radau: Implicit Runge-Kutta RADAU IIA
In deSolve: Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')
radau R Documentation
Implicit Runge-Kutta RADAU IIA
Description
Solves the initial value problem for stiff or nonstiff systems of
ordinary differential equations (ODE) in the form:
dy/dt =
f(t,y)
or linearly implicit differential algebraic equations in the
form:
M dy/dt = f(t,y)
.
The R function radau provides an interface to the Fortran solver
RADAU5, written by Ernst Hairer and G. Wanner, which implements the 3-stage
RADAU IIA method.
It implements the implicit Runge-Kutta method of order 5 with step size
control and continuous output.
The system of ODEs or DAEs is written as an R function or can be defined in
compiled code that has been dynamically loaded.
Usage
radau(y, times, func, parms, nind = c(length(y), 0, 0),
rtol = 1e-6, atol = 1e-6, jacfunc = NULL, jactype = "fullint",
mass = NULL, massup = NULL, massdown = NULL, rootfunc = NULL,
verbose = FALSE, nroot = 0, hmax = NULL, hini = 0, ynames = TRUE,
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 the right-hand side of the equation
M dy/dt = f(t,y)
if a DAE. (if mass is supplied then the problem is assumed a DAE).
func can also be 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
radau() is called. See deSolve package vignette "compiledCode"
for more details.
parms
vector or list of parameters used in func or
jacfunc.
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.
This has implications on the scaling of the variables,
i.e. index 2 variables are scaled by 1/h,
index 3 variables are scaled by 1/h^2.
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" from package deSolve, 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 example.
jactype
the structure of the Jacobian, one of
"fullint", "fullusr", "bandusr" or
"bandint" - either full or banded and estimated internally or
by user.
mass
the mass matrix.
If not NULL, the problem is a linearly
implicit DAE and defined as M\, dy/dt = f(t,y).
If the mass-matrix M is full, it should be of dimension
n^2 where n is the number of y-values;
if banded the number of rows should be less than n,
and the mass-matrix is stored diagonal-wise with element (i, j)
stored in mass(i - j + mumas + 1, j).
If mass = NULL then the model is an ODE (default)
massup
number of non-zero bands above the diagonal of the mass
matrix, in case it is banded.
massdown
number of non-zero bands below the diagonal of the mass
matrix, in case it is banded.
rootfunc
if not NULL, an R function that computes the
function whose root has to be estimated or a string giving the name
of a function or subroutine in ‘dllname’ that computes the root
function. The R calling sequence for rootfunc is identical
to that of func. rootfunc should return a vector with
the function values whose root is sought.
verbose
if TRUE: full output to the screen, e.g. will
print the diagnostiscs of the integration - see details.
nroot
only used if ‘dllname’ is specified: the number of
constraint functions whose roots are desired during the integration;
if rootfunc is an R-function, the solver estimates the number
of roots.
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 set equal to 1e-6. Usually 1e-3 to 1e-5 is good for stiff equations
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.
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
average maximal number of steps per output interval
taken by the solver. This argument is defined such as to ensure
compatibility with the Livermore-solvers. RADAU only accepts the maximal
number of steps for the entire integration, and this is calculated
as length(times) * maxsteps.
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 vignette "compiledCode"
from package deSolve.
initfunc
if not NULL, the name of the initialisation function
(which initialises values of parameters), as provided in
‘dllname’. See vignette "compiledCode"
from package deSolve.
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 calculed in the DLL - you have
to perform this check in the code - See vignette "compiledCode"
from package deSolve.
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.
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 and
jacfunc allowing this to be a generic function.
Details
The work is done by the FORTRAN subroutine RADAU5, whose
documentation should be consulted for details. The implementation
is based on the Fortran 77 version from January 18, 2002.
There are four standard choices for the Jacobian which can be specified with
jactype.
The options for jactype are
- jactype = "fullint"
a full Jacobian, calculated internally by
the solver.
