# bimd: Blended Implicit Method for DAE In deTestSet: Test Set for Differential Equations

## Description

Solves the initial value problem for stiff or nonstiff systems of either:

• a system of ordinary differential equations (ODE) of the form

y' = f(t,y,...)

or

• a system of linearly implicit DAES in the form

M y' = f(t,y)

The R function `bimd` provides an interface to the Fortran DAE solver bimd, written by Cecilia Magherini and Luigi Bugnano.

It implements a Blended Implicit Methods of order 4-6-8-10-12 with step size control and continuous output.

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

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```bimd(y, times, func, parms, nind = c(length(y), 0, 0), rtol = 1e-6, atol = 1e-6, jacfunc = NULL, jactype = "fullint", mass = NULL, massup = NULL, massdown = NULL, verbose = FALSE, hmax = NULL, hini = 0, ynames = TRUE, minord = NULL, maxord = NULL, bandup = NULL, banddown = NULL, maxsteps = 1e4, maxnewtit = c(10, 12, 14, 16, 18), wrkpars = NULL, dllname = NULL, initfunc = dllname, initpar = parms, rpar = NULL, ipar = NULL, nout=0, outnames = NULL, forcings = NULL, initforc = NULL, fcontrol = NULL, ...) ```

## Arguments

 `y ` the initial (state) values for the DAE or ODE system. 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; if only one step is to be taken; set `times` = `NULL`. `func ` either an R-function that computes the values of the derivatives in the DAE or 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 should be specified in the same order as the state variables `y`. If `func` is a string, then `dllname` must give the name of the shared library (without extension) which must be loaded before `bimd()` is called. See deSolve package vignette `"compiledCode"` for more details. `parms ` vector or list of parameters used in `func` or `jacfunc`. `nind ` if a DAE system: a three-valued vector with the number of variables of index 1, 2, 3 respectively. The equations must be defined such that the index 1 variables precede the index 2 variables which in turn precede the index 3 variables. The sum of the variables of different index should equal N, the total number of variables. `rtol ` relative error tolerance, either a scalar or an array as long as `y`. See details. `atol ` absolute error tolerance, either a scalar or an array as long as `y`. See details. `jacfunc ` if not `NULL`, an R function that computes the Jacobian of the system of differential equations dydot(i)/dy(j), or a string giving the name of a function or subroutine in ‘dllname’ that computes the Jacobian (see vignette `"compiledCode"` from package deSolve, for more about this option). In some circumstances, supplying `jacfunc` can speed up the computations, if the system is stiff. The R calling sequence for `jacfunc` is identical to that of `func`. If the Jacobian is a full matrix, `jacfunc` should return a matrix dydot/dy, where the ith row contains the derivative of dy_i/dt with respect to y_j, or a vector containing the matrix elements by columns (the way R and FORTRAN store matrices). If the Jacobian is banded, `jacfunc` should return a matrix containing only the nonzero bands of the Jacobian, rotated row-wise. See first example. `jactype ` the structure of the Jacobian, one of `"fullint"`, `"fullusr"`, `"bandusr"` or `"bandint"` - either full or banded and estimated internally or by user. `mass ` the mass matrix. If not `NULL`, the problem is a linearly implicit DAE and defined as M dy/dt = f(t,y). If the mass-matrix M is full, it should be of dimension n*n where n is the number of y-values; if banded the number of rows should be less than n, and the mass-matrix is stored diagonal-wise with element (i, j) stored in `mass(i - j + mumas + 1, j)`. If `mass = NULL` then the model is an ODE (default) `massup ` number of non-zero bands above the diagonal of the `mass` matrix, in case it is banded. `massdown ` number of non-zero bands below the diagonal of the `mass` matrix, in case it is banded. `verbose ` if `TRUE`: full output to the screen, e.g. will print the `diagnostiscs` of the integration - see details. `hmax ` an optional maximum value of the integration stepsize. If not specified, `hmax` is set to the largest difference in `times`, to avoid that the simulation possibly ignores short-term events. If 0, no maximal size is specified. `hini ` initial step size to be attempted; if 0, the initial step size is set equal to 1e-6. Usually 1e-3 to 1e-5 is good for stiff equations `ynames ` logical, if `FALSE` names of state variables are not passed to function `func`; this may speed up the simulation especially for multi-D models. `minord ` the minimum order to be allowed, >= 3 and <= 9. `NULL` uses the default, 3. `maxord ` the maximum order to be allowed, >= `minord` and <= 9. `NULL` uses the default, 9. `bandup ` number of non-zero bands above the diagonal, in case the Jacobian is banded. `banddown ` number of non-zero bands below the diagonal, in case the Jacobian is banded. `maxsteps ` maximal number of steps taken by the solver, for the entire integration. This is different from the settings of this argument in the solvers from package deSolve! `maxnewtit ` A five-valued integer vector, with the maximal number of splitting-Newton iterations for the solution of the iplicit system in each step for order 4, 6, 8, 10 and 12 respectively. The default is c(10, 12, 14, 16, 18) `wrkpars ` A 12-valued real vector, with extra input parameters, put in the work vector work, at position work[3:14] in the fortran code - see details in fortran code. `NULL` uses the defaults `dllname ` a string giving the name of the shared library (without extension) that contains all the compiled function or subroutine definitions refered to in `func` and `jacfunc`. See vignette `"compiledCode"` from package `deSolve`. `initfunc ` if not `NULL`, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See vignette `"compiledCode"` from package `deSolve`. `initpar ` only when ‘dllname’ is specified and an initialisation function `initfunc` is in the dll: the parameters passed to the initialiser, to initialise the common blocks (FORTRAN) or global variables (C, C++). `rpar ` only when ‘dllname’ is specified: a vector with double precision values passed to the dll-functions whose names are specified by `func` and `jacfunc`. `ipar ` only when ‘dllname’ is specified: a vector with integer values passed to the dll-functions whose names are specified by `func` and `jacfunc`. `nout ` only used if `dllname` is specified and the model is defined in compiled code: the number of output variables calculated in the compiled function `func`, present in the shared library. Note: it is not automatically checked whether this is indeed the number of output variables calculed in the dll - you have to perform this check in the code - See vignette `"compiledCode"` from package `deSolve`. `outnames ` only used if ‘dllname’ is specified and `nout` > 0: the names of output variables calculated in the compiled function `func`, present in the shared library. These names will be used to label the output matrix. `forcings ` only used if ‘dllname’ is specified: a list with the forcing function data sets, each present as a two-columned matrix, with (time,value); interpolation outside the interval [min(`times`), max(`times`)] is done by taking the value at the closest data extreme. See forcings or package vignette `"compiledCode"`. `initforc ` if not `NULL`, the name of the forcing function initialisation function, as provided in ‘dllname’. It MUST be present if `forcings` has been given a value. See forcings or package vignette `"compiledCode"`. `fcontrol ` A list of control parameters for the forcing functions. See forcings or vignette `compiledCode`. `... ` additional arguments passed to `func` and `jacfunc` allowing this to be a generic function.

