R: Solver for 2-Dimensional Ordinary Differential Equations

ode.2D {deSolve} | R Documentation

## 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,
      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` | 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.
`names` | the names of the components; used for plotting.
`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). If `"lsodes"` is used, then also the size of the work array should be specified (`lrw`) (see lsodes). 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 2-dimensional models, that are only subjected to transport between adjacent layers.

Based on the dimension of the problem, and if `lsodes` is used as the integrator, 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.

If the model is not stiff, then it is more efficient to use one of the explicit integration routines

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.

If `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] == length(y)`).

Do not use this method for problems that are not 2D!

### 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.3D` for integrating 3-D models

- `lsodes` for 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(nrow = N, ncol = N,state[1:NN])
      Pred <- matrix(nrow = N, ncol = 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...  use Cash-Karp Runge-Kutta method
    times   <- seq(0, 50, by = 1)
    out <- ode.2D(y = yini, times = times, func = lvmod2D, parms = pars,
                  dimens = c(N, N), names = c("Prey", "Pred"),
                  N = N, dx = dx, Da = Da, method = rkMethod("rk45ck"))

    diagnostics(out)
    summary(out)

    # Mean of prey concentration at each time step
    Prey <- subset(out, select = "Prey", arr = TRUE)
    dim(Prey)
    MeanPrey <- apply(Prey, MARGIN = 3, FUN = mean)
    plot(times, MeanPrey)

    ## Not run:
    ## plot results
    Col <- colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan",
                              "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000"))

    for (i in seq(1, length(times), by = 1))
       image(Prey[ , ,i],
       col = Col(100), xlab = , zlim = range(out[,2:(NN+1)]))

    ## similar, plotting both and adding a margin text with times:
    image(out, xlab = "x", ylab = "y", mtext = paste("time = ", times))

    ## End(Not run)

    select <- c(1, 40)
    image(out, xlab = "x", ylab = "y", mtext = "Lotka-Volterra in 2-D",
          subset = select, mfrow = c(2,2), legend =  TRUE)

    # plot prey and pred at t = 10; first use subset to select data
    prey10 <-  matrix (nrow = N, ncol = N,
         data = subset(out, select = "Prey", subset = (time == 10)))
    pred10 <-  matrix (nrow = N, ncol = N,
         data = subset(out, select = "Pred", subset = (time == 10)))

    mf <- par(mfrow = c(1, 2))
    image(prey10)
    image(pred10)
    par (mfrow = mf)

    # same, using deSolve's image:
    image(out, subset = (time == 10))

    ## =======================================================================
    ## 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(nrow = nx, ncol = 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(nrow = nx, ncol = ny, 1)

    ## model most efficiently solved with lsodes - need to specify lrw

    print(system.time(
      ST3 <- ode.2D(y, times = 1:100, func = diffusion2D, parms = NULL,
                    dimens = c(nx, ny), verbose = TRUE, names = "Y",
                    lrw = 400000, atol = 1e-10, rtol = 1e-10, cyclicBnd = 2)
    ))

    # summary of 2-D variable
    summary(ST3)

    # plot output at t = 10
    t10 <-  matrix (nrow = nx, ncol = ny,
         data = subset(ST3, select = "Y", subset = (time == 10)))

    persp(t10, theta = 30, border = NA, phi = 70,
         col = "lightblue", shade = 0.5, box = FALSE)

    # image plot, using deSolve's image function
    image(ST3, subset = time == 10, method = "persp",
          theta = 30, border = NA, phi = 70, main = "",
          col = "lightblue", shade = 0.5, box = FALSE)

    ## Not run:
    zlim <- range(ST3[, -1])
    for (i in 2:nrow(ST3)) {
      y <- matrix(nrow = nx, ncol = ny, data = ST3[i, -1])
      filled.contour(y, zlim = zlim, main = i)
    }

    # same
    image(ST3, method = "filled.contour")

    ## End(Not run)

[Package deSolve version 1.28 Index]