deSolve (version 1.5-1)

ode.2D: Solver for 2-Dimensional Ordinary Differential Equations

Description

Solves a system of ordinary differential equations resulting from 2-Dimensional partial differential equations that have been converted to ODEs by numerical differencing.

Usage

ode.2D(y, times, func, parms, nspec = NULL, dimens,
  cyclicBnd = NULL, ...)

Arguments

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

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.

Value

• 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.

Details

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.

See Also

• odefor a general interface to most of the ODE solvers,
• ode.bandfor integrating models with a banded Jacobian
• ode.1Dfor integrating 1-D models
• ode.3Dfor integrating 3-D models
• lsodesfor the integration options.

diagnostics to print diagnostic messages.

Examples

## =======================================================================
## 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"))

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)]))

## =======================================================================
## 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)
 ))

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)
  }