing function data
Flux <- matrix(ncol = 2, byrow = TRUE, data = c(
1, 0.654, 11, 0.167, 21, 0.060, 41, 0.070, 73, 0.277, 83, 0.186,
93, 0.140,103, 0.255, 113, 0.231,123, 0.309,133, 1.127,143, 1.923,
153,1.091,163, 1.001, 173, 1.691,183, 1.404,194, 1.226,204, 0.767,
214,0.893,224, 0.737, 234, 0.772,244, 0.726,254, 0.624,264, 0.439,
274,0.168,284, 0.280, 294, 0.202,304, 0.193,315, 0.286,325, 0.599,
335,1.889,345, 0.996, 355, 0.681,365, 1.135))
parms <- c(k = 0.01)
meanDepo <- mean(approx(Flux[,1], Flux[,2], xout = seq(1, 365, by = 1))$y)
Yini <- c(y = as.double(meanDepo/parms))
times <- 1:365
out <- vode(Yini, times, func = "scocder",
parms = parms, dllname = "deSolve",
initforc = "scocforc", forcings = Flux,
initfunc = "scocpar", nout = 2,
outnames = c("Mineralisation", "Depo"))
matplot(out[,1], out[,c("Depo", "Mineralisation")],
type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")
## Constant interpolation of forcing function - left side of interval
fcontrol <- list(method = "constant")
out2 <- vode(Yini, times, func = "scocder",
parms = parms, dllname = "deSolve",
initforc = "scocforc", forcings = Flux, fcontrol = fcontrol,
initfunc = "scocpar", nout = 2,
outnames = c("Mineralisation", "Depo"))
matplot(out2[,1], out2[,c("Depo", "Mineralisation")],
type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")
## Constant interpolation of forcing function - middle of interval
fcontrol <- list(method = "constant", f = 0.5)
out3 <- vode(Yini, times, func = "scocder",
parms = parms, dllname = "deSolve",
initforc = "scocforc", forcings = Flux, fcontrol = fcontrol,
initfunc = "scocpar", nout = 2,
outnames = c("Mineralisation", "Depo"))
matplot(out3[,1], out3[,c("Depo", "Mineralisation")],
type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")
plot(out, out2, out3)
deSolve documentation built on Nov. 28, 2023, 1:11 a.m.
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| vode | R Documentation |
Solver for Ordinary Differential Equations (ODE)
Description
Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form:
dy/dt = f(t,y)
The R function vode provides an interface to the FORTRAN ODE
solver of the same name, written by Peter N. Brown, Alan C. Hindmarsh
and George D. Byrne.
The system of ODE's is written as an R function or be defined in compiled code that has been dynamically loaded.
In contrast to lsoda, the user has to specify whether or
not the problem is stiff and choose the appropriate solution method.
vode is very similar to lsode, but uses a
variable-coefficient method rather than the fixed-step-interpolate
methods in lsode. In addition, in vode it is possible
to choose whether or not a copy of the Jacobian is saved for reuse in
the corrector iteration algorithm; In lsode, a copy is not
kept.
Usage
vode(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
jacfunc = NULL, jactype = "fullint", mf = NULL, verbose = FALSE,
tcrit = NULL, hmin = 0, hmax = NULL, hini = 0, ynames = TRUE,
maxord = NULL, 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 ODE system. If |
times |
time sequence for which output is wanted; the first
value of |
func |
either an R-function that computes the values of the
derivatives in the ODE system (the model definition) at time
If The return value of If |
parms |
vector or list of parameters used in |
rtol |
relative error tolerance, either a scalar or an array as
long as |
atol |
absolute error tolerance, either a scalar or an array as
long as |
jacfunc |
if not In some circumstances, supplying
If the Jacobian is a full matrix, If the Jacobian is banded, |
jactype |
the structure of the Jacobian, one of
|
mf |
the "method flag" passed to function vode - overrules
|
verbose |
if TRUE: full output to the screen, e.g. will
print the |
tcrit |
if not |
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
|
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. |
bandup |
number of non-zero bands above the diagonal, in case the Jacobian is banded. |
banddown |
number of non-zero bands below the diagonal, in case the Jacobian is banded. |
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 refered 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 |
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.
forcings or package 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
Before using the integrator vode, the user has to decide
whether or not the problem is stiff.
