Solver for 3-Dimensional Ordinary Differential Equations
Solves a system of ordinary differential equations resulting from 3-Dimensional partial differential equations that have been converted to ODEs by numerical differencing.
ode.3D(y, times, func, parms, nspec = NULL, dimens,
method = c("lsodes", "euler", "rk4", "ode23", "ode45", "adams", "iteration"),
names = NULL, cyclicBnd = NULL, ...)y |
the initial (state) values for the ODE system, a vector. 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 |
parms |
parameters passed to |
nspec |
the number of species (components) in the model. |
dimens |
3-valued vector with the number of boxes in three dimensions in the model. |
names |
the names of the components; used for plotting. |
cyclicBnd |
if not |
method |
the integrator. Use Method |
... |
additional arguments passed to |
This is the method of choice for 3-dimensional models, that are only subjected to transport between adjacent layers.
Based on the dimension of the problem, the method first calculates the
sparsity pattern of the Jacobian, under the assumption that transport
is only occurring between adjacent layers. Then lsodes is
called to solve the problem.
As lsodes is used to integrate, it will probably be necessary
to specify the length of the real work array, lrw.
Although a reasonable guess of lrw is made, it is likely that
this will be too low.
In this case, ode.2D will return with an
error message telling the size of the work array actually needed. In
the second try then, set lrw equal to this number.
For instance, if you get the error:
DLSODES- RWORK length is insufficient to proceed. Length needed is .ge. LENRW (=I1), exceeds LRW (=I2) In above message, I1 = 27627 I2 = 25932
set lrw equal to 27627 or a higher value.
See lsodes for the additional options.
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 second element of the return from func, plus an
additional column (the first) for the time value. There will be one
row for each element in times unless the integrator returns
with an unrecoverable error. If y has a names attribute, it
will be used to label the columns of the output value.
The output will have the attributes istate, and rstate,
two vectors with several useful elements. The first element of istate
returns the conditions under which the last call to the integrator
returned. Normal is istate = 2. If verbose = TRUE, the
settings of istate and rstate will be written to the screen. See the
help for the selected integrator for details.
It is advisable though not mandatory to specify both
nspec and dimens. In this case, the solver can check
whether the input makes sense (as nspec*dimens[1]*dimens[2]*dimens[3]
== length(y)).
Do not use this method for problems that are not 3D!
Karline Soetaert <karline.soetaert@nioz.nl>
diagnostics to print diagnostic messages.
## =======================================================================
## Diffusion in 3-D; imposed boundary conditions
## =======================================================================
diffusion3D <- function(t, Y, par) {
## function to bind two matrices to an array
mbind <- function (Mat1, Array, Mat2, along = 1) {
dimens <- dim(Array) + c(0, 0, 2)
if (along == 3)
array(dim = dimens, data = c(Mat1, Array, Mat2))
else if (along == 1)
aperm(array(dim = dimens,
data=c(Mat1, aperm(Array, c(3, 2, 1)), Mat2)), c(3, 2, 1))
else if (along == 2)
aperm(array(dim = dimens,
data = c(Mat1, aperm(Array, c(1, 3, 2)), Mat2)), c(1, 3, 2))
}
yy <- array(dim=c(n, n, n), data = Y) # vector to 3-D array
dY <- -r*yy # consumption
BND <- matrix(nrow = n, ncol = n, data = 1) # boundary concentration
## diffusion in x-direction
## new array including boundary concentrations in X-direction
BNDx <- mbind(BND, yy, BND, along = 1)
## diffusive Flux
Flux <- -Dx * (BNDx[2:(n+2),,] - BNDx[1:(n+1),,])/dx
## rate of change = - flux gradient
dY[] <- dY[] - (Flux[2:(n+1),,] - Flux[1:n,,])/dx
## diffusion in y-direction
BNDy <- mbind(BND, yy, BND, along = 2)
Flux <- -Dy * (BNDy[,2:(n+2),] - BNDy[,1:(n+1),])/dy
dY[] <- dY[] - (Flux[,2:(n+1),] - Flux[,1:n,])/dy
## diffusion in z-direction
BNDz <- mbind(BND, yy, BND, along = 3)
Flux <- -Dz * (BNDz[,,2:(n+2)] - BNDz[,,1:(n+1)])/dz
dY[] <- dY[] - (Flux[,,2:(n+1)] - Flux[,,1:n])/dz
return(list(as.vector(dY)))
}
## parameters
dy <- dx <- dz <-1 # grid size
Dy <- Dx <- Dz <-1 # diffusion coeff, X- and Y-direction
r <- 0.025 # consumption rate
n <- 10
y <- array(dim=c(n,n,n),data=10.)
## use lsodes, the default (for n>20, Runge-Kutta more efficient)
print(system.time(
RES <- ode.3D(y, func = diffusion3D, parms = NULL, dimens = c(n, n, n),
times = 1:20, lrw = 120000, atol = 1e-10,
rtol = 1e-10, verbose = TRUE)
))
y <- array(dim = c(n, n, n), data = RES[nrow(RES), -1])
filled.contour(y[, , n/2], color.palette = terrain.colors)
summary(RES)
## Not run:
for (i in 2:nrow(RES)) {
y <- array(dim=c(n,n,n),data=RES[i,-1])
filled.contour(y[,,n/2],main=i,color.palette=terrain.colors)
}
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.