R: Solver For Multicomponent 1-D Ordinary Differential Equations

ode.1D {deSolve} | R Documentation

## Solver For Multicomponent 1-D Ordinary Differential Equations

### Description

Solves a system of ordinary differential equations resulting from 1-Dimensional partial differential equations that have been converted to ODEs by numerical differencing.

### Usage

    ode.1D(y, times, func, parms, nspec = NULL, dimens = NULL,
       method= c("lsoda", "lsode", "lsodes", "lsodar", "vode", "daspk",
       "euler", "rk4", "ode23", "ode45", "radau", "bdf", "adams", "impAdams",
       "iteration"),
       names = NULL, bandwidth = 1, restructure = FALSE, ...)

### 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`. If `func` is a character string then integrator `lsodes` will be used. See details.
`parms` | parameters passed to `func`.
`nspec` | the number of species (components) in the model. If `NULL`, then `dimens` should be specified.
`dimens` | the number of boxes in the model. If `NULL`, then `nspec` should be specified.
`method` | the integrator. Use `"vode", "lsode", "lsoda", "lsodar", "daspk"`, or `"lsodes"` if the model is very stiff; `"impAdams"` or `"radau"` 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`)
`names` | the names of the components; used for plotting.
`bandwidth` | the number of adjacent boxes over which transport occurs. Normally equal to 1 (box i only interacts with box i-1, and i+1). Values larger than 1 will not work with `method = "lsodes"`. Ignored if the method is explicit.
`restructure` | whether or not the Jacobian should be restructured. Only used if the `method` is an integrator function. Should be `TRUE` if the method is implicit, `FALSE` if explicit.
`...` | additional arguments passed to the integrator.

### Details

This is the method of choice for multi-species 1-dimensional models, that are only subjected to transport between adjacent layers.

More specifically, this method is to be used if the state variables are arranged per species:

A[1], A[2], A[3],.... B[1], B[2], B[3],.... (for species A, B))

Two methods are implemented.

- The default method rearranges the state variables as A[1], B[1], ... A[2], B[2], ... A[3], B[3], .... This reformulation leads to a banded Jacobian with (upper and lower) half bandwidth = number of species.

  Then the selected integrator solves the banded problem.

- The second method uses `lsodes`. 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 may be necessary to specify the length of the real work array, `lrw`.

  Although a reasonable guess of `lrw` is made, it is possible that this will be too low. In this case, `ode.1D` 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

If the model is specified in compiled code (in a DLL), then option 2, based on `lsodes` is the only solution method.

For single-species 1-D models, you may also use `ode.band`.

See the selected integrator 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 (i.e. if `nspec * dimens == length(y)`).

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

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

- `lsodes`,`lsode`, `lsoda`, `lsodar`,`vode` for the integration options.

`diagnostics` to print diagnostic messages.

### Examples

    ## =======================================================================
    ## example 1
    ## a predator and its prey diffusing on a flat surface
    ## in concentric circles
    ## 1-D model with using cylindrical coordinates
    ## Lotka-Volterra type biology
    ## =======================================================================

    ## ================
    ## Model equations
    ## ================

    lvmod <- function (time, state, parms, N, rr, ri, dr, dri) {
      with (as.list(parms), {
        PREY <- state[1:N]
        PRED <- state[(N+1):(2*N)]

        ## Fluxes due to diffusion
        ## at internal and external boundaries: zero gradient
        FluxPrey <- -Da * diff(c(PREY[1], PREY, PREY[N]))/dri
        FluxPred <- -Da * diff(c(PRED[1], PRED, PRED[N]))/dri

        ## Biology: Lotka-Volterra model
        Ingestion     <- rIng  * PREY * PRED
        GrowthPrey    <- rGrow * PREY * (1-PREY/cap)
        MortPredator  <- rMort * PRED

        ## Rate of change = Flux gradient + Biology
        dPREY    <- -diff(ri * FluxPrey)/rr/dr   +
                    GrowthPrey - Ingestion
        dPRED    <- -diff(ri * FluxPred)/rr/dr   +
                    Ingestion * assEff - MortPredator

        return (list(c(dPREY, dPRED)))
      })
    }

    ## ==================
    ## Model application
    ## ==================

    ## model parameters:

    R  <- 20                        # total radius of surface, m
    N  <- 100                       # 100 concentric circles
    dr <- R/N                       # thickness of each layer
    r  <- seq(dr/2,by = dr,len = N) # distance of center to mid-layer
    ri <- seq(0,by = dr,len = N+1)  # distance to layer interface
    dri <- dr                       # dispersion distances

    parms <- c(Da     = 0.05,       # m2/d, dispersion coefficient
               rIng   = 0.2,        # /day, rate of ingestion
               rGrow  = 1.0,        # /day, growth rate of prey
               rMort  = 0.2 ,       # /day, mortality rate of pred
               assEff = 0.5,        # -, assimilation efficiency
               cap    = 10)         # density, carrying capacity

