deSolve (version 1.5-1)

ode.3D: Solver for 3-Dimensional Ordinary Differential Equations

Description

Solves a system of ordinary differential equations resulting from 3-Dimensional partial differential equations that have been converted to ODEs by numerical differencing.

Usage

ode.3D(y, times, func, parms, nspec = NULL, dimens,
   ...)

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

3-valued vector with the number of boxes in three dimensions in the model.

...

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 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.

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.2Dfor integrating 2-D models
• lsodesfor the integration options.

diagnostics to print diagnostic messages.

Examples

## =======================================================================
## 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(nr=n,nc=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.)

  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)

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)
  }