## Interactive session for prepared example 6

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*******************************************************
****  Welcome to the PDE2D 9.6 Interactive Driver  ****
*******************************************************

PDE2D can solve systems of (linear or nonlinear) steady-state,
time-dependent and eigenvalue partial differential equations
in 1D intervals, general 2D regions, and in a wide range of simple
3D regions.  Ordinary differential equation systems can also be
solved.

problem.  The answers you give will be used to construct a PDE2D
FORTRAN program, which can then be compiled and linked with the PDE2D
runtime routines to produce an executable program.  The FORTRAN
driver program created will be well-documented and highly readable
(most of the interactive messages are repeated in the comments),
so that minor modifications or corrections can be made directly to
the FORTRAN program, without the need to work through a new
interactive session.

You can alternatively create your PDE2D FORTRAN program using the
PDE2D graphical user interface (GUI) ("pde2d_gui [progname]").  It
[RETURN]
is extraordinarily easy to set up problems using the PDE2D GUI,
which handles 0D and 1D problems, and 2D and 3D problems in "a wide
range of simple regions".  However, the PDE2D GUI cannot handle
complex regions, so if you have a complex 2D region you must use
this Interactive Driver.

If this is your first time to use PDE2D, you may want to work
through an example problem before trying one of your own.  Do you
want to work through a prepared example?
|---- Enter yes or no
yes
Several prepared examples are available.  Enter:

1 - to see a simple problem: a simply-supported elastic plate
equation, with a unit load concentrated at the midpoint of
a square.
2 - to see a more complex problem: a non-linear, steady-state
PDE, solved in an annulus.  Dirichlet (U = ...) boundary
conditions are imposed on part of the boundary, and Neumann
(dU/dn = ...) conditions are imposed on the other part, in
this example.  The initial triangulation is generated
automatically, and adaptive grid refinement is illustrated.
3 - to see an eigenvalue problem.  The region has a curved
interface across which material properties vary abruptly, in
this example.
4 - to see the first part of a thermal stress problem.  In this
part, the temperature distribution in a V-notched block is
calculated by solving the time-dependent heat conduction
equation, using adaptive time step control.
5 - to see the second part of a thermal stress problem.  In
this part, the stresses induced in the V-notched block by
thermal expansion are calculated, using the temperature
distribution output by example 4.  You must run example 4
[RETURN]
and save the tabular output before you can run example 5.
Examples 4 and 5 illustrate communication between problems.
6 - to see a 1D time-dependent integro-differential equation for
a financial math application.  In this problem there is a
term involving an integral of the solution, which requires
that we use PDE2D's feature for interpolating the solution
at the last saved time step, for use in the integral term.
7 - to see a waveguide problem (an eigenvalue problem in which
the eigenvalue appears nonlinearly).  This example shows how
to handle boundary conditions of different types on the same
arc, and how to produce a plot of a computed integral vs time.
8 - to see the Navier-Stokes equations (penalty formulation)
solved for a fluid flowing around a bend.
9 - to see a 3D elasticity problem, solved in a torus.  This
example illustrates the use of user-defined coordinate
transformations to handle more general 3D regions.
10 - to see a time-dependent wave equation (reduced to a system
of two PDEs), solved in a 3D box.
11 - to see a 3D eigenvalue problem (the Schrodinger equation
in a hydrogen atom).  This example illustrates the use of
spherical coordinates and periodic boundary conditions.
12 - to see a 3D eigenvalue problem, solved in a composite region
[RETURN]
consisting of two cylinders of different material properties.
13 - to see the axisymmetric Navier-Stokes equations solved in a
non-rectangular channel, using the collocation FEM.
14 - to see a 1D saturated/unsaturated water flow problem.
15 - to see a 1D version of the Schrodinger eigenvalue equation
of example 11.
0 - (no example)