    ## Initial conditions: both present in central circle (box 1) only
    state    <- rep(0, 2 * N)
    state[1] <- state[N + 1] <- 10

    ## RUNNING the model:
    times  <- seq(0, 200, by = 1)   # output wanted at these time intervals

    ## the model is solved by the two implemented methods:
    ## 1. Default: banded reformulation
    print(system.time(
      out <- ode.1D(y = state, times = times, func = lvmod, parms = parms,
                    nspec = 2, names = c("PREY", "PRED"),
                    N = N, rr = r, ri = ri, dr = dr, dri = dri)
    ))

    ## 2. Using sparse method
    print(system.time(
      out2 <- ode.1D(y = state, times = times, func = lvmod, parms = parms,
                     nspec = 2, names = c("PREY","PRED"),
                     N = N, rr = r, ri = ri, dr = dr, dri = dri,
                     method = "lsodes")
    ))

    ## ================
    ## Plotting output
    ## ================
    # the data in 'out' consist of: 1st col times, 2-N+1: the prey
    # N+2:2*N+1: predators

    PREY   <- out[, 2:(N + 1)]

    filled.contour(x = times, y = r, PREY, color = topo.colors,
                   xlab = "time, days", ylab = "Distance, m",
                   main = "Prey density")
    # similar:
    image(out, which = "PREY", grid = r, xlab = "time, days",
          legend = TRUE, ylab = "Distance, m", main = "Prey density")

    image(out2, grid = r)

    # summaries of 1-D variables
    summary(out)

    # 1-D plots:
    matplot.1D(out, type = "l", subset = time == 10)
    matplot.1D(out, type = "l", subset = time > 10 & time < 20)

    ## =======================================================================
    ## Example 2.
    ## Biochemical Oxygen Demand (BOD) and oxygen (O2) dynamics
    ## in a river
    ## =======================================================================

    ## ================
    ## Model equations
    ## ================
    O2BOD <- function(t, state, pars) {
      BOD <- state[1:N]
      O2  <- state[(N+1):(2*N)]

      ## BOD dynamics
      FluxBOD <- v * c(BOD_0, BOD)   # fluxes due to water transport
      FluxO2  <- v * c(O2_0, O2)

      BODrate <- r * BOD             # 1-st order consumption

      ## rate of change = flux gradient  - consumption + reaeration (O2)
      dBOD         <- -diff(FluxBOD)/dx - BODrate
      dO2          <- -diff(FluxO2)/dx  - BODrate      +  p * (O2sat-O2)

      return(list(c(dBOD = dBOD, dO2 = dO2)))
    }

    ## ==================
    ## Model application
    ## ==================
    ## parameters
    dx      <- 25        # grid size of 25 meters
    v       <- 1e3       # velocity, m/day
    x       <- seq(dx/2, 5000, by = dx)  # m, distance from river
    N       <- length(x)
    r       <- 0.05      # /day, first-order decay of BOD
    p       <- 0.5       # /day, air-sea exchange rate
    O2sat   <- 300       # mmol/m3 saturated oxygen conc
    O2_0    <- 200       # mmol/m3 riverine oxygen conc
    BOD_0   <- 1000      # mmol/m3 riverine BOD concentration

    ## initial conditions:
    state <- c(rep(200, N), rep(200, N))
    times <- seq(0, 20, by = 0.1)

    ## running the model
    ##  step 1  : model spinup
    out <- ode.1D(y = state, times, O2BOD, parms = NULL,
                  nspec = 2, names = c("BOD", "O2"))

    ## ================
    ## Plotting output
    ## ================
    ## select oxygen (first column of out:time, then BOD, then O2
    O2   <- out[, (N + 2):(2 * N + 1)]
    color = topo.colors

    filled.contour(x = times, y = x, O2, color = color, nlevels = 50,
                   xlab = "time, days", ylab = "Distance from river, m",
                   main = "Oxygen")

    ## or quicker plotting:
    image(out, grid = x,  xlab = "time, days", ylab = "Distance from river, m")

[Package deSolve version 1.21 Index]