## Details

The work is done by the FORTRAN 77 subroutine `bimd`, whose documentation should be consulted for details.

There are four standard choices for the jacobian which can be specified with `jactype`.

The options for jactype are

jactype = "fullint"

a full Jacobian, calculated internally by the solver.

jactype = "fullusr"

a full Jacobian, specified by user function `jacfunc`.

jactype = "bandusr"

a banded Jacobian, specified by user function `jacfunc`; the size of the bands specified by `bandup` and `banddown`.

jactype = "bandint"

a banded Jacobian, calculated by bimd; the size of the bands specified by `bandup` and `banddown`.

Inspection of the example below shows how to specify both a banded and full Jacobian.

The input parameters `rtol`, and `atol` determine the error control performed by the solver, which roughly keeps the local error of y(i) below rtol(i)*abs(y(i))+atol(i).

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") from the `deSolve` package 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"` from package `deSolve` for details.

Information about linking forcing functions to compiled code is in forcings (from package `deSolve`).

## 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 ‘bimd’ returns with an unrecoverable error. If `y` has a names attribute, it will be used to label the columns of the output value.

## Author(s)

Karline Soetaert <[email protected]>

Francesca Mazzia

## References

L.BRUGNANO, C.MAGHERINI, F.MUGNAI. Blended Implicit Methods for the Numerical Solution of DAE problems. Jour. CAM 189 (2006) 34-50.

L.BRUGNANO, C.MAGHERINI The BiM code for the numerical solution of ODEs Jour. CAM 164-165 (2004) 145-158.

L.BRUGNANO, C.MAGHERINI Some Linear Algebra issues concerning the implementation of Blended Implicit Methods Numer. Linear Alg. Appl. 12 (2005) 305-314.

L.BRUGNANO, C.MAGHERINI Economical Error Estimates for Block Implicit Methods for ODEs via Deferred Correction. Appl. Numer. Math. 56 (2006) 608-617.

L.BRUGNANO, C.MAGHERINI Blended Implementation of Block Implicit Methods for ODEs Appl. Numer. Math. 42 (2002) 29-45.

• `gamd` another DAE solver from package `deTestSet`,

• `mebdfi` another DAE solver from package `deTestSet`,

• `daspk` another DAE solver from package `deSolve`,

• `ode` for a general interface to most of the ODE solvers from package `deSolve`,

• `ode.1D` for integrating 1-D models,

• `ode.2D` for integrating 2-D models,

• `ode.3D` for integrating 3-D models,

• `mebdfi` for integrating DAE models,

• `dopri853` for the Dormand-Prince Runge-Kutta method of order 8(53)