If you select one of the examples, the correct answer for each
interactive question will be supplied after the question.
|---- Enter an integer value in the range 0 to 15
6
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ The example problem is a 1D time-dependent integro-differential      \$
\$ equation from a financial mathematics application:                   \$
\$                                                                      \$
\$    -Vt = 0.5*Sigma**2*x**2*Vxx + (R-Rlam*Rk)*x*Vx - (Rlam+R)*V       \$
\$            + Rlam*Integral(V(x*y,t)g(y)dy)                           \$
\$                                                                      \$
\$ where  g(y) = exp(-(y-1)**2/(2*var))/sqrt(2*pi*var)                  \$
\$ with initial conditions:                                             \$
\$       V(x,Tfinal) = max(x-E,0)                                       \$
\$ and boundary conditions:                                             \$
\$       V(0,t) = 0                                                     \$
\$       Vx(Smax,t) = 1                                                 \$
\$ We will solve this problem backward in time, to t=0.                 \$
\$                                                                      \$
\$ Parameter values used will be Sigma=0.15, R=0.05, Rlam=5, Smax=200,  \$
\$ Tfinal=1.0, Rk=0, E=100 and var=0.09.                                \$
\$                                                                      \$
\$ The solution V(x,t) represents the value of a European call option   \$
\$ at time t, if the current price of an asset is x.  The option holder \$
\$ has the right to buy, if desired, the asset at price E, at time      \$
\$ Tfinal.                                                              \$
[RETURN]
\$                                                                      \$
\$ To handle the integral term, we use the PDE2D function DOLDSOL1 to   \$
\$ interpolate the solution at the last saved time value, and a         \$
\$ midpoint rule to integrate this function.                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
[RETURN]
In what follows, when you are told to enter a 'FORTRAN expression',
this means any valid FORTRAN expression of 65 characters or less.
In this expression, you may include references to FORTRAN function
subprograms.  You may define these functions line by line at the end
of the interactive session, when prompted, or add them later using an
editor.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you enter a "#" in the first column of any input line, this       +
+ instructs the interactive driver to read this and subsequent input   +
+ lines from the file "pde2d.in".  A "#" in the first column of an     +
+ input line in the file "pde2d.in" (or an end-of-file) instructs the  +
+ driver to switch back to interactive input.                          +
+                                                                      +
+ All lines input during an interactive session are echo printed to    +
+ a file "echo.out".  You may want to modify this file and rename it   +
+ "pde2d.in", and read some or all of your input from this file during +
+ your next interactive session.                                       +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++

PDE2D can solve problems with 0,1,2 or 3 space variables.  Enter the
[RETURN]

0 - to solve a time-dependent ordinary differential equation system,
or an algebraic or algebraic eigenvalue system
1 - to solve problems in 1D intervals
2 - to solve problems in general 2D regions
3 - to solve problems in a wide range of simple 3D regions

\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: 1                                                             \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter an integer value in the range 0 to 3
1
Which finite element method do you want to use:

1. Galerkin method
2. Collocation method

Enter 1 or 2 to select a finite element method.
+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you have a problem with discontinuous material parameters, you    +
+ should use the Galerkin method.  If your partial differential        +
+ equations and boundary conditions are difficult to put into the      +
+ "divergence" form required by the Galerkin method, or if you have    +
+ periodic boundary conditions, use the collocation method.  The       +
+ collocation method produces approximate solutions with continuous    +
+ first derivatives; the Galerkin solutions are continuous but their   +
+ first derivatives are not.                                           +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: 2                                                             \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter an integer value in the range 1 to 2
2
Is double precision mode to be used?  Double precision is recommended.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If double precision mode is used, variables and functions assigned   +
+ names beginning with a letter in the range A-H or O-Z will be DOUBLE +
+ PRECISION, and you should use double precision constants and FORTRAN +
+ expressions throughout; otherwise such variables and functions will  +
+ be of type REAL.  In either case, variables and functions assigned   +
+ names beginning with I,J,K,L,M or N will be of INTEGER type.         +
+                                                                      +
+ It is possible to convert a single precision PDE2D program to double +
+ precision after it has been created, using an editor.  Just change   +
+ all occurrences of "real" to "double precision"                      +
+                    " tdp" to "dtdp"  (note leading blank)            +
+ Any user-written code or routines must be converted "by hand", of    +
+ course.  To convert from double to single, reverse the changes.      +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: yes                                                           \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
yes
If you don't want to read the FINE PRINT, default NPROB.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you want to solve several similar problems in the same run, set   +
+ NPROB equal to the number of problems you want to solve.  Then NPROB +
+ loops through the main program will be done, with IPROB=1,...,NPROB, +
+ and you can make the problem parameters vary with IPROB.  NPROB      +
+ defaults to 1.                                                       +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ press [RETURN] to default NPROB                                      \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
NPROB =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

PDE2D solves the time-dependent system (note: U,F,G,U0 may be vectors,
C,RHO may be matrices):

C(X,T,U,Ux)*d(U)/dT = F(X,T,U,Ux,Uxx)

F(X,U,Ux,Uxx) = 0

or the linear and homogeneous eigenvalue system:

F(X,U,Ux,Uxx) = lambda*RHO(X)*U

with boundary conditions:

G(X,[T],U,Ux) = 0
(periodic boundary conditions are also permitted)

at two X values.

