EXPLANATION FILE OF PROGRAM STORMER
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  Solve Y"=f(x,y,y') with initial conditions by Stormer's method
  --------------------------------------------------------------

    The differential equation of order 2 can be replaced by a system of two
  equations of order 1:

  Given y"=f(x,y,y') with y(a) and y'(a), by calling u = y', the problem becomes

                    | u'=f(x,y,u)
                    | y'=u
                    | with y(a), u(a) given


    We start from a particular form of the Taylor's formula, where the remainder
  is under the form of an integral:

                                     x+h
           y(x+h) = y(x) + h y'(x) + Sum (x+h-t) y"(t) dt
                                      x

  In the same way

                                     x-h
           y(x-h) = y(x) - h y'(x) + Sum (x-h-t) y"(t) dt
                                      x

  By summing

                                     x+h       x-h
           y(x+h) - 2y(x) + y(x-h) = Sum ... + Sum ...
                                      x         x

  For the second integral, we change the variable: u = 2x - t

  So       x-h                      x+h
           Sum (x-h-t) y"(t) dt = - Sum (x+h-u) y"(u) du
            x                        x

                                    x+h
           du=-dt ==>           =   Sum (x+h-t) y"(2x-t) dt
                                     x

  Finally
                                     x+h
           y(x+h) - 2y(x) + y(x-h) = Sum (x+h-t)[y"(t)+y"(2x-t)] dt
                                      x

  We now use the interpolation polynomial recalling that y"(x)=f(x,y,y'):

                              x
                               n+1
           y    - y  + y    = Sum  (x   - t)[P(t)+P(2x - 1)] dt
            n+1    n    n-1   x      n+1              n
                               n
              x
            2  n+1                 0             1            2
         = h  Sum (x   - t) [O0 Div (fn) + O1 Div (fn) +O2 Div (fn)] dt
              x     n+1
               n
                       m  1        |(-s)   (m)|
         with O  = (-1)  Sum (1-s) |(  ) + ( )| ds
               m          0        |(m )   (s)|

                                 (see file Adambash.txt).  


  This leads to Stormer's formulas:


    Explicit:  y    - 2 y  + y    = (h²/12)[13f - 2 f    + f   ]
                n+1      n    n-1              n     n-1    n-2

    Implicit:  y    - 2 y  + y    = (h²/12)[f   + 10 f  + f   ]
                n+1      n    n-1            n+1      n    n-1


  From [BIBLI 04]
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End of file Stormer.txt