Chapter 5: Differential Equations

IVOAM

Solves an initial-value problem for a system of ordinary differential equations of order one or two using a variable order Adams method.

Required Arguments

IDO — Flag indicating the state of the computation.   (Input/Output)

IDO     State

  1        Initial entry input value.

  2        Normal re-entry input value. On output, if IDO = 2 then the integration is finished. If the integrator is called with a new value for TEND, the integration continues. If the integrator is called with TEND unchanged, an error message is issued.

  3        Input value to use on final call to release workspace.

>3        Output value that indicates that a fatal error has occurred.

            The initial call is made with IDO = 1.  The routine then sets IDO = 2, and this value is used for all but the last call that is made with IDO = 3. This final call is only used to release workspace which was automatically allocated by the initial call with IDO = 1.

FCN — User-supplied subroutine to evaluate functions.
The usage is CALL FCN (IDO, T, Y, HIDRVS[, …]), where

Required Arguments

IDO — Flag indicating the state of the computation.   (Input)
This flag corresponds to the IDO argument described above. If FCN has complicated subexpressions, which depend only weakly or not at all on Y then these subexpressions need only be computed when IDO = 1 and their values then reused when IDO = 2.

T — Independent variable, t.   (Input)

Y — Array of length k containing the dependent variable values, y, and first derivatives, if any.  k will be the sum of the orders of the equations in the system of equations to solve. (Input)

HIDRVS — Array of length n = NEQ, where n is the number of equations in the system to solve, containing the values of the highest order derivatives evaluated at (t, y).   (Output)

       IVOAM uses size(HIDRVS) to set the default value of NEQ unless the optional argument NEQ is used.

Optional Arguments

FCN_DATA — A derived type, s_fcn_data, which may be used to pass additional integer or floating point information to or from the user-supplied subroutine.   (Input/Output)

                For a detailed description of this argument see FCN_DATA below.

     FCN must be declared EXTERNAL in the calling program.

T — Independent variable, t.  (Input/Output)
On input, T contains the initial independent variable value.  On output, T is replaced by TEND unless error conditions arise.  See IDO for details.   (Input/Output)

TEND — Value of t = tend where the solution is required.   (Input)

Y — Array of length k containing the dependent variables, y(t), and first derivatives, if any. (Input/Output)
k will be the sum of the orders of the equations in the system of equations to solve. On input, Y contains the initial values, y(t0) and y’(t0) (if needed).  On output, Y contains the approximate solution, y(t). For example, for a system of first order equations, Y(i) is the i-th dependent variable. For a system of second order equations, Y(2i-1) is the i-th dependent variable and Y(2i) is the derivative of the i-th dependent variable. For systems of equations in which one or more equations is of order 2, optional argument KORDER must be used to denote the order of each equation so that the derivatives in Y can be identified. By default it is assumed that all equations are of order 1 and Y contains only dependent variables.

HIDRVS — Array of length n = NEQ, where n is the number of equations in the system to solve, containing the highest order derivatives at the point Y.   (Output)
IVOAM uses size(HIDRVS) to set the default value of NEQ unless the optional argument NEQ is used.

Optional Arguments

NEQ — Number of differential equations in the system of equations to solve.   (Input) Default: NEQ = size (HIDRVS).

KORDER — An array of length NEQ specifying the orders of the equations in the system of equations to solve. The elements of  KORDER can be 1 or 2. KORDER must be used with argument Y to define systems of mixed or higher order. (Input)
Default: KORDER = (1,1,1,...,1).

EQNERR — An array of length NEQ specifying the error tolerance for each equation.   (Input)
Let e(i) be the error tolerance for equation i. Then

 

 

Value

Explanation

e(i) > 0

Implies an absolute error tolerance of e(i) is to be used for equation i.

e(i) = 0

implies no error checking is to be performed for equation i.

e(i) < 0

Implies a relative error test is to be performed for equation i. In this case, the base error  tolerance used will be |e(i)| and the relative  error factor used will be (15/16 * |e(i)|). Thus the actual absolute error tolerance used will be |e(i)| * ( (15/16 * |e(i)|).

 

            Default: An absolute error tolerance of 1.E-5 is used for single precision and 1.D-10 for double precision for all equations.