- jactype = "fullusr"
a full Jacobian, specified by user
function jacfunc.
- jactype = "bandusr"
a banded Jacobian, specified by user
function jacfunc; the size of the bands specified by
bandup and banddown.
- jactype = "bandint"
a banded Jacobian, calculated by radau;
the size of the bands specified by bandup and
banddown.
Inspection of the example below shows how to specify both a banded and
full Jacobian.
The input parameters rtol, and atol determine the
error control performed by the solver, which roughly keeps the
local error of y(i) below rtol(i)*abs(y(i))+atol(i).
The diagnostics of the integration can be printed to screen
by calling diagnostics. If verbose = TRUE,
the diagnostics will be written to the screen at the end of the integration.
See vignette("deSolve") from the deSolve package 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" from package
deSolve for details.
Information about linking forcing functions to compiled code is in
forcings (from package deSolve).
radau can find the root of at least one of a set of constraint functions
rootfunc of the independent and dependent variables. It then returns the
solution at the root if that occurs sooner than the specified stop
condition, and otherwise returns the solution according the specified
stop condition.
Caution: Because of numerical errors in the function
rootfun due to roundoff and integration error, radau may
return false roots, or return the same root at two or more
nearly equal values of time.
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
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
References
E. Hairer and G. Wanner, 1996. Solving Ordinary Differential Equations II.
Stiff and Differential-algebraic problems. Springer series in computational
mathematics 14, Springer-Verlag, second edition.
See Also
-
ode for a general interface to most of the ODE solvers ,
-
ode.1D for integrating 1-D models,
-
ode.2D for integrating 2-D models,
-
ode.3D for integrating 3-D models,
-
daspk for integrating DAE models up to index 1
diagnostics to print diagnostic messages.
Examples
## =======================================================================
## Example 1: ODE
## Various ways to solve the same model.
## =======================================================================
## the model, 5 state variables
f1 <- function (t, y, parms) {
ydot <- vector(len = 5)
ydot[1] <- 0.1*y[1] -0.2*y[2]
ydot[2] <- -0.3*y[1] +0.1*y[2] -0.2*y[3]
ydot[3] <- -0.3*y[2] +0.1*y[3] -0.2*y[4]
ydot[4] <- -0.3*y[3] +0.1*y[4] -0.2*y[5]
ydot[5] <- -0.3*y[4] +0.1*y[5]
return(list(ydot))
}
## the Jacobian, written as a full matrix
fulljac <- function (t, y, parms) {
jac <- matrix(nrow = 5, ncol = 5, byrow = TRUE,
data = c(0.1, -0.2, 0 , 0 , 0 ,
-0.3, 0.1, -0.2, 0 , 0 ,
0 , -0.3, 0.1, -0.2, 0 ,
0 , 0 , -0.3, 0.1, -0.2,
0 , 0 , 0 , -0.3, 0.1))
return(jac)
}
## the Jacobian, written in banded form
bandjac <- function (t, y, parms) {
jac <- matrix(nrow = 3, ncol = 5, byrow = TRUE,
data = c( 0 , -0.2, -0.2, -0.2, -0.2,
0.1, 0.1, 0.1, 0.1, 0.1,
-0.3, -0.3, -0.3, -0.3, 0))
return(jac)
}
## initial conditions and output times
yini <- 1:5
times <- 1:20
## default: stiff method, internally generated, full Jacobian
out <- radau(yini, times, f1, parms = 0)
plot(out)
## stiff method, user-generated full Jacobian
out2 <- radau(yini, times, f1, parms = 0, jactype = "fullusr",
jacfunc = fulljac)
## stiff method, internally-generated banded Jacobian
## one nonzero band above (up) and below(down) the diagonal
out3 <- radau(yini, times, f1, parms = 0, jactype = "bandint",
bandup = 1, banddown = 1)
## stiff method, user-generated banded Jacobian
out4 <- radau(yini, times, f1, parms = 0, jactype = "bandusr",
jacfunc = bandjac, bandup = 1, banddown = 1)
## =======================================================================
## Example 2: ODE
## stiff problem from chemical kinetics
## =======================================================================
Chemistry <- function (t, y, p) {
dy1 <- -.04*y[1] + 1.e4*y[2]*y[3]
dy2 <- .04*y[1] - 1.e4*y[2]*y[3] - 3.e7*y[2]^2
dy3 <- 3.e7*y[2]^2
list(c(dy1, dy2, dy3))
}
times <- 10^(seq(0, 10, by = 0.1))
yini <- c(y1 = 1.0, y2 = 0, y3 = 0)
out <- radau(func = Chemistry, times = times, y = yini, parms = NULL)
plot(out, log = "x", type = "l", lwd = 2)