If the problem is nonstiff, use method flag mf = 10, which
selects a nonstiff (Adams) method, no Jacobian used.
If the problem is stiff, there are four standard choices which can be
specified with jactype or mf.
The options for jactype are
- jac = "fullint":
a full Jacobian, calculated internally by vode, corresponds to
mf= 22,- jac = "fullusr":
a full Jacobian, specified by user function
jacfunc, corresponds tomf= 21,- jac = "bandusr":
a banded Jacobian, specified by user function
jacfunc; the size of the bands specified bybandupandbanddown, corresponds tomf= 24,- jac = "bandint":
a banded Jacobian, calculated by vode; the size of the bands specified by
bandupandbanddown, corresponds tomf= 25.
More options are available when specifying mf directly.
The legal values of mf are 10, 11, 12, 13, 14, 15, 20, 21, 22,
23, 24, 25, -11, -12, -14, -15, -21, -22, -24, -25.
mf is a signed two-digit integer, mf = JSV*(10*METH +
MITER), where
- JSV = SIGN(mf)
indicates the Jacobian-saving strategy: JSV = 1 means a copy of the Jacobian is saved for reuse in the corrector iteration algorithm. JSV = -1 means a copy of the Jacobian is not saved.
- METH
indicates the basic linear multistep method: METH = 1 means the implicit Adams method. METH = 2 means the method based on backward differentiation formulas (BDF-s).
- MITER
indicates the corrector iteration method: MITER = 0 means functional iteration (no Jacobian matrix is involved).
MITER = 1 means chord iteration with a user-supplied full (NEQ by NEQ) Jacobian.
MITER = 2 means chord iteration with an internally generated (difference quotient) full Jacobian (using NEQ extra calls to
funcper df/dy value).MITER = 3 means chord iteration with an internally generated diagonal Jacobian approximation (using 1 extra call to
funcper df/dy evaluation).MITER = 4 means chord iteration with a user-supplied banded Jacobian.
MITER = 5 means chord iteration with an internally generated banded Jacobian (using ML+MU+1 extra calls to
funcper df/dy evaluation).
If MITER = 1 or 4, the user must supply a subroutine jacfunc.
The example for integrator lsode demonstrates how to
specify both a banded and full Jacobian.
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, vode will return an
error code. See lsoda for details.
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,
plus and additional column for the time value. There will be a row
for each element in times unless the FORTRAN routine ‘vode’
returns with an unrecoverable error. If y has a names
attribute, it will be used to label the columns of the output value.
Note
From version 1.10.4, the default of atol was changed from 1e-8 to 1e-6,
to be consistent with the other solvers.
Author(s)
Karline Soetaert <karline.soetaert@nioz.nl>
References
P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, 1989. VODE: A Variable
Coefficient ODE Solver, SIAM J. Sci. Stat. Comput., 10, pp. 1038-1051.
Also, LLNL Report UCRL-98412, June 1988.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/0910062")}
G. D. Byrne and A. C. Hindmarsh, 1975. A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations. ACM Trans. Math. Software, 1, pp. 71-96. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1145/355626.355636")}
A. C. Hindmarsh and G. D. Byrne, 1977. EPISODE: An Effective Package for the Integration of Systems of Ordinary Differential Equations. LLNL Report UCID-30112, Rev. 1.
G. D. Byrne and A. C. Hindmarsh, 1976. EPISODEB: An Experimental Package for the Integration of Systems of Ordinary Differential Equations with Banded Jacobians. LLNL Report UCID-30132, April 1976.