HINC — Factor used for increasing the stepsize.   (Input)
One should set HINC such that 9/8 <= HINC <= 4.
Default: HINC = 2.0.

HDEC — Factor used for decreasing the stepsize.   (Input)
One should set HDEC such that 1/4 <= HDEC <= 7/8.
Default: HDEC = 0.5.

HMIN — Absolute value of the minimum stepsize permitted.   (Input)
Default: HMIN = 10.0/amach(2) for single precision and 10.0/dmach(2) for double precision.

HMAX — Absolute value of the maximum stepsize permitted.   (Input)
Default: HMAX = amach(2) for single precision and dmach(2) for double precision.

FCN_DATA – A derived type, s_fcn_data, which may be used to pass additional information to/from the user-supplied subroutine.   (Input/Output)
The derived type, s_fcn_data, is defined as:

type s_fcn_data

   real(kind(1e0)), pointer, dimension(:) :: rdata

   integer, pointer, dimension(:) :: idata

end type

in module mp_types. The double precision counterpart to s_fcn_data is named d_fcn_data. The user must include a use mp_types statement in the calling program to define this derived type.

Note that if this optional argument is present then FCN_DATA must also be defined as an optional argument in the user-supplied subroutine.

Fortran 90 Interface

Generic:                              CALL IVOAM (IDO, FCN, T, TEND, Y, HIDRVS [,…])

Specific:            The specific interface names are S_IVOAM and D_IVOAM.

Description

Routine IVOAM is based on the JPL Library routine SIVA. IVOAM uses a variable order Adams method to solve the initial value problem

 

or more generally

where y is the vector

  is the  derivative of  with respect to ,  is the order of the  differential equation, and  is a vector with the same dimension as .

Note that the systems of equations solved by IVOAM can be of order one, order two, or mixed order one and two.

Comments

Informational errors

Type                      Code

3                             1  The requested error tolerance is too small.

3                             2  The stepsize has been reduced too rapidly. The integrator is going to do a restart.

Example 1

In this example a system of two equations of order two is solved.

The initial conditions are

Since the system is of order two, optional argument korder must be used to specify the orders of the equations. Also, because the system is of order two, Y(1) contains the first dependent variable, Y(2) contains the derivative of the first dependent variable, Y(3) contains the second dependent variable, and Y(4) contains the derivative of the second dependent variable.

 

      USE IVOAM_INT

      USE UMACH_INT

      USE CONST_INT

 

      IMPLICIT   NONE

      INTEGER    IDO, IEND, NOUT, KORDER(2)

      REAL       T, TEND, Y(4), HIDRVS(2), DELTA

 

      EXTERNAL   FCN

!                                 Initialize

      IDO  = 1

      T    = 0.0

      Y(1) = 1.0

      Y(2) = 0.0

      Y(3) = 0.0

      Y(4) = 1.0

      KORDER = 2

!                                 Write title

      CALL UMACH (2, NOUT)

      WRITE (NOUT,99997)

!                                 Integrate ODE

      IEND = 0

      DELTA = CONST('PI')

      DELTA = 2.0*DELTA

      DO

         IEND = IEND + 1

         TEND = T + DELTA

         IF(TEND .GT. 20.0) TEND = 20.0

         CALL IVOAM (IDO, FCN, T, TEND, Y, HIDRVS, KORDER=KORDER)

         IF (IEND .LE. 4) THEN

            WRITE (NOUT,99998) T, Y(1), Y(2), HIDRVS(1)

            WRITE (NOUT,99999) Y(3), Y(4), HIDRVS(2)

!                                 Finish up

            IF (IEND .EQ. 4) IDO = 3

            CYCLE

         END IF

         EXIT

      END DO

99997 FORMAT (11X, 'T', 12X, 'Y1/Y2', 9X, 'Y1P/Y2P', 7X, 'Y1PP/Y2PP')

99998 FORMAT (4F15.4)

99999 FORMAT (15X, 3F15.4)

      END

 

      SUBROUTINE FCN (IDO, T, Y, HIDRVS)

      INTEGER    IDO

      REAL       T, Y(*), HIDRVS(*)

      REAL       TP

 

      TP = Y(1)*Y(1) + Y(3)*Y(3)

