| ode.2D {deSolve} | R Documentation |
Solves a system of ordinary differential equations resulting from 2-Dimensional partial differential equations that have been converted to ODEs by numerical differencing.
ode.2D(y, times, func, parms, nspec = NULL, dimens, cyclicBnd = NULL, ...)
y |
the initial (state) values for the ODE system, a vector. 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.
|
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
should be specified in the same order as the state variables y.
|
parms |
parameters passed to func. |
nspec |
the number of species (components) in the model. |
dimens |
2-valued vector with the number of boxes in two dimensions in the model. |
cyclicBnd |
if not NULL then a number or a 2-valued vector
with the dimensions where a cyclic boundary is used - 1: x-dimension,
2: y-dimension; see details.
|
... |
additional arguments passed to lsodes. |
This is the method of choice for 2-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.
In some cases, a cyclic boundary condition exists. This is when the first
boxes in x-or y-direction interact with the last boxes. In this case, there
will be extra non-zero fringes in the Jacobian which need to be taken
into account. The occurrence of cyclic boundaries can be
toggled on by specifying argument cyclicBnd. For innstance,
cyclicBnd = 1 indicates that a cyclic boundary is required only for
the x-direction, whereas cyclicBnd = c(1,2) imposes a cyclic boundary
for both x- and y-direction. The default is no cyclic boundaries.
See lsodes for the additional options.
A matrix 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]
== length(y)).
Do not use this method for problems that are not 2D!
Karline Soetaert <k.soetaert@nioo.knaw.nl>
ode for a general interface to most of the ODE solvers,
ode.band for integrating models with a banded Jacobian
ode.1D for integrating 1-D models
ode.3D for integrating 3-D models
lsodes for the integration options.
diagnostics to print diagnostic messages.
## =======================================================================
## A Lotka-Volterra predator-prey model with predator and prey
## dispersing in 2 dimensions
## =======================================================================
## ==================
## Model definitions
## ==================
lvmod2D <- function (time, state, pars, N, Da, dx)
{
NN <- N*N
Prey <- matrix(nr = N,nc = N,state[1:NN])
Pred <- matrix(nr = N,nc = N,state[(NN+1):(2*NN)])
with (as.list(pars),
{
## Biology
dPrey <- rGrow* Prey *(1- Prey/K) - rIng* Prey *Pred
dPred <- rIng* Prey *Pred*assEff -rMort* Pred
zero <- rep(0,N)
## 1. Fluxes in x-direction; zero fluxes near boundaries
FluxPrey <- -Da * rbind(zero,(Prey[2:N,]-Prey[1:(N-1),]), zero)/dx
FluxPred <- -Da * rbind(zero,(Pred[2:N,]-Pred[1:(N-1),]), zero)/dx
## Add flux gradient to rate of change
dPrey <- dPrey - (FluxPrey[2:(N+1),]-FluxPrey[1:N,])/dx
dPred <- dPred - (FluxPred[2:(N+1),]-FluxPred[1:N,])/dx
## 2. Fluxes in y-direction; zero fluxes near boundaries
FluxPrey <- -Da * cbind(zero,(Prey[,2:N]-Prey[,1:(N-1)]), zero)/dx
FluxPred <- -Da * cbind(zero,(Pred[,2:N]-Pred[,1:(N-1)]), zero)/dx
## Add flux gradient to rate of change
dPrey <- dPrey - (FluxPrey[,2:(N+1)]-FluxPrey[,1:N])/dx
dPred <- dPred - (FluxPred[,2:(N+1)]-FluxPred[,1:N])/dx
return (list(c(as.vector(dPrey), as.vector(dPred))))
})
}
## ===================
## Model applications
## ===================
pars <- c(rIng = 0.2, # /day, rate of ingestion
rGrow = 1.0, # /day, growth rate of prey
rMort = 0.2 , # /day, mortality rate of predator
assEff = 0.5, # -, assimilation efficiency
K = 5 ) # mmol/m3, carrying capacity
R <- 20 # total length of surface, m
N <- 50 # number of boxes in one direction
dx <- R/N # thickness of each layer
Da <- 0.05 # m2/d, dispersion coefficient
NN <- N*N # total number of boxes
## initial conditions
yini <- rep(0, 2*N*N)
cc <- c((NN/2):(NN/2+1)+N/2, (NN/2):(NN/2+1)-N/2)
yini[cc] <- yini[NN+cc] <- 1
## solve model (5000 state variables...
times <- seq(0, 50, by = 1)
out <- ode.2D(y = yini, times = times, func = lvmod2D, parms = pars,
dimens = c(N, N), N = N, dx = dx, Da = Da, lrw = 5000000)
## plot results
Col <- colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan",
"#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000"))
## Not run:
for (i in seq(1, length(times), by = 1))
image(matrix(nr = N, nc = N, out[i, 2:(NN+1)]),
col = Col(100), xlab = "x", ylab = "y", zlim = range(out[,2:(NN+1)]))
## End(Not run)
## =======================================================================
## An example with a cyclic boundary condition.
## Diffusion in 2-D; extra flux on 2 boundaries,
## cyclic boundary in y
## =======================================================================
diffusion2D <- function(t,Y,par)
{
y <- matrix(nr=nx,nc=ny,data=Y) # vector to 2-D matrix
dY <- -r*y # consumption
BNDx <- rep(1,nx) # boundary concentration
BNDy <- rep(1,ny) # boundary concentration
#diffusion in X-direction; boundaries=imposed concentration
Flux <- -Dx * rbind(y[1,]-BNDy,(y[2:nx,]-y[1:(nx-1),]),BNDy-y[nx,])/dx
dY <- dY - (Flux[2:(nx+1),]-Flux[1:nx,])/dx
#diffusion in Y-direction
Flux <- -Dy * cbind(y[,1]-BNDx,(y[,2:ny]-y[,1:(ny-1)]),BNDx-y[,ny])/dy
dY <- dY - (Flux[,2:(ny+1)]-Flux[,1:ny])/dy
# extra flux on two sides
dY[,1] <- dY[,1]+ 10
dY[1,] <- dY[1,]+ 10
# and exchange between sides on y-direction
dY[,ny] <- dY[,ny]+ (y[,1]-y[,ny])*10
return(list(as.vector(dY)))
}
# parameters
dy <- dx <- 1 # grid size
Dy <- Dx <- 1 # diffusion coeff, X- and Y-direction
r <- 0.05 # consumption rate
nx <- 50
ny <- 100
y <- matrix(nr=nx,nc=ny,1.)
print(system.time(
ST3 <- ode.2D(y, times=1:100, func=diffusion2D, parms=NULL,
dimens=c(nx,ny), verbose=TRUE,
lrw=400000, atol=1e-10, rtol=1e-10, cyclicBnd=2)
))
## Not run:
zlim <- range(ST3[,-1])
for (i in 2:nrow(ST3)) {
y <- matrix(nr=nx,nc=ny,data=ST3[i,-1])
filled.contour(y,zlim=zlim,main=i)
}
## End(Not run)