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\chapter[FIDE: Finite differences for PDEs]%
        {FIDE: Finite difference method for partial differential equations}
\label{FIDE}
\typeout{[FIDE: Finite differences for PDEs]}

{\footnotesize
\begin{center}
Richard Liska \\
Faculty of Nuclear Science and Physical Engineering \\
Technical University of Prague \\
Brehova 7, 115 19 Prague 1, Czech Republic \\[0.05in]
e--mail: tjerl@aci.cvut.cz
\end{center}
}

\ttindex{FIDE}

The FIDE package performs automation of the process of numerical
solving partial differential equations systems (PDES) by generating
finite difference methods.  In the process one can find several stages
in which computer algebra can be used for performing routine
analytical calculations, namely: transforming differential equations
into different coordinate systems, discretisation of differential
equations, analysis of difference schemes and generation of numerical
programs.  The FIDE package consists of the following modules:

\begin{description}
\item[EXPRES]  for transforming PDES into any orthogonal coordinate system.
\item[IIMET]   for discretisation of PDES by integro-interpolation method.
\item[APPROX]  for determining the order of approximation of
difference scheme.
\item[CHARPOL] for calculation of amplification matrix and
characteristic polynomial of difference scheme, which are needed in
Fourier stability analysis.\
\item[HURWP] for polynomial roots locating necessary in verifying the
von Neumann stability condition.
\item[LINBAND] for generating the block of FORTRAN code, which solves
a system of linear algebraic equations with band matrix appearing
quite often in difference schemes.
\end{description}

For more details on this package are given in the FIDE documentation,
and in the examples.  A flavour of its capabilities can be seen from
the following simple example.

\begin{verbatim}
off exp;

factor diff;

on rat,eqfu;

% Declare which indexes will be given to coordinates
coordinates x,t into j,m;

% Declares uniform grid in x coordinate
grid uniform,x;

% Declares dependencies of functions on coordinates
dependence eta(t,x),v(t,x),eps(t,x),p(t,x);

% Declares p as known function
given p;

same eta,v,p;

iim a, eta,diff(eta,t)-eta*diff(v,x)=0,
    v,diff(v,t)+eta/ro*diff(p,x)=0,
    eps,diff(eps,t)+eta*p/ro*diff(v,x)=0;


*****************************
*****      Program      *****          IIMET Ver 1.1.2
*****************************

      Partial Differential Equations
      ==============================

diff(eta,t) - diff(v,x)*eta    =    0

 diff(p,x)*eta
--------------- + diff(v,t)    =    0
      ro

               diff(v,x)*eta*p
diff(eps,t) + -----------------    =    0
                     ro


 Backtracking needed in grid optimalization
0 interpolations are needed in x coordinate
  Equation for eta variable is integrated in half grid point
  Equation for v variable is integrated in half grid point
  Equation for eps variable is integrated in half grid point
0 interpolations are needed in t coordinate
  Equation for eta variable is integrated in half grid point
  Equation for v variable is integrated in half grid point
  Equation for eps variable is integrated in half grid point

         Equations after Discretization Using IIM :
         ==========================================

(4*(eta(j,m + 1) - eta(j,m) - eta(j + 1,m)

     + eta(j + 1,m + 1))*hx - (

    (eta(j + 1,m + 1) + eta(j,m + 1))

    *(v(j + 1,m + 1) - v(j,m + 1))

     + (eta(j + 1,m) + eta(j,m))*(v(j + 1,m) - v(j,m)))

 *(ht(m + 1) + ht(m)))/(4*(ht(m + 1) + ht(m))*hx)   =   0


(4*(v(j,m + 1) - v(j,m) - v(j + 1,m) + v(j + 1,m + 1))*hx*ro

  + ((eta(j + 1,m + 1) + eta(j,m + 1))

     *(p(j + 1,m + 1) - p(j,m + 1))

      + (eta(j + 1,m) + eta(j,m))*(p(j + 1,m) - p(j,m)))

 *(ht(m + 1) + ht(m)))/(4*(ht(m + 1) + ht(m))*hx*ro)   =   0


(4*(eps(j,m + 1) - eps(j,m) - eps(j + 1,m)

     + eps(j + 1,m + 1))*hx*ro + ((

       eta(j + 1,m + 1)*p(j + 1,m + 1)

        + eta(j,m + 1)*p(j,m + 1))

    *(v(j + 1,m + 1) - v(j,m + 1)) +

    (eta(j + 1,m)*p(j + 1,m) + eta(j,m)*p(j,m))

    *(v(j + 1,m) - v(j,m)))*(ht(m + 1) + ht(m)))/(4

   *(ht(m + 1) + ht(m))*hx*ro)   =   0


clear a;

clearsame;

cleargiven;
\end{verbatim}