      TP = 1.0E0/(TP*SQRT(TP))

      HIDRVS(1) = -Y(1)*TP

      HIDRVS(2) = -Y(3)*TP

      RETURN

      END

Output

 

    T            Y1/Y2        Y1P/Y2P      Y1PP/Y2PP
  6.2832         1.0000       -0.0000       -1.0000

                 0.0000        1.0000        0.0000

 12.5664         1.0000       -0.0000       -1.0000

                 0.0000        1.0000       -0.0000

 18.8496         1.0000       -0.0000       -1.0000

                 0.0000        1.0000       -0.0000

 20.0000         0.4081       -0.9129       -0.4081

                 0.9129        0.4081       -0.9129
 

Example 2

This contrived example illustrates how to use IVOAM to solve a system of equations of mixed order.

The height, y(t), of an object of mass m above the surface of the Earth can be modelled using Newton's second law as:

 

or

     (1)

where -mg is the downward force of gravity and -ky' is the force due to air resistance, in a direction opposing the velocity.  If the object is a meteor, the mass, m, and air resistance, k, will decrease as the meteor  burns up in the atmosphere.  The mass is proportional to r3 (r=radius) and the air resistance, presumably dependent on the surface area, may be assumed to be proportional to r2, so that k/m = k0/r. The rate at which the meteor's radius decreases as it burns up may depend on r, on the velocity y', and, since the density of the atmosphere depends on y, on y itself.  However, we will construct a very simple model where the rate is just proportional to the square of the velocity,

       (2)

We solve (1) and (2), with k0 = 0.005, c0 = 10-8, g = 9.8 and initial conditions y(0) = 100,000 meters, y'(0) = -1000 meters/second, r(0) = 1 meter.

 

      USE IVOAM_INT

      USE UMACH_INT

 

      IMPLICIT   NONE

      INTEGER    IDO, IEND, NOUT, KORDER(2)

      REAL       T, TEND, Y(3), HIDRVS(2), DELTA, EQNERR(2)

      EXTERNAL   FCN

!                                 Initialize

      IDO  = 1

      T    = 0.0

      Y(1) = 100000.0

      Y(2) = -1000.0

      Y(3) = 1.0

      KORDER(1) = 2

      KORDER(2) = 1

      EQNERR = .003

!                                 Write title

      CALL UMACH (2, NOUT)

      WRITE (NOUT,99997)

!                                 Integrate ODE

      IEND = 0

      DELTA = 10.0

      DO

         IEND = IEND + 1

         TEND = T + DELTA

         IF(TEND .GT. 50.0) TEND = 50.0

         CALL IVOAM (IDO, FCN, T, TEND, Y, HIDRVS, &

                     KORDER=KORDER, EQNERR=EQNERR)

         IF (IEND .LE. 5) THEN

            WRITE (NOUT,99998) T, Y(1), Y(2), HIDRVS(1)

            WRITE (NOUT,99999) Y(3), HIDRVS(2)

!                                 Finish up

            IF (IEND .EQ. 5) IDO = 3

            CYCLE

         END IF

         EXIT

      END DO

99997 FORMAT (11X, 'T', 10X, 'Y1/Y2', 11X, 'Y1P', 11X, 'Y1PP/Y2PP')

99998 FORMAT (4F15.4)

99999 FORMAT (2(15X, F15.4))

      END

 

      SUBROUTINE FCN (IDO, T, Y, HIDRVS)

      INTEGER    IDO

      REAL       T, Y(*), HIDRVS(*)

 

      HIDRVS(1) = -9.8 - .005/y(3)*y(2)

      HIDRVS(2) = -1.0E-8 * y(2)*y(2)

      RETURN

      END

Output

 

           T          Y1/Y2          Y1P          Y1PP/Y2PP

       10.0000     89773.0391    -1044.0096        -3.9701

                       0.8954                      -0.0109

       20.0000     79150.9844    -1078.6334        -2.9083

                       0.7826                      -0.0116

       30.0000     68240.9453    -1101.0380        -1.5031

                       0.6635                      -0.0121

       40.0000     57184.9062    -1106.9635         0.4253

                       0.5413                      -0.0121

       50.0000     46178.1367    -1089.8292         3.1700

                       0.4201                      -0.0119



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