% m57_leapfrog2Dcheb.m - second-order wave eq.
%         in 2D via Chebyshev spectral
%         (compare m48_leapfrog2D.m, based on finite differences)

% Grid and initial data:
  N = 32;
  [D,x] = cheb(N);
  D2 = D^2; D2p = D2';
  x = x'; y = x'; k = 6/N^2;
  [xx,yy] = meshgrid(x,y);

% Initial data and plot setup:
  vv = exp(-12.5*(xx.^2+yy.^2));
  vvold = vv; vvnew = 0*vv;
  clf, shg
  plt = mesh(xx,yy,vv); view(-20,30)
  axis([-1 1 -1 1 -1.2 1.2])
  colormap(1e-6*[1 1 1]), drawnow

% Time-stepping by leap frog formula:
  plotgap = round(.01/k);
  I = 2:N;
  for n = 1:20/k
    if mod(n,plotgap)==0
      set(plt,'zdata',vv)
      drawnow
    end
    uxx = vv*D2p; uyy = D2*vv;
    vvnew(I,I) = 2*vv(I,I) - vvold(I,I) + k^2*(uxx(I,I)+uyy(I,I)); 
    vvold = vv; vv = vvnew;
    title(sprintf('t = %6.3f', n*k), 'fontsize',12)
  end