## =============================================================================
## Example 3: 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 from https://archimede.uniba.it/~testset/
## =============================================================================
## Problem is written as M*y' = f(t,y,p).
## caraxisfun implements the right-hand side:
caraxisfun <- function(t, y, parms) {
with(as.list(y), {
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)
dxl <- ul; dyl <- vl; dxr <- ur; dyr <- vr
dul <- (L0-Ll) * xl/Ll + 2 * lam2 * (xl-xr) + lam1*xb
dvl <- (L0-Ll) * yl/Ll + 2 * lam2 * (yl-yr) + lam1*yb - k * g
dur <- (L0-Lr) * (xr-xb)/Lr - 2 * lam2 * (xl-xr)
dvr <- (L0-Lr) * (yr-yb)/Lr - 2 * lam2 * (yl-yr) - k * g
c1 <- xb * xl + yb * yl
c2 <- (xl - xr)^2 + (yl - yr)^2 - L * L
list(c(dxl, dyl, dxr, dyr, dul, dvl, dur, dvr, c1, c2))
})
}
eps <- 0.01; M <- 10; k <- M * eps^2/2;
L <- 1; L0 <- 0.5; r <- 0.1; w <- 10; g <- 1
yini <- c(xl = 0, yl = L0, xr = L, yr = L0,
ul = -L0/L, vl = 0,
ur = -L0/L, vr = 0,
lam1 = 0, lam2 = 0)
# the mass matrix
Mass <- diag(nrow = 10, 1)
Mass[5,5] <- Mass[6,6] <- Mass[7,7] <- Mass[8,8] <- M * eps * eps/2
Mass[9,9] <- Mass[10,10] <- 0
Mass
# index of the variables: 4 of index 1, 4 of index 2, 2 of index 3
index <- c(4, 4, 2)
times <- seq(0, 3, by = 0.01)
out <- radau(y = yini, mass = Mass, times = times, func = caraxisfun,
parms = NULL, nind = index)
plot(out, which = 1:4, type = "l", lwd = 2)
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| radau | R Documentation |
Implicit Runge-Kutta RADAU IIA
Description
Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form:
dy/dt =
f(t,y)
or linearly implicit differential algebraic equations in the form:
M dy/dt = f(t,y)
.
The R function radau provides an interface to the Fortran solver
RADAU5, written by Ernst Hairer and G. Wanner, which implements the 3-stage
RADAU IIA method.
It implements the implicit Runge-Kutta method of order 5 with step size
control and continuous output.
The system of ODEs or DAEs is written as an R function or can be defined in
compiled code that has been dynamically loaded.
Usage
radau(y, times, func, parms, nind = c(length(y), 0, 0),
rtol = 1e-6, atol = 1e-6, jacfunc = NULL, jactype = "fullint",
mass = NULL, massup = NULL, massdown = NULL, rootfunc = NULL,
verbose = FALSE, nroot = 0, hmax = NULL, hini = 0, ynames = TRUE,
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 t, or the right-hand side of the equation
if a DAE. (if
If
The return value of If |
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. This has implications on the scaling of the variables, i.e. index 2 variables are scaled by 1/h, index 3 variables are scaled by 1/h^2. |
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,
|
jactype |
the structure of the Jacobian, one of
|
mass |
the mass matrix.