For time-dependent problems there are also initial conditions:

[RETURN]
U = U0(X)   at T=T0

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If your PDEs involve the solution at points other than X, the        +
+ function                                                             +
+             (D)OLDSOL1(IDER,IEQ,XX,KDEG)                             +
+ will interpolate (using interpolation of degree KDEG=1,2 or 3) to XX +
+ the function saved in UOUT(*,IDER,IEQ,ISET) on the last time step or +
+ iteration (ISET) for which it has been saved.  Thus, for example, if +
+ IDER=1, this will return the latest value of component IEQ of the    +
+ solution at XX, assuming this has not been modified using UPRINT...  +
+ If your equations involve integrals of the solution, for example,    +
+ you can use (D)OLDSOL1 to approximate these using the solution from  +
+ the last time step or iteration.                                     +
+                                                                      +
+ CAUTION: For a steady-state or eigenvalue problem, you must reset    +
+ NOUT=1 if you want to save the solution each iteration.              +
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ A system of NEQN complex partial differential equations must be      +
+ written as a system of 2*NEQN real equations, by separating the      +
+ equations into their real and imaginary parts.  However, note that   +
+ the complex arithmetic abilities of FORTRAN can be used to simplify  +
[RETURN]
+ this separation.  For example, the complex PDE:                      +
+      I*Uxx - 1/(1+U**10) = 0,   where U = UR + UI*I                  +
+ would be difficult to split up analytically, but using FORTRAN       +
+ expressions it is easy:                                              +
+   F1 = -UIxx -  REAL(1.0/(1.0+CMPLX(UR,UI)**10))                     +
+   F2 =  URxx - AIMAG(1.0/(1.0+CMPLX(UR,UI)**10))                     +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
You may now define global parameters, which may be referenced in any
of the "FORTRAN expressions" you input throughout the rest of this
interactive session.  You will be prompted alternately for parameter
names and their values; enter a blank name when you are finished.

Parameter names are valid FORTRAN variable names, starting in
column 1.  Thus each name consists of 1 to 6 alphanumeric characters,
the first of which must be a letter.  If the first letter is in the
range I-N, the parameter must be an integer.

Parameter values are either FORTRAN constants or FORTRAN expressions
involving only constants and global parameters defined on earlier
lines.  They may also be functions of the problem number IPROB, if
you are solving several similar problems in one run (NPROB > 1).  Note
that you are defining global CONSTANTS, not functions; e.g., parameter
[RETURN]
values may not reference any of the independent or dependent variables

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you define other parameters here later, using an editor, you must +
+ add them to COMMON block /PARM8Z/ everywhere this block appears, if  +
+ they are to be "global" parameters.                                  +
+                                                                      +
+ The variable PI is already included as a global parameter, with an   +
+ accurate value 3.14159...                                            +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: Sigma                                                         \$
\$          0.15                                                        \$
\$        R                                                             \$
\$          0.05                                                        \$
\$        Rlam                                                          \$
\$          5                                                           \$
\$        Smax                                                          \$
\$          200                                                         \$
\$        Tfinal                                                        \$
\$          1.0                                                         \$
[RETURN]
\$        Rk                                                            \$
\$          0                                                           \$
\$        E                                                             \$
\$          100                                                         \$
\$      [blank line]                                                    \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
Parameter name = (type blank line to terminate)
Sigma
Sigma  =
|----Enter constant or FORTRAN expression-----------------------|
0.15
Parameter name = (type blank line to terminate)
R
R      =
|----Enter constant or FORTRAN expression-----------------------|
0.05
Parameter name = (type blank line to terminate)
Rlam
Rlam   =
|----Enter constant or FORTRAN expression-----------------------|
5
Parameter name = (type blank line to terminate)
Smax
Smax   =
|----Enter constant or FORTRAN expression-----------------------|
200
Parameter name = (type blank line to terminate)
Tfinal
Tfinal =
|----Enter constant or FORTRAN expression-----------------------|
1.0
Parameter name = (type blank line to terminate)
Rk
Rk     =
|----Enter constant or FORTRAN expression-----------------------|
0
Parameter name = (type blank line to terminate)
E
E      =
|----Enter constant or FORTRAN expression-----------------------|
100
Parameter name = (type blank line to terminate)

