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 multi-component transport-reaction models that include transport only between adjacent layers.

Usage

ode.1D(y, times, func, parms, nspec=NULL, dimens=NULL, 
       method="lsode", ...)

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: yprime = func(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.
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 to use, one of "vode", "lsode", "lsoda", "lsodar", "lsodes"
... 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.

    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, use ode.band.

    See the selected integrator for the additional options

    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.

    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 <k.soetaert@nioo.knaw.nl>

    See Also

    ode,

  • ode.band for solving models with a banded Jacobian
  • ode.2D for integrating 2-D models
  • lsoda, lsode, lsodes, lsodar, vode, daspk.

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

    [Package deSolve version 1.2-2 Index]