D02NDF is a forward communication routine for integrating stiff systems of explicit ordinary differential equations when the Jacobian is a sparse matrix.
D02NDF is a general purpose routine for integrating the initial value problem for a stiff system of explicit ordinary differential equations,
It is designed specifically for the case where the Jacobian
is a sparse matrix.
Both interval and step oriented modes of operation are available and also modes designed to permit intermediate output within an interval oriented mode.
An outline of a typical calling program for D02NDF is given below. It calls the sparse matrix linear algebra setup routine D02NUF
, the Backward Differentiation Formula (BDF) integrator setup routine D02NVF
, its diagnostic counterpart D02NYF
, and the sparse linear algebra diagnostic routine D02NXF
EXTERNAL FCN, JAC, MONITR
IFAIL = 0
CALL D02NUF(NEQ, NEQMAX, JCEVAL, NWKJAC, IA, NIA, JA, NJA, &
JACPVT, NJCPVT, SENS, U, ETA, LBLOCK, ISPLIT, &
IFAIL = -1
CALL D02NDF(NEQ, NEQMAX, T, TOUT, Y, YDOT, RWORK, RTOL, &
ATOL, ITOL, INFORM, FCN, YSAVE, NY2DIM, JAC, &
WKJAC,NWKJAC, JACPVT, NJCPVT, MONITR, ITASK, &
IF(IFAIL.EQ.1 .OR. IFAIL.GE.14) STOP
IFAIL = 0
The linear algebra setup routine D02NUF
and one of the integrator setup routines, D02NVF
, must be called prior to the call of D02NDF. Either or both of the integrator diagnostic routine D02NYF
, or the sparse matrix linear algebra diagnostic routine D02NXF
, may be called after the call to D02NDF. There is also a routine, D02NZF
, designed to permit you to change step size on a continuation call to D02NDF without restarting the integration process.
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
The accuracy of the numerical solution may be controlled by a careful choice of the parameters RTOL
, and to a much lesser extent by the choice of norm. You are advised to use scalar error control unless the components of the solution are expected to be poorly scaled. For the type of decaying solution typical of many stiff problems, relative error control with a small absolute error threshold will be most appropriate (that is, you are advised to choose
small but positive).
Since numerical stability and memory are often conflicting requirements when solving ordinary differential systems where the Jacobian matrix is sparse, we provide a diagnostic routine, D02NXF
, whose aim is to inform you how much memory is required to solve the problem and to give you some indication of numerical stability.
In general, you are advised to choose the Backward Differentiation Formula option (setup routine D02NVF
) but if efficiency is of great importance and especially if it is suspected that
has complex eigenvalues near the imaginary axis for some part of the integration, you should try the BLEND option (setup routine D02NWF
This example solves the well-known stiff Robertson problem
over the range
with initial conditions
using scalar error control (
). The solution is computed up to
by overshooting and interpolating (
) and the intermediate solution computed on an equispaced mesh through MONITR
. The integration algorithm used is the BDF method (setup routine D02NVF
) and a modified Newton method is also used. The use of the 'N' (Numerical) and 'S' (Structural) options are illustrated in turn for calculating the Jacobian.