If not If |
massup |
number of non-zero bands above the diagonal of the |
massdown |
number of non-zero bands below the diagonal of the |
rootfunc |
if not |
verbose |
if |
nroot |
only used if ‘dllname’ is specified: the number of
constraint functions whose roots are desired during the integration;
if |
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 set equal to 1e-6. Usually 1e-3 to 1e-5 is good for stiff equations |
ynames |
logical, if |
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 |
average maximal number of steps per output interval
taken by the solver. This argument is defined such as to ensure
compatibility with the Livermore-solvers. RADAU only accepts the maximal
number of steps for the entire integration, and this is calculated
as |
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.
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 work is done by the FORTRAN subroutine RADAU5, whose
documentation should be consulted for details. The implementation
is based on the Fortran 77 version from January 18, 2002.
There are four standard choices for the Jacobian which can be specified with
jactype.
The options for jactype are
- jactype = "fullint"
a full Jacobian, calculated internally by the solver.
- jactype = "fullusr"
a full Jacobian, specified by user function
jacfunc.- jactype = "bandusr"
a banded Jacobian, specified by user function
jacfunc; the size of the bands specified bybandupandbanddown.- jactype = "bandint"
a banded Jacobian, calculated by radau; the size of the bands specified by
bandupandbanddown.
Inspection of the example below shows how to specify both a banded and full Jacobian.
The input parameters rtol, and atol determine the
error control performed by the solver, which roughly keeps the
local error of y(i) below rtol(i)*abs(y(i))+atol(i).
The diagnostics of the integration can be printed to screen
by calling diagnostics. If verbose = TRUE,
the diagnostics will be written to the screen at the end of the integration.
See vignette("deSolve") from the deSolve package 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" from package
deSolve for details.
Information about linking forcing functions to compiled code is in
forcings (from package deSolve).
radau can find the root of at least one of a set of constraint functions
rootfunc of the independent and dependent variables. It then returns the
solution at the root if that occurs sooner than the specified stop
condition, and otherwise returns the solution according the specified
stop condition.
Caution: Because of numerical errors in the function
rootfun due to roundoff and integration error, radau may
return false roots, or return the same root at two or more
nearly equal values of time.
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
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
References
E. Hairer and G. Wanner, 1996. Solving Ordinary Differential Equations II. Stiff and Differential-algebraic problems. Springer series in computational mathematics 14, Springer-Verlag, second edition.
See Also
-
odefor a general interface to most of the ODE solvers , -
ode.1Dfor integrating 1-D models, -
ode.2Dfor integrating 2-D models, -
ode.3Dfor integrating 3-D models, -
daspkfor integrating DAE models up to index 1
diagnostics to print diagnostic messages.
