% p19.m - 2nd-order wave eq. on Chebyshev grid (compare p6.m)

% Time-stepping by leap frog formula:
  N = 80; x = cos(pi*(0:N)/N); dt = 8/N^2;
  v = exp(-200*x.^2); vold = exp(-200*(x-dt).^2);
  tmax = 4; tplot = .075; 
  plotgap = round(tplot/dt); dt = tplot/plotgap;
  nplots = round(tmax/tplot);
  plotdata = [v; zeros(nplots,N+1)]; tdata = 0;
  clf, drawnow, h = waitbar(0,'please wait...');
  set(gcf,'renderer','zbuffer')
  for i = 1:nplots, waitbar(i/nplots)
    for n = 1:plotgap
      w = chebfft(chebfft(v))'; w(1) = 0; w(N+1) = 0;
      vnew = 2*v - vold + dt^2*w; vold = v; v = vnew;
    end
    plotdata(i+1,:) = v; tdata = [tdata; dt*i*plotgap];
  end

% Plot results:
  clf, drawnow, waterfall(x,tdata,plotdata)
  axis([-1 1 0 tmax -2 2]), view(10,70), grid off
  colormap(1e-6*[1 1 1]); ylabel t, zlabel u, close(h)