D02NGF is a general purpose routine for integrating the initial value problem for a stiff system of implicit ordinary differential equations coupled with algebraic equations, written in the form
It is designed specifically for the case where the resulting Jacobian is a full matrix (see the description of JAC).
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 D02NGF is given below. It calls the full matrix linear algebra setup routine D02NSF, the Backward Differentiation Formula (BDF) integrator setup routine D02NVF, and its diagnostic counterpart D02NYF.
The linear algebra setup routine D02NSF and one of the integrator setup routines, D02MVF, D02NVF or D02NWF, must be called prior to the call of D02NGF. The integrator diagnostic routine D02NYF may be called after the call to D02NGF. There is also a routine, D02NZF, designed to permit you to change step size on a continuation call to D02NGF without restarting the integration process.
On entry: the number of differential equations to be solved.
2: LDYSAV – INTEGERInput
On entry: a bound on the maximum number of equations to be solved during the integration.
3: T – REAL (KIND=nag_wp)Input/Output
On entry: , the value of the independent variable. The input value of T is used only on the first call as the initial point of the integration.
On exit: the value at which the computed solution is returned (usually at TOUT).
4: TOUT – REAL (KIND=nag_wp)Input/Output
On entry: the next value of at which a computed solution is desired. For the initial , the input value of TOUT is used to determine the direction of integration. Integration is permitted in either direction (see also ITASK).
On exit: normally unchanged. However, when , then TOUT contains the value of T at which initial values have been computed without performing any integration. See descriptions of ITASK and LDERIV.
On entry: a value to indicate the form of the local error test. ITOL indicates to D02NGF whether to interpret either or both of RTOL or ATOL as a vector or a scalar. The error test to be satisfied is , where is defined as follows:
On exit: must contain the th component of , for , where
and where the definition of is determined by the input value of IRES.
6: IRES – INTEGERInput/Output
On entry: the form of the residual that must be returned in array R.
The residual defined in equation (2) must be returned.
The residual defined in equation (1) must be returned.
On exit: should be unchanged unless one of the following actions is required of the integrator, in which case IRES should be set accordingly.
Indicates to the integrator that control should be passed back immediately to the calling (sub)program with the error indicator set to .
Indicates to the integrator that an error condition has occurred in the solution vector, its time derivative or in the value of . The integrator will use a smaller time step to try to avoid this condition. If this is not possible, the integrator returns to the calling (sub)program with the error indicator set to .
Indicates to the integrator to stop its current operation and to enter MONITR immediately with parameter .
RESID must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02NGF is called. Parameters denoted as Input must not be changed by this procedure.
13: YSAV(LDYSAV,SDYSAV) – REAL (KIND=nag_wp) arrayCommunication Array
14: SDYSAV – INTEGERInput
On entry: the second dimension of the array YSAV as declared in the (sub)program from which D02NGF is called. An appropriate value for SDYSAV is described in the specifications of the integrator setup routines D02MVF, D02NVF and D02NWF. This value must be the same as that supplied to the integrator setup routine.
15: JAC – SUBROUTINE, supplied by the NAG Library or the user.External Procedure
JAC must evaluate the Jacobian of the system. If this option is not required, the actual argument for JAC must be the dummy routine D02NGZ. (D02NGZ 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 full linear algebra setup routine D02NSF.
First we must define the system of nonlinear equations which is solved internally by the integrator. The time derivative, , generated internally, has the form
where is the current step size and is a parameter that depends on the integration method in use. The vector is the current solution and the vector depends on information from previous time steps. This means that . The system of nonlinear equations that is solved has the form
but it is solved in the form
where is the function defined by
It is the Jacobian matrix that you must supply in JAC as follows:
On entry: with , contains the weight used for the th equation when the norm is evaluated, and contains the estimated local error for the th equation. The scaled local error at the end of a timestep may be obtained by calling the real function D02ZAF as follows:
On entry: a flag indicating under what circumstances MONITR was called:
Entry from the integrator after (set in RESID) caused an early termination (this facility could be used to locate discontinuities).
The current step failed repeatedly.
Entry after a call to the internal nonlinear equation solver (see INLN).
The current step was successful.
On exit: may be reset to determine subsequent action in D02NGF.
Integration is to be halted. A return will be made from the integrator to the calling (sub)program with .
Allow the integrator to continue with its own internal strategy. The integrator will try up to three restarts unless on exit.
Return to the internal nonlinear equation solver, where the action taken is determined by the value of INLN (see INLN).
Normal exit to the integrator to continue integration.
Restart the integration at the current time point. The integrator will restart from order when this option is used. The solution Y, provided by MONITR, will be used for the initial conditions.
Try to continue with the same step size and order as was to be used before the call to MONITR. HMIN and HMAX may be altered if desired.
Continue the integration but using a new value of HNEXT and possibly new values of HMIN and HMAX.
12: INLN – INTEGEROutput
On exit: the action to be taken by the internal nonlinear equation solver when MONITR is exited with . By setting and returning to the integrator, the residual vector is evaluated and placed in the array R, and then MONITR is called again. At present this is the only option available: INLN must not be set to any other value.
