# ode.1D: Solver For Multicomponent 1-D Ordinary Differential Equations In deSolve: Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')

## 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

 ```1 2 3 4 5``` ```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 <[email protected]>

• `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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167``` ```## ======================================================================= ## 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") ```

deSolve documentation built on May 10, 2018, 3 p.m.