R: Solver for 3-Dimensional Ordinary Differential Equations

ode.3D {deSolve} | R Documentation

## 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,
      method = c("lsodes", "euler", "rk4", "ode23", "ode45", "adams", "iteration"),
      names = NULL, 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 `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`.
`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.
`names` | the names of the components; used for plotting.
`cyclicBnd` | if not `NULL` then a number or a 3-valued vector with the dimensions where a cyclic boundary is used - `1`: x-dimension, `2`: y-dimension; `3`: z-dimension.
`method` | the integrator. Use `"lsodes"` if the model is very stiff; "impAdams" may be best suited for mildly stiff problems; `"euler", "rk4", "ode23", "ode45", "adams"` are most efficient for non-stiff problems. Also allowed is to pass an integrator `function`. Use one of the other Runge-Kutta methods via `rkMethod`. For instance, `method = rkMethod("ode45ck")` will trigger the Cash-Karp method of order 4(5). Method `"iteration"` is special in that here the function `func` should return the new value of the state variables rather than the rate of change. This can be used for individual based models, for difference equations, or in those cases where the integration is performed within `func`)
`...` | additional arguments passed to `lsodes`.

### 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.

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.

### 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 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.

### Note

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!

### Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

### See Also

- `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.2D` for integrating 2-D models

- `lsodes` for 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(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)

[Package deSolve version 1.21 Index]