If you don't want to read the FINE PRINT, enter 'no'.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ Do you want to be given a chance to write a FORTRAN block before the +
+ definitions of many functions?  If you answer 'no', you will still   +
+ be given a chance to write code before the definition of the PDE     +
+ coefficients, but not other functions.  Of course, you can always    +
+ add code later directly to the resulting program, using an editor.   +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: no                                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
no
A collocation finite element method is used, with cubic Hermite
basis functions on the subintervals defined by the grid points:
XGRID(1),XGRID(2),...,XGRID(NXGRID)
You will first be prompted for NXGRID, the number of X-grid points,
then for XGRID(1),...,XGRID(NXGRID).  Any points defaulted will be
uniformly spaced between the points you define; the first and last
points cannot be defaulted.  The interval over which the PDE system
is to be solved is then:
XGRID(1) < X < XGRID(NXGRID)

\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: NXGRID = 51                                                   \$
\$        XGRID(1) = 0                                                  \$
\$        XGRID(26) = E                                                 \$
\$        XGRID(NXGRID) = Smax                                          \$
\$ and default the other XGRID points.                                  \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
NXGRID =
|---- Enter an integer value in the range 2 to +INFINITY
51
XGRID(1) =
|----Enter constant or FORTRAN expression-----------------------|
0
XGRID(2) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(3) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(4) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(5) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(6) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(7) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(8) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(9) =              (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(10) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(11) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(12) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(13) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(14) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(15) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(16) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(17) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(18) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(19) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(20) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(21) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(22) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(23) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(24) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(25) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(26) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|
E
XGRID(27) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(28) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(29) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(30) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(31) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(32) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(33) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(34) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(35) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(36) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(37) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(38) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(39) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(40) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(41) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(42) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(43) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(44) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(45) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(46) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(47) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(48) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(49) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(50) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XGRID(NXGRID) =
|----Enter constant or FORTRAN expression-----------------------|
Smax
What type of PDE problem do you want to solve?

2. a time-dependent problem
3. a linear, homogeneous eigenvalue problem

Enter 1,2 or 3 to select a problem type.
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: 2                                                             \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter an integer value in the range 1 to 3
2
Enter the initial time value (T0) and the final time value (TF), for
this time-dependent problem.  T0 defaults to 0.

TF is not required to be greater than T0.
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: T0 = Tfinal                                                   \$
\$        TF = 0.0                                                      \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
T0 =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|
Tfinal
TF =
|----Enter constant or FORTRAN expression-----------------------|
0.0
Is this a linear problem? ("linear" means all differential equations
and all boundary conditions are linear).  If you aren't sure, it is
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: yes                                                           \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
yes
Do you want the time step to be chosen adaptively?  If you answer
'yes', you will then be prompted to enter a value for TOLER(1), the
local relative time discretization error tolerance.  The default is
TOLER(1)=0.01.  If you answer 'no', a user-specified constant time step
will be used.  We suggest that you answer 'yes' and default TOLER(1)
(although for certain linear problems, a constant time step may be much
more efficient).

