D02NEF : NAG Library, Mark 22

3 Description

D02NEF is a general purpose routine for integrating the initial value problem for a stiff system of implicit ordinary differential equations with coupled algebraic equations written in the form

F t,y,y′ = 0 .

An outline of a typical calling program for D02NEF is given below. It calls the DASSL implementation of the BDF integrator setup routine D02MWF and the banded matrix setup routine D02NPF (if required), and, if the integration needs to proceed, calls D02MCF before continuing the integration.

C     Declarations
C
      EXTERNAL RES, JAC
             .
             .
             .
C     Initialize the integrator
      CALL D02MWF(...)
C     Is the Jacobian matrix banded?
      IF (BANDED) CALL D02NPF(...)

C     Set DT to the required temporal resolution
C     Set TEND to the final time
C     Call the integrator for each temporal value:
1000  CALL D02NEF(...,RES,JAC,...)
C     Continue integration?
      IF (TOUT.LT.TEND .AND. ITASK.GE.0) THEN
        IF (ITASK.NE.1) TOUT = MIN(TOUT+DT,TEND)
C       Print solution
        CALL D02MCF(...)
        GO TO 1000
      ENDIF
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             .
             .

If this option is not required, the actual argument for JAC must be the dummy routine D02NEZ. (D02NEZ is included in the NAG Library.) You must indicate to the integrator whether this option is to be used by setting the parameter JCEVAL appropriately in a call to the setup routine D02MWF.

The specification of JAC is:

SUBROUTINE JAC (	NEQ, T, Y, YDOT, PD, CJ, IUSER, RUSER)
INTEGER	NEQ, IUSER(*)
*double precision*	T, Y(NEQ), YDOT(NEQ), PD(), CJ, RUSER()

1: NEQ – INTEGERInput: On entry: the number of differential-algebraic equations being solved.
2: T – double precisionInput: On entry: t, the current value of the independent variable.
3: Y(NEQ) – double precision arrayInput: On entry: yi, the current solution component, for i=1,2,…,NEQ.
4: YDOT(NEQ) – double precision arrayInput: On entry: the derivative of the solution at the current point t.
5: PD(*) – double precision arrayInput/Output: Note: the dimension of the array PD is NEQ×NEQ when the Jacobian is full and will be 2×ML+MU+1×NEQ when the Jacobian is banded (that is, a prior call to D02NPF has been made) (see D02NPF).
On entry: PD is preset to zero before the call to JAC.

On exit: if the Jacobian is full then PDi-j×NEQ+i=Jij, for i,j=1,2,…,NEQ; if the Jacobian is banded then PDj-1×NEQ+ML+MU+i-j=Jij, for max1,j-MU≤i≤minn,j+ML; (see also in F07BDF (DGBTRF)).
6: CJ – double precisionInput: On entry: CJ is a scalar constant which will be defined in D02NEF.
7: IUSER(*) – INTEGER arrayUser Workspace
8: RUSER(*) – double precision arrayUser Workspace: You are free to use the arrays IUSER and RUSER to supply information to JAC.

7 Accuracy

The accuracy of the numerical solution may be controlled by a careful choice of the parameters RTOL and ATOL. 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 ITOL=0 with ATOL1 small but positive).

8 Further Comments

The cost of computing a solution depends critically on the size of the differential system and to a lesser extent on the degree of stiffness of the problem. For banded systems the cost is proportional to NEQ× ML+ MU+1 2 , while for full systems the cost is proportional to NEQ3. Note however that for moderately sized problems which are only mildly nonlinear the cost may be dominated by factors proportional to NEQ×ML+MU+1 and NEQ2 respectively.

SUBROUTINE RES (	NEQ, T, Y, YDOT, R, IRES, IUSER, RUSER)
INTEGER	NEQ, IRES, IUSER(*)
*double precision*	T, Y(NEQ), YDOT(NEQ), R(NEQ), RUSER(*)

On entry,	TOUT=T,
or	TOUT is too close to T to start integration,
or	TOUT is behind T in the direction of H0 (see D02MWF),

On entry,	RTOLi<0.0 for i=1,2,…,n, where n=NEQ when ITOL=1 and n=1 otherwise,
or	RTOLi=ATOLi=0.0 for all relevant i.

On entry,	LCOM<40+maxorder+4×NEQ+NEQ×NEQ, and the Jacobian is full;
or	LCOM<40+maxorder+4×NEQ+NEQ×2×ML+MU+1+q, and the Jacobian is banded,

NAG Library Routine Document

D02NEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

SUBROUTINE D02NEF (	NEQ, T, TOUT, Y, YDOT, RTOL, ATOL, ITASK, RES, JAC, ICOM, COM, LCOM, IUSER, RUSER, IFAIL)
INTEGER	NEQ, ITASK, ICOM(50+NEQ), LCOM, IUSER(*), IFAIL
*double precision*	T, TOUT, Y(NEQ), YDOT(NEQ), RTOL(), ATOL(), COM(LCOM), RUSER(*)
EXTERNAL	RES, JAC

NAG Library Routine DocumentD02NEF

+− Contents

NAG Library Routine Document

D02NEF