# ode.2D: Solver for 2-Dimensional 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 2-Dimensional partial differential equations that have been converted to ODEs by numerical differencing.

## Usage

 ```1 2 3``` ```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:

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

• `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

 ``` 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185``` ```## ======================================================================= ## 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) ```

deSolve documentation built on July 17, 2017, 3 a.m.