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If a negative value is specified for TOLER(1), then ABS(TOLER(1)) is +
+ taken to be the "absolute" error tolerance.  If a system of PDEs is  +
+ solved, by default the error tolerance specified in TOLER(1) applies +
+ to all variables, but the error tolerance for the J-th variable can  +
+ be set individually by specifying a value for TOLER(J) using an      +
+ editor, after the end of the interactive session.                    +
+                                                                      +
+ Each time step, two steps of size dt/2 are taken, and that solution  +
+ is compared with the result when one step of size dt is taken.  If   +
+ the maximum difference between the two answers is less than the      +
+ tolerance (for each variable), the time step dt is accepted (and the +
+ next step dt is doubled, if the agreement is "too" good); otherwise  +
+ dt is halved and the process is repeated.  Note that forcing the     +
[RETURN]
+ local (one-step) error to be less than the tolerance does not        +
+ guarantee that the global (cumulative) error is less than that value.+
+ However, as the tolerance is decreased, the global error should      +
+ decrease correspondingly.                                            +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: no                                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
no
If you don't want to read the FINE PRINT, it is safe (though possibly
very inefficient) to enter 'no'.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If your time-dependent problem is linear with all PDE and boundary   +
+ condition coefficients independent of time except inhomogeneous      +
+ terms, then a large savings in execution time may be possible if     +
+ this is recognized (the LU decomposition computed on the first step  +
+ can be used on subsequent steps).  Is this the case for your         +
+ problem?  (Caution: if you answer 'yes' when you should not, you     +
+ will get incorrect results with no warning.)                         +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: yes                                                           \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
yes
The time stepsize will be constant, DT = (TF-T0)/NSTEPS.  Enter a
value for NSTEPS, the number of time steps.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you later turn on adaptive step control, the time stepsize will be+
+ chosen adaptively, between an upper limit of DTMAX = (TF-T0)/NSTEPS  +
+ and a lower limit of 0.0001*DTMAX.                                   +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: NSTEPS = 1000                                                 \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
NSTEPS =
|----Enter constant or FORTRAN expression-----------------------|
1000
If you don't want to read the FINE PRINT, enter 'no'.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ Is the Crank-Nicolson scheme to be used to discretize time?  If you  +
+ answer 'no', a backward Euler scheme will be used.                   +
+                                                                      +
+ If a user-specified constant time step is chosen, the second order   +
+ Crank Nicolson method is recommended only for problems with very     +
+ well-behaved solutions, and the first order backward Euler scheme    +
+ should be used for more difficult problems.  In particular, do not   +
+ use the Crank Nicolson method if the left hand side of any PDE is    +
+ zero, for example, if a mixed elliptic/parabolic problem is solved.  +
+                                                                      +
+ If adaptive time step control is chosen, however, an extrapolation   +
+ is done between the 1-step and 2-step answers which makes the Euler  +
+ method second order, and the Crank-Nicolson method strongly stable.  +
+ Thus in this case, both methods have second order accuracy, and both +
+ are strongly stable.                                                 +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: no                                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
[RETURN]
|---- Enter yes or no
no
How many differential equations (NEQN) are there in your problem?
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: NEQN = 1                                                      \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
NEQN =
|---- Enter an integer value in the range 1 to 99
1
You may now choose names for the component(s) of the (possibly vector)
solution U.  Each must be an alphanumeric string of one to three
characters, beginning with a letter in the range A-H or O-Z.  The
variable names X and T must not be used.  The name should start in
column 1.
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: U1 = V                                                        \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
U1 =
V
You may calculate one or more integrals (over the entire region) of
some functions of the solution and its derivatives.  How many integrals
(NINT), if any, do you want to calculate?

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ In the FORTRAN program created by the preprocessor, the computed     +
+ values of the integrals will be returned in the vector SINT8Z.  If   +
+ several iterations or time steps are done, only the last computed    +
+ values are saved in SINT8Z (all values are printed).                 +
+                                                                      +
+ A limiting value, SLIM8Z(I), for the I-th integral can be set        +
+ below in the main program.  The computations will then stop          +
+ gracefully whenever SINT8Z(I) > SLIM8Z(I), for any I=1...NINT.       +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: NINT = 0                                                      \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
NINT =
|---- Enter an integer value in the range 0 to 20
0
You may calculate one or more boundary integrals (over the entire
boundary) of some functions of the solution and its derivatives.  How
many boundary integrals (NBINT), if any, do you want to calculate?

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ In the FORTRAN program created by the preprocessor, the computed     +
+ values of the integrals will be returned in the vector BINT8Z.  If   +
+ several iterations or time steps are done, only the last computed    +
+ values are saved in BINT8Z (all values are printed).                 +
+                                                                      +
+ A limiting value, BLIM8Z(I), for the I-th boundary integral can be   +
+ set below in the main program.  The computations will then stop      +
+ gracefully whenever BINT8Z(I) > BLIM8Z(I), for any I=1...NBINT.      +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: NBINT = 0                                                     \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
NBINT =
|---- Enter an integer value in the range 0 to 20
0
Now enter FORTRAN expressions to define the PDE coefficients, which
may be functions of

