# vode: Solver for Ordinary Differential Equations (ODE) In deSolve: Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')

## Description

Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form:

dy/dt = f(t,y)

The R function vode provides an interface to the FORTRAN ODE solver of the same name, written by Peter N. Brown, Alan C. Hindmarsh and George D. Byrne.

The system of ODE's is written as an R function or be defined in compiled code that has been dynamically loaded.

In contrast to lsoda, the user has to specify whether or not the problem is stiff and choose the appropriate solution method.

vode is very similar to lsode, but uses a variable-coefficient method rather than the fixed-step-interpolate methods in lsode. In addition, in vode it is possible to choose whether or not a copy of the Jacobian is saved for reuse in the corrector iteration algorithm; In lsode, a copy is not kept.

## Usage

 1 2 3 4 5 6 7 vode(y, times, func, parms, rtol = 1e-6, atol = 1e-6, jacfunc = NULL, jactype = "fullint", mf = NULL, verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL, hini = 0, ynames = TRUE, maxord = NULL, bandup = NULL, banddown = NULL, maxsteps = 5000, dllname = NULL, initfunc = dllname, initpar = parms, rpar = NULL, ipar = NULL, nout = 0, outnames = NULL, forcings=NULL, initforc = NULL, fcontrol=NULL, events=NULL, lags = NULL,...)

## Details

Before using the integrator vode, the user has to decide whether or not the problem is stiff.

If the problem is nonstiff, use method flag mf = 10, which selects a nonstiff (Adams) method, no Jacobian used.

If the problem is stiff, there are four standard choices which can be specified with jactype or mf.

The options for jactype are

jac = "fullint":

a full Jacobian, calculated internally by vode, corresponds to mf = 22,

jac = "fullusr":

a full Jacobian, specified by user function jacfunc, corresponds to mf = 21,

jac = "bandusr":

a banded Jacobian, specified by user function jacfunc; the size of the bands specified by bandup and banddown, corresponds to mf = 24,

jac = "bandint":

a banded Jacobian, calculated by vode; the size of the bands specified by bandup and banddown, corresponds to mf = 25.

More options are available when specifying mf directly.

The legal values of mf are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, -11, -12, -14, -15, -21, -22, -24, -25.

mf is a signed two-digit integer, mf = JSV*(10*METH + MITER), where

JSV = SIGN(mf)

indicates the Jacobian-saving strategy: JSV = 1 means a copy of the Jacobian is saved for reuse in the corrector iteration algorithm. JSV = -1 means a copy of the Jacobian is not saved.

METH

indicates the basic linear multistep method: METH = 1 means the implicit Adams method. METH = 2 means the method based on backward differentiation formulas (BDF-s).

MITER

indicates the corrector iteration method: MITER = 0 means functional iteration (no Jacobian matrix is involved).

MITER = 1 means chord iteration with a user-supplied full (NEQ by NEQ) Jacobian.

MITER = 2 means chord iteration with an internally generated (difference quotient) full Jacobian (using NEQ extra calls to func per df/dy value).

MITER = 3 means chord iteration with an internally generated diagonal Jacobian approximation (using 1 extra call to func per df/dy evaluation).

MITER = 4 means chord iteration with a user-supplied banded Jacobian.

MITER = 5 means chord iteration with an internally generated banded Jacobian (using ML+MU+1 extra calls to func per df/dy evaluation).

If MITER = 1 or 4, the user must supply a subroutine jacfunc.

The example for integrator lsode demonstrates how to specify both a banded and full Jacobian.

The input parameters rtol, and atol determine the error control performed by the solver. If the request for precision exceeds the capabilities of the machine, vode will return an error code. See lsoda for details.

The diagnostics of the integration can be printed to screen by calling diagnostics. If verbose = TRUE, the diagnostics will written to the screen at the end of the integration.

See vignette("deSolve") for an explanation of each element in the vectors containing the diagnostic properties and how to directly access them.

Models may be defined in compiled C or FORTRAN code, as well as in an R-function. See package vignette "compiledCode" for details.

Examples in both C and FORTRAN are in the ‘dynload’ subdirectory of the deSolve package directory.

## 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 next elements of the return from func, plus and additional column for the time value. There will be a row for each element in times unless the FORTRAN routine ‘vode’ returns with an unrecoverable error. If y has a names attribute, it will be used to label the columns of the output value.

## Note

From version 1.10.4, the default of atol was changed from 1e-8 to 1e-6, to be consistent with the other solvers.

## Author(s)

Karline Soetaert <[email protected]>

## References

P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, 1989. VODE: A Variable Coefficient ODE Solver, SIAM J. Sci. Stat. Comput., 10, pp. 1038-1051.
Also, LLNL Report UCRL-98412, June 1988.