13: HMIN – REAL (KIND=nag_wp)Input/Output
On entry: the minimum step size to be taken on the next step.
On exit: the minimum step size to be used. If this is different from the input value, then IMON must be set to or .
14: HMAX – REAL (KIND=nag_wp)Input/Output
On entry: the maximum step size to be taken on the next step.
On exit: the maximum step size to be used. If this is different from the input value, then IMON must be set to or . If HMAX is set to zero, no limit is assumed.
15: NQU – INTEGERInput
On entry: the order of the integrator used on the last step. This is supplied to enable you to carry out interpolation using either of the routines D02XJF or D02XKF.
MONITR must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02NGF is called. Parameters denoted as Input must not be changed by this procedure.
19: LDERIV() – LOGICAL arrayInput/Output
On entry: must be set to .TRUE. if you have supplied both an initial and an initial . must be set to .FALSE. if only the initial has been supplied.
must be set to .TRUE. if the integrator is to use a modified Newton method to evaluate the initial and . Note that and , if supplied, are used as initial estimates. This method involves taking a small step at the start of the integration, and if on entry, T and TOUT will be set to the result of taking this small step. must be set to .FALSE. if the integrator is to use functional iteration to evaluate the initial and , and if this fails a modified Newton method will then be attempted. is recommended if there are implicit equations or the initial and are zero.
On exit: is normally unchanged. However if and internal initialization was successful then .
, if implicit equations were detected. Otherwise .
20: ITASK – INTEGERInput
On entry: the task to be performed by the integrator.
Normal computation of output values of at (by overshooting and interpolating).
Take one step only and return.
Stop at the first internal integration point at or beyond and return.
Normal computation of output values of at but without overshooting . TCRIT must be specified as an option in one of the integrator setup routines before the first call to the integrator, or specified in the optional input routine before a continuation call. TCRIT may be equal to or beyond TOUT, but not before it, in the direction of integration.
Take one step only and return, without passing TCRIT. TCRIT must be specified as under .
The integrator will solve for the initial values of and only and then return to the calling (sub)program without doing the integration. This option can be used to check the initial values of and . Functional iteration or a ‘small’ backward Euler method used in conjunction with a damped Newton iteration is used to calculate these values (see LDERIV). Note that if a backward Euler step is used then the value of will have been advanced a short distance from the initial point.
Note: if D02NGF is recalled with a different value of ITASK (and TOUT altered), then the initialization procedure is repeated, possibly leading to different initial conditions.
21: ITRACE – INTEGERInput
On entry: the level of output that is printed by the integrator. ITRACE may take the value , , , or .
is assumed and similarly if , then is assumed.
No output is generated.
Only warning messages are printed on the current error message unit (see X04AAF).
Warning messages are printed as above, and on the current advisory message unit (see X04ABF) output is generated which details Jacobian entries, the nonlinear iteration and the time integration. The advisory messages are given in greater detail the larger the value of ITRACE.
22: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to , . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if on exit, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6 Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
An illegal input was detected on entry, or after an internal call to MONITR. If , then the form of the error will be detailed on the current error message unit (see X04AAF).
The maximum number of steps specified has been taken (see the description of optional inputs in the integrator setup routines and the optional input continuation routine, D02NZF).
With the given values of RTOL and ATOL no further progress can be made across the integration range from the current point T. The components contain the computed values of the solution at the current point T.
There were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as T. The problem may have a singularity, or the local error requirements may be inappropriate.
There were repeated convergence test failures on an attempted step, before completing the requested task, but the integration was successful as far as T. This may be caused by an inaccurate Jacobian matrix or one which is incorrectly computed.
Some error weight became zero during the integration (see the description of ITOL). Pure relative error control () was requested on a variable (the th) which has now vanished. The integration was successful as far as T.
RESID set its error flag () continually despite repeated attempts by the integrator to avoid this.
on entry but the internal initialization routine was unable to initialize (more detailed information may be directed to the current error message unit, see X04AAF).
A singular Jacobian has been encountered. You should check the problem formulation and Jacobian calculation.
An error occurred during Jacobian formulation or back-substitution (a more detailed error description may be directed to the current error message unit, see X04AAF).
RESID signalled the integrator to halt the integration and return (). Integration was successful as far as T.
MONITR set and so forced a return but the integration was successful as far as T.
The requested task has been completed, but it is estimated that a small change in RTOL and ATOL is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when or .)
The values of RTOL and ATOL are so small that D02NGF is unable to start the integration.
The linear algebra setup routine D02NSF was not called before the call to D02NGF.
The accuracy of the numerical solution may be controlled by a careful choice of the parameters RTOL and ATOL, 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 with 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 D02NGF the cost is proportional to , though for problems which are only mildly nonlinear the cost may be dominated by factors proportional to except for very large problems.
In general, you are advised to choose the BDF 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 written in implicit form
with initial conditions and over the range with vector error control (), the BDF method (setup routine D02NVF) and functional iteration. The Jacobian is calculated numerically if the functional iteration encounters difficulty and the integration is in one-step mode (), with interpolation to calculate the solution at intervals of using D02XJF externally. D02NBY is used for MONITR.