X,T,V,Vx,Vxx

Recall that the PDE has the form

C*d(V)/dT = F

\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ The differential equation is                                         \$
\$                                                                      \$
\$  -Vt = 0.5*Sigma**2*x**2*Vxx+(R-Rlam*Rk)*x*Vx-(Rlam+R)*V + Rlam*AINT \$
\$                                                                      \$
\$ where AINT is the integral from y=0 to y=2 (g(y) is almost 0 outside \$
\$ this range) of V(x*y,t)*g(y)dy.                                      \$
\$                                                                      \$
\$ The midpoint rule is used to approximate this integral, using 200    \$
\$ subintervals.  We use the function DOLDSOL1(1,1,X*Y,2) to return     \$
\$ V(X*Y,t_n) at the last saved time value t_n, using quadratic         \$
\$ interpolation.                                                       \$
\$                                                                      \$
[RETURN]
\$ Note that since y may be greater than 1, X*Y may be greater than     \$
\$ Smax, in which case we have to extrapolate a value for V beyond the  \$
\$ right boundary using the fact that dV/dx = 1 at x = Smax.            \$
\$                                                                      \$
\$ When asked if you want to write a FORTRAN block,                     \$
\$ enter: yes                                                           \$
\$   then, when prompted, enter:                                        \$
\$          VAR = 0.09                                                  \$
\$          YB = 2.0                                                    \$
\$          NY = 200                                                    \$
\$          HY = YB/NY                                                  \$
\$          AINT = 0.0                                                  \$
\$          DO 10 K=1,NY                                                \$
\$             Y = (K-0.5)*HY                                           \$
\$             GY = EXP(-(Y-1.0)**2/(2*VAR))/SQRT(2*PI*VAR)             \$
\$             IF (X*Y.LE.Smax) THEN                                    \$
\$                VXY = DOLDSOL1(1,1,X*Y,2)                             \$
\$             ELSE                                                     \$
\$                VXY = DOLDSOL1(1,1,Smax,2) + (X*Y-Smax)               \$
\$             ENDIF                                                    \$
\$             AINT = AINT + VXY*GY*HY                                  \$
\$       10 CONTINUE                                                    \$
[RETURN]
\$        [blank line]                                                  \$
\$   then, when prompted, enter:                                        \$
\$    C = -1                                                            \$
\$    F = 0.5*Sigma**2*x**2*Vxx+(R-Rlam*Rk)*x*Vx-(Rlam+R)*V + Rlam*AINT \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$

Do you want to write a FORTRAN block to define some parameters to be
used in the definition of these coefficients?
|---- Enter yes or no
yes
Remember to begin FORTRAN statements in column 7
|-----7-----Input FORTRAN now (type blank line to terminate)-----------|
VAR = 0.09
YB = 2.0
NY = 200
HY = YB/NY
AINT = 0.0
DO 10 K=1,NY
Y = (K-0.5)*HY
GY = EXP(-(Y-1.0)**2/(2*VAR))/SQRT(2*PI*VAR)
IF (X*Y.LE.Smax) THEN
VXY = DOLDSOL1(1,1,X*Y,2)
ELSE
VXY = DOLDSOL1(1,1,Smax,2) + (X*Y-Smax)
ENDIF
AINT = AINT + VXY*GY*HY
10 CONTINUE

C =         (Press [RETURN] to default to 0)
|----Enter constant or FORTRAN expression-----------------------|
-1
F =
|----Enter constant or FORTRAN expression-----------------------|
0.5*Sigma**2*x**2*Vxx+(R-Rlam*Rk)*x*Vx-(Rlam+R)*V + Rlam*AINT
Now the initial values must be defined using FORTRAN expressions.
They may be functions of X, and may also reference the initial time T0.
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: V0 = max(X-E,0.D0)                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
V0 =         (Press [RETURN] to default to 0)
|----Enter constant or FORTRAN expression-----------------------|
max(X-E,0.D0)
If you don't want to read the FINE PRINT, enter 'no'.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ Normally, interpolation is done to approximate the initial values    +
+ using cubic Hermites.  Since some derivatives must be interpolated,  +
+ if the initial values are not smooth (ie, have large or infinite     +
+ derivatives), the resulting cubic interpolants may have undesired    +
+ noise or large spikes.  Do you want to compute a least squares       +
+ approximation to the initial values, rather than an interpolant?     +
+ The least squares fit is generally much smoother, but requires one   +
+ extra linear system solution.                                        +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: no                                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
no
If you don't want to read the FINE PRINT, enter 'no'.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ Do you want to read the initial conditions from the restart file,    +
+ if it exists (and use the conditions supplied above if it does not   +
+ exist)?                                                              +
+                                                                      +
+ If so, PDE2D will dump the final solution at the end of each run     +
+ into a restart file "pde2d.res".  Thus the usual procedure for       +
+ using this dump/restart option is to make sure there is no restart   +
+ file in your directory left over from a previous job, then the       +
+ first time you run this job, the initial conditions supplied above   +
+ will be used, but on the second and subsequent runs the restart file +
+ from the previous run will be used to define the initial conditions. +
+                                                                      +
+ You can do all the "runs" in one program, by setting NPROB > 1.      +
+ Each pass through the DO loop, T0,TF,NSTEPS and possibly other       +
+ parameters may be varied, by making them functions of IPROB.         +
+                                                                      +
+ If the 2D or 3D collocation method is used, the coordinate           +
+ transformation should not change between dump and restart.           +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
[RETURN]
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: no                                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
no
If you do not have periodic boundary conditions, enter IPERDC=0.