A. C. Hindmarsh, 1983. ODEPACK, a Systematized Collection of ODE Solvers. in Scientific Computing, R. S. Stepleman et al., eds., North-Holland, Amsterdam, pp. 55-64.
K. R. Jackson and R. Sacks-Davis, 1980. An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs. ACM Trans. Math. Software, 6, pp. 295-318. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1145/355900.355903")}
Netlib: https://netlib.org
See Also
-
rk, -
rk4andeulerfor Runge-Kutta integrators. -
lsoda,lsode,lsodes,lsodar,daspkfor 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
## =======================================================================
## ex. 1
## The famous Lorenz equations: chaos in the earth's atmosphere
## Lorenz 1963. J. Atmos. Sci. 20, 130-141.
## =======================================================================
chaos <- function(t, state, parameters) {
with(as.list(c(state)), {
dx <- -8/3 * x + y * z
dy <- -10 * (y - z)
dz <- -x * y + 28 * y - z
list(c(dx, dy, dz))
})
}
state <- c(x = 1, y = 1, z = 1)
times <- seq(0, 100, 0.01)
out <- vode(state, times, chaos, 0)
plot(out, type = "l") # all versus time
plot(out[,"x"], out[,"y"], type = "l", main = "Lorenz butterfly",
xlab = "x", ylab = "y")
## =======================================================================
## ex. 2
## SCOC model, in FORTRAN - to see the FORTRAN code:
## browseURL(paste(system.file(package="deSolve"),
## "/doc/examples/dynload/scoc.f",sep=""))
## example from Soetaert and Herman, 2009, chapter 3. (simplified)
## =======================================================================
## Forcing function data
Flux <- matrix(ncol = 2, byrow = TRUE, data = c(
1, 0.654, 11, 0.167, 21, 0.060, 41, 0.070, 73, 0.277, 83, 0.186,
93, 0.140,103, 0.255, 113, 0.231,123, 0.309,133, 1.127,143, 1.923,
153,1.091,163, 1.001, 173, 1.691,183, 1.404,194, 1.226,204, 0.767,
214,0.893,224, 0.737, 234, 0.772,244, 0.726,254, 0.624,264, 0.439,
274,0.168,284, 0.280, 294, 0.202,304, 0.193,315, 0.286,325, 0.599,
335,1.889,345, 0.996, 355, 0.681,365, 1.135))
parms <- c(k = 0.01)
meanDepo <- mean(approx(Flux[,1], Flux[,2], xout = seq(1, 365, by = 1))$y)
Yini <- c(y = as.double(meanDepo/parms))
times <- 1:365
out <- vode(Yini, times, func = "scocder",
parms = parms, dllname = "deSolve",
initforc = "scocforc", forcings = Flux,
initfunc = "scocpar", nout = 2,
outnames = c("Mineralisation", "Depo"))
matplot(out[,1], out[,c("Depo", "Mineralisation")],
type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")
## Constant interpolation of forcing function - left side of interval
fcontrol <- list(method = "constant")
out2 <- vode(Yini, times, func = "scocder",
parms = parms, dllname = "deSolve",
initforc = "scocforc", forcings = Flux, fcontrol = fcontrol,
initfunc = "scocpar", nout = 2,
outnames = c("Mineralisation", "Depo"))
matplot(out2[,1], out2[,c("Depo", "Mineralisation")],
type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")
## Constant interpolation of forcing function - middle of interval
fcontrol <- list(method = "constant", f = 0.5)
out3 <- vode(Yini, times, func = "scocder",
parms = parms, dllname = "deSolve",
initforc = "scocforc", forcings = Flux, fcontrol = fcontrol,
initfunc = "scocpar", nout = 2,
outnames = c("Mineralisation", "Depo"))
matplot(out3[,1], out3[,c("Depo", "Mineralisation")],
type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")
plot(out, out2, out3)
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