Solver For Multicomponent 1-D Ordinary Differential Equations

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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

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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

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

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