Enter IPERDC=1 for periodic conditions at X = XGRID(1),XGRID(NXGRID)
+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ When periodic boundary conditions are selected, they apply to all    +
+ variables by default.  To turn off periodic boundary conditions on   +
+ the I-th variable, set PERDC(I) to 0 below in the main program and   +
+ set the desired boundary conditions in subroutine GB8Z, "by hand".   +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: IPERDC = 0                                                    \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
IPERDC =
|---- Enter an integer value in the range 0 to 1
0
Enter FORTRAN expressions to define the boundary condition functions,
which may be functions of

X,V,Vx and (if applicable) T

Recall that the boundary conditions have the form

G = 0

Enter NONE to indicate "no" boundary condition.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If "no" boundary condition is specified, the PDE is enforced at a    +
+ point just inside the boundary (exactly on the boundary, if EPS8Z    +
+ is set to 0 in the main program).                                    +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++

First define the boundary conditions at the point X = XGRID(1).
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: G = V                                                         \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
G =
|----Enter constant or FORTRAN expression-----------------------|
V

Now define the boundary conditions at the point X = XGRID(NXGRID).
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: G = Vx-1.0                                                    \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
G =
|----Enter constant or FORTRAN expression-----------------------|
Vx-1.0
If you don't want to read the FINE PRINT, default all of the following
variables.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ Normally, PDE2D saves the values of V,Vx at the output points.       +
+ If different variables are to be saved (for later printing or        +
+ plotting) the following functions can be used to re-define the       +
+ output variables:                                                    +
+    define UPRINT(1) to replace  V                                    +
+           UXPRINT(1)            Vx                                   +
+ Each function may be a function of                                   +
+                                                                      +
+    X,V,Vx,Vxx and (if applicable) T                                  +
+                                                                      +
+ Each may also be a function of the integral estimates SINT(1),...,   +
+ BINT(1),...                                                          +
+                                                                      +
+ The default for each variable is no change, for example, UPRINT(1)   +
+ defaults to V.  Enter FORTRAN expressions for each of the            +
+ following functions (or default).                                    +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
[RETURN]
\$ press [RETURN] to default all output modification variables          \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
Replace V for postprocessing?
UPRINT(1) =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

Replace Vx for postprocessing?
UXPRINT(1) =            (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

The solution is saved on a uniform grid of NX+1 points
XA + I*(XB-XA)/NX
I=0,...,NX.  Enter a value for NX (suggested value = 50).

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you want to save the solution at an arbitrary user-specified set  +
+ of NX+1 points, enter -NX.                                           +
+                                                                      +
+ If you set NEAR8Z=1 in the main program, the values saved at each    +
+ output point will actually be the solution as evaluated at a nearby  +
+ collocation or integration point.  For most problems this obviously  +
+ will produce less accurate output or plots, but for certain (rare)   +
+ problems, a solution component may be much less noisy when plotted   +
+ only at collocation or integration points.                           +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ NX needs to be fairly large, since the function DOLDSOL1 interpolates\$
\$ between points on the output grid.                                   \$
\$ enter: 200                                                           \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter an integer value
200
The solution is saved on a uniform grid of NX+1 points, covering the
interval (XA,XB).  Enter values for XA,XB.  These variables are usually
defaulted.

The defaults are XA = XGRID(1), XB = XGRID(NXGRID)

\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ press [RETURN] to default XA,XB                                      \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
XA =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

XB =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

The solution will be saved (for possible postprocessing) at the NSAVE+1
time points
T0 + K*(TF-T0)/NSAVE
K=0,...,NSAVE.  Enter a value for NSAVE.

If a user-specified constant time step is used, NSTEPS must be an
integer multiple of NSAVE.
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ NSAVE should be fairly large, so UOUT is updated frequently, and     \$
\$ the old solution accessed by DOLDSOL1 will not be too out-of-date.   \$
\$ enter: NSAVE = 100                                                   \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
NSAVE =
|---- Enter an integer value in the range 1 to +INFINITY
100
To generate tabular or graphical output, choose an output option from
the list below.