Examples
## =======================================================================
## Example 1: ODE
## Various ways to solve the same model.
## =======================================================================
## the model, 5 state variables
f1 <- function (t, y, parms) {
ydot <- vector(len = 5)
ydot[1] <- 0.1*y[1] -0.2*y[2]
ydot[2] <- -0.3*y[1] +0.1*y[2] -0.2*y[3]
ydot[3] <- -0.3*y[2] +0.1*y[3] -0.2*y[4]
ydot[4] <- -0.3*y[3] +0.1*y[4] -0.2*y[5]
ydot[5] <- -0.3*y[4] +0.1*y[5]
return(list(ydot))
}
## the Jacobian, written as a full matrix
fulljac <- function (t, y, parms) {
jac <- matrix(nrow = 5, ncol = 5, byrow = TRUE,
data = c(0.1, -0.2, 0 , 0 , 0 ,
-0.3, 0.1, -0.2, 0 , 0 ,
0 , -0.3, 0.1, -0.2, 0 ,
0 , 0 , -0.3, 0.1, -0.2,
0 , 0 , 0 , -0.3, 0.1))
return(jac)
}
## the Jacobian, written in banded form
bandjac <- function (t, y, parms) {
jac <- matrix(nrow = 3, ncol = 5, byrow = TRUE,
data = c( 0 , -0.2, -0.2, -0.2, -0.2,
0.1, 0.1, 0.1, 0.1, 0.1,
-0.3, -0.3, -0.3, -0.3, 0))
return(jac)
}
## initial conditions and output times
yini <- 1:5
times <- 1:20
## default: stiff method, internally generated, full Jacobian
out <- radau(yini, times, f1, parms = 0)
plot(out)
## stiff method, user-generated full Jacobian
out2 <- radau(yini, times, f1, parms = 0, jactype = "fullusr",
jacfunc = fulljac)
## stiff method, internally-generated banded Jacobian
## one nonzero band above (up) and below(down) the diagonal
out3 <- radau(yini, times, f1, parms = 0, jactype = "bandint",
bandup = 1, banddown = 1)
## stiff method, user-generated banded Jacobian
out4 <- radau(yini, times, f1, parms = 0, jactype = "bandusr",
jacfunc = bandjac, bandup = 1, banddown = 1)
## =======================================================================
## Example 2: ODE
## stiff problem from chemical kinetics
## =======================================================================
Chemistry <- function (t, y, p) {
dy1 <- -.04*y[1] + 1.e4*y[2]*y[3]
dy2 <- .04*y[1] - 1.e4*y[2]*y[3] - 3.e7*y[2]^2
dy3 <- 3.e7*y[2]^2
list(c(dy1, dy2, dy3))
}
times <- 10^(seq(0, 10, by = 0.1))
yini <- c(y1 = 1.0, y2 = 0, y3 = 0)
out <- radau(func = Chemistry, times = times, y = yini, parms = NULL)
plot(out, log = "x", type = "l", lwd = 2)
## =============================================================================
## Example 3: 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 from https://archimede.uniba.it/~testset/
## =============================================================================
## Problem is written as M*y' = f(t,y,p).
## caraxisfun implements the right-hand side:
caraxisfun <- function(t, y, parms) {
with(as.list(y), {
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)
dxl <- ul; dyl <- vl; dxr <- ur; dyr <- vr
dul <- (L0-Ll) * xl/Ll + 2 * lam2 * (xl-xr) + lam1*xb
dvl <- (L0-Ll) * yl/Ll + 2 * lam2 * (yl-yr) + lam1*yb - k * g
dur <- (L0-Lr) * (xr-xb)/Lr - 2 * lam2 * (xl-xr)
dvr <- (L0-Lr) * (yr-yb)/Lr - 2 * lam2 * (yl-yr) - k * g
c1 <- xb * xl + yb * yl
c2 <- (xl - xr)^2 + (yl - yr)^2 - L * L
list(c(dxl, dyl, dxr, dyr, dul, dvl, dur, dvr, c1, c2))
})
}
eps <- 0.01; M <- 10; k <- M * eps^2/2;
L <- 1; L0 <- 0.5; r <- 0.1; w <- 10; g <- 1
yini <- c(xl = 0, yl = L0, xr = L, yr = L0,
ul = -L0/L, vl = 0,
ur = -L0/L, vr = 0,
lam1 = 0, lam2 = 0)
# the mass matrix
Mass <- diag(nrow = 10, 1)
Mass[5,5] <- Mass[6,6] <- Mass[7,7] <- Mass[8,8] <- M * eps * eps/2
Mass[9,9] <- Mass[10,10] <- 0
Mass
# index of the variables: 4 of index 1, 4 of index 2, 2 of index 3
index <- c(4, 4, 2)
times <- seq(0, 3, by = 0.01)
out <- radau(y = yini, mass = Mass, times = times, func = caraxisfun,
parms = NULL, nind = index)
plot(out, which = 1:4, type = "l", lwd = 2)
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