PARTIAL DIFFERENTIAL EQUATIONS 3
                                        
This slide show consists of graphs of numerical solutions of a particular
partial differential equation, viz. the wave equation

                              U  +  U  = 0,
                               t     x

with an initial condition of a hump "resolved" with 1 point.  The weak solution
to this problem consists of an initial profile moving with speed 1 to the right.

The numerical approximation is given by the following iterative map

      u*(i) = u(i) - «ùLù[u(i+1) - u(i-1)] + åù[u(i+1) - 2u(i) + u(i-1)],

where L = t/x is called the Courant number (and is usually denoted by lambda),
and å is the numerical diffusion.  Consistency and stability of the above
numerical scheme require 0 < L ó 1,  Lý ó å ó 1.

In the demonstrations, x is set to 1, and the following values of L and å were
used

                    å =  0   (central)
                    å = -L   (downwind)
                    å =  1   (Lax-Friedrichs)
                    å =  L   (upwind)
                    å =  Lý  (Lax-Wendroff).

L is 0.8, 1.0, or 1.2, and is shown on the appropriate slide with the initial
conditions.

The demonstration illustrates notions of stability, accuracy, numerical
diffusion (e.g. damping), and numerical dispersion (e.g. phase errors).

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