0. No further output is desired
1. Table of values at output points
The tabulated output is saved in a file.
2. One dimensional cross-sectional plots (versus X or T)
or, if applicable:
3. Surface plot of variable as function of X and T

Enter 0,1,2 or 3 to select an output option.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you decide later that you want additional types of plots not      +
+ requested during this interactive session, you will have to work     +
+ through a new interactive session, so it is recommended that you     +
+ request all output or plots you think you MIGHT eventually want now, +
+ during this session.                                                 +
+                                                                      +
+ Regardless of the options you select, a dummy subroutine POSTPR      +
+ will be included in the program created by the interactive driver;   +
+ you can add your own postprocessing code to this subroutine.         +
[RETURN]
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: 2, the first time you see this message and                    \$
\$        0, the second time                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter an integer value in the range 0 to 3
2
Enter a value for IVAR, to select the variable to be plotted or
printed:
IVAR = 1 means V  (possibly as modified by UPRINT,..)
2       Vx
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: IVAR = 1                                                      \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
IVAR =
|---- Enter an integer value in the range 1 to 2
1
Which type of cross-sectional plots do you want?

1. Plots of output variable as function of X (constant [T])
or, if applicable:
2. Plots of output variable as function of T (constant X)

Enter 1 or 2 to select a plot type.
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: 1                                                             \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter an integer value in the range 1 to 2
1
If you don't want to read the FINE PRINT, default ISET1,ISET2,ISINC.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ The tabular output or plots will be made at times:                   +
+        T(K) = T0 + K*(TF-T0)/NSAVE                                   +
+ for    K = ISET1, ISET1+ISINC, ISET1+2*ISINC,..., ISET2              +
+ Enter values for ISET1, ISET2 and ISINC.                             +
+                                                                      +
+ The default is ISET1=0, ISET2=NSAVE, ISINC=1, that is, the tabular   +
+ output or plots will be made at all time values for which the        +
+ solution has been saved.                                             +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: ISINC=10 and default ISET1,ISET2                              \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
ISET1 =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

ISET2 =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|

ISINC =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|
10
Specify the range (UMIN,UMAX) for the dependent variable axis.  UMIN
and UMAX are often defaulted.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ By default, each plot will be scaled to just fit in the plot area.   +
+ For a common scaling, you may want to set UMIN=ALOW, UMAX=AHIGH.     +
+ ALOW and AHIGH are the minimum and maximum values over all output    +
+ points and over all saved time steps or iterations.                  +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: UMIN =   0.0                                                  \$
\$        UMAX = 120.0                                                  \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
UMIN =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|
0.0
UMAX =             (Press [RETURN] to default)
|----Enter constant or FORTRAN expression-----------------------|
120.0
Enter a title, WITHOUT quotation marks.  A maximum of 40 characters
are allowed.  The default is no title.
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: V                                                             \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
TITLE =             (Press [RETURN] to default)
|----Enter title or name---------------|
V
To generate tabular or graphical output, choose an output option from
the list below.

0. No further output is desired
1. Table of values at output points
The tabulated output is saved in a file.
2. One dimensional cross-sectional plots (versus X or T)
or, if applicable:
3. Surface plot of variable as function of X and T

Enter 0,1,2 or 3 to select an output option.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you decide later that you want additional types of plots not      +
+ requested during this interactive session, you will have to work     +
+ through a new interactive session, so it is recommended that you     +
+ request all output or plots you think you MIGHT eventually want now, +
+ during this session.                                                 +
+                                                                      +
+ Regardless of the options you select, a dummy subroutine POSTPR      +
+ will be included in the program created by the interactive driver;   +
+ you can add your own postprocessing code to this subroutine.         +
[RETURN]
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: 2, the first time you see this message and                    \$
\$        0, the second time                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter an integer value in the range 0 to 3
0
Do you want to define any FORTRAN function subprograms used in any of
the FORTRAN 'expressions' entered earlier, entering them line by line?

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ If you selected double precision accuracy earlier, be sure to        +
+ declare these functions and their arguments DOUBLE PRECISION.        +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: no                                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
no
If you don't want to read the FINE PRINT, enter 'no'.

+++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++
+ Do you want to define any FORTRAN function subprograms used in any   +
+ of the FORTRAN 'expressions' entered earlier, by interpolating the   +
+ tabular output saved in a file created on an earlier PDE2D run?      +
++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$ enter: no                                                            \$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ EXAMPLE 6 \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
|---- Enter yes or no
no
More detailed information about PDE2D can be found in the book
"Solving Partial Differential Equation Applications with PDE2D,"
Granville Sewell, John Wiley & Sons, 2018.

*******************************************
***** Input program has been created  *****
*******************************************

```