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512 chapter 194u(x,t)0012x200tFigure 19.1 Formation of a shock wave. Wave height versus position for increasing times asvisualized with Gnuplot (left) and with OpenDX (right).solution in terms of present and past ones:[ u 2 (x +∆x, t) − u 2 (x − ∆x, t)u(x, t +∆t)=u(x, t − ∆t) − β2[ ]u2i+1,j − u 2 i−1,ju i,j+1 = u i,j−1 − β,β=2],ɛ∆x/∆t . (19.9)Here u 2 is the square of u and is not its second derivative, and β is a ratio of constantsknown as the Courant–Friedrichs–Lewy (CFL) number. As you should provefor yourself, β