`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``` ```## ======================================================================= ## Example 1: ## Various ways to solve the same model. ## ======================================================================= ## the model, 5 state variables f1 <- function (t, y, parms) { ydot <- vector(len = 5) ydot[1] <- 0.1*y[1] -0.2*y[2] ydot[2] <- -0.3*y[1] +0.1*y[2] -0.2*y[3] ydot[3] <- -0.3*y[2] +0.1*y[3] -0.2*y[4] ydot[4] <- -0.3*y[3] +0.1*y[4] -0.2*y[5] ydot[5] <- -0.3*y[4] +0.1*y[5] return(list(ydot)) } ## the Jacobian, written as a full matrix fulljac <- function (t, y, parms) { jac <- matrix(nrow = 5, ncol = 5, byrow = TRUE, data = c(0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1) ) return(jac) } ## the Jacobian, written in banded form bandjac <- function (t, y, parms) { jac <- matrix(nrow = 3, ncol = 5, byrow = TRUE, data = c( 0 , -0.2, -0.2, -0.2, -0.2, 0.1, 0.1, 0.1, 0.1, 0.1, -0.3, -0.3, -0.3, -0.3, 0) ) return(jac) } ## initial conditions and output times yini <- 1:5 times <- 1:20 ## default: stiff method, internally generated, full Jacobian out <- bimd(yini, times, f1, parms = 0, jactype = "fullint") plot(out) ## stiff method, user-generated full Jacobian out2 <- bimd(yini, times, f1, parms = 0, jactype = "fullusr", jacfunc = fulljac) ## stiff method, internally-generated banded Jacobian ## one nonzero band above (up) and below(down) the diagonal out3 <- bimd(yini, times, f1, parms = 0, jactype = "bandint", bandup = 1, banddown = 1) ## stiff method, user-generated banded Jacobian out4 <- bimd(yini, times, f1, parms = 0, jactype = "bandusr", jacfunc = bandjac, bandup = 1, banddown = 1) ## ======================================================================= ## Example 2: ## stiff problem from chemical kinetics ## ======================================================================= Chemistry <- function (t, y, p) { dy1 <- -.04*y[1] + 1.e4*y[2]*y[3] dy2 <- .04*y[1] - 1.e4*y[2]*y[3] - 3.e7*y[2]^2 dy3 <- 3.e7*y[2]^2 list(c(dy1,dy2,dy3)) } times <- 10^(seq(0, 10, by = 0.1)) yini <- c(y1 = 1.0, y2 = 0, y3 = 0) out <- bimd(func = Chemistry, times = times, y = yini, parms = NULL) plot(out, log = "x", type = "l", lwd = 2) ## ============================================================================= ## Example 3: DAE ## Car axis problem, index 3 DAE, 8 differential, 2 algebraic equations ## from ## F. Mazzia and C. Magherini. Test Set for Initial Value Problem Solvers, ## release 2.4. Department ## of Mathematics, University of Bari and INdAM, Research Unit of Bari, ## February 2008. ## Available at http://www.dm.uniba.it/~testset. ## ============================================================================= ## Problem is written as M*y = f(t,y,p). ## caraxisfun implements the right-hand side: caraxisfun <- function(t, y, parms) { with(as.list(y), { yb <- r * sin(w * t) xb <- sqrt(L * L - yb * yb) Ll <- sqrt(xl^2 + yl^2) Lr <- sqrt((xr - xb)^2 + (yr - yb)^2) dxl <- ul; dyl <- vl; dxr <- ur; dyr <- vr dul <- (L0-Ll) * xl/Ll + 2 * lam2 * (xl-xr) + lam1*xb dvl <- (L0-Ll) * yl/Ll + 2 * lam2 * (yl-yr) + lam1*yb - k * g dur <- (L0-Lr) * (xr-xb)/Lr - 2 * lam2 * (xl-xr) dvr <- (L0-Lr) * (yr-yb)/Lr - 2 * lam2 * (yl-yr) - k * g c1 <- xb * xl + yb * yl c2 <- (xl - xr)^2 + (yl - yr)^2 - L * L list(c(dxl, dyl, dxr, dyr, dul, dvl, dur, dvr, c1, c2)) }) } eps <- 0.01; M <- 10; k <- M * eps^2/2; L <- 1; L0 <- 0.5; r <- 0.1; w <- 10; g <- 1 yini <- c(xl = 0, yl = L0, xr = L, yr = L0, ul = -L0/L, vl = 0, ur = -L0/L, vr = 0, lam1 = 0, lam2 = 0) # the mass matrix Mass <- diag(nrow = 10, 1) Mass[5,5] <- Mass[6,6] <- Mass[7,7] <- Mass[8,8] <- M * eps * eps/2 Mass[9,9] <- Mass[10,10] <- 0 Mass # index of the variables: 4 of index 1, 4 of index 2, 2 of index 3 index <- c(4, 4, 2) times <- seq(0, 3, by = 0.01) out <- bimd(y = yini, mass = Mass, times = times, func = caraxisfun, parms = NULL, nind = index) plot(out, which = 1:4, type = "l", lwd = 2) ```

deTestSet documentation built on May 29, 2017, 2:56 p.m.