G. D. Byrne and A. C. Hindmarsh, 1975. A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations. ACM Trans. Math. Software, 1, pp. 71-96.

A. C. Hindmarsh and G. D. Byrne, 1977. EPISODE: An Effective Package for the Integration of Systems of Ordinary Differential Equations. LLNL Report UCID-30112, Rev. 1.

G. D. Byrne and A. C. Hindmarsh, 1976. EPISODEB: An Experimental Package for the Integration of Systems of Ordinary Differential Equations with Banded Jacobians. LLNL Report UCID-30132, April 1976.

A. C. Hindmarsh, 1983. ODEPACK, a Systematized Collection of ODE Solvers. in Scientific Computing, R. S. Stepleman et al., eds., North-Holland, Amsterdam, pp. 55-64.

K. R. Jackson and R. Sacks-Davis, 1980. An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs. ACM Trans. Math. Software, 6, pp. 295-318.

Netlib: http://www.netlib.org

• rk,

• rk4 and euler for Runge-Kutta integrators.

• lsoda, lsode, lsodes, lsodar, daspk for other solvers of the Livermore family,

• ode for a general interface to most of the ODE solvers,

• ode.band for solving models with a banded Jacobian,

• ode.1D for integrating 1-D models,

• ode.2D for integrating 2-D models,

• ode.3D for integrating 3-D models,

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 ## ======================================================================= ## ex. 1 ## The famous Lorenz equations: chaos in the earth's atmosphere ## Lorenz 1963. J. Atmos. Sci. 20, 130-141. ## ======================================================================= chaos <- function(t, state, parameters) { with(as.list(c(state)), { dx <- -8/3 * x + y * z dy <- -10 * (y - z) dz <- -x * y + 28 * y - z list(c(dx, dy, dz)) }) } state <- c(x = 1, y = 1, z = 1) times <- seq(0, 100, 0.01) out <- vode(state, times, chaos, 0) plot(out, type = "l") # all versus time plot(out[,"x"], out[,"y"], type = "l", main = "Lorenz butterfly", xlab = "x", ylab = "y") ## ======================================================================= ## ex. 2 ## SCOC model, in FORTRAN - to see the FORTRAN code: ## browseURL(paste(system.file(package="deSolve"), ## "/doc/examples/dynload/scoc.f",sep="")) ## example from Soetaert and Herman, 2009, chapter 3. (simplified) ## ======================================================================= ## Forcing function data Flux <- matrix(ncol = 2, byrow = TRUE, data = c( 1, 0.654, 11, 0.167, 21, 0.060, 41, 0.070, 73, 0.277, 83, 0.186, 93, 0.140,103, 0.255, 113, 0.231,123, 0.309,133, 1.127,143, 1.923, 153,1.091,163, 1.001, 173, 1.691,183, 1.404,194, 1.226,204, 0.767, 214,0.893,224, 0.737, 234, 0.772,244, 0.726,254, 0.624,264, 0.439, 274,0.168,284, 0.280, 294, 0.202,304, 0.193,315, 0.286,325, 0.599, 335,1.889,345, 0.996, 355, 0.681,365, 1.135)) parms <- c(k = 0.01) meanDepo <- mean(approx(Flux[,1], Flux[,2], xout = seq(1, 365, by = 1))\$y) Yini <- c(y = as.double(meanDepo/parms)) times <- 1:365 out <- vode(Yini, times, func = "scocder", parms = parms, dllname = "deSolve", initforc = "scocforc", forcings = Flux, initfunc = "scocpar", nout = 2, outnames = c("Mineralisation", "Depo")) matplot(out[,1], out[,c("Depo", "Mineralisation")], type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo") ## Constant interpolation of forcing function - left side of interval fcontrol <- list(method = "constant") out2 <- vode(Yini, times, func = "scocder", parms = parms, dllname = "deSolve", initforc = "scocforc", forcings = Flux, fcontrol = fcontrol, initfunc = "scocpar", nout = 2, outnames = c("Mineralisation", "Depo")) matplot(out2[,1], out2[,c("Depo", "Mineralisation")], type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo") ## Constant interpolation of forcing function - middle of interval fcontrol <- list(method = "constant", f = 0.5) out3 <- vode(Yini, times, func = "scocder", parms = parms, dllname = "deSolve", initforc = "scocforc", forcings = Flux, fcontrol = fcontrol, initfunc = "scocpar", nout = 2, outnames = c("Mineralisation", "Depo")) matplot(out3[,1], out3[,c("Depo", "Mineralisation")], type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo") plot(out, out2, out3)

### Example output

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