function prob_2_35
% 
% Written by:
% -- 
% John L. Weatherwax                2005-05-07
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----
  
global c_h
% Spatial grid, initial values:
N = 1000;
h = 2*pi/N;
x = h*(1:N);
v0 = exp(-100*(x - 1) .^ 2);
c_h = - (0.2 + sin(x - 1) .^ 2)' / h;

% Sparsity pattern for the Jacobian:
S = sparse(N,N);
for m = 2:N
    S(m,m-1) = 1;
    S(m,m) = 1;
end
% WWX: for periodic BC's:
S(1,1) = 1; S(1,N) = 1; 

options = odeset('JPattern',S);
% Integrate and plot:
%t = 0:0.30:8;
% WWX: Run for longer to see the trend.
t = 0:0.30:24;
[t,v] = ode15s(@f,t,v0,options);
pltspace = ceil(N/128);
x = x(1:pltspace:end);
v = v(:,1:pltspace:end);
surf(x,t,v), %view(10,70), 
axis([0 2*pi 0 t(end) 0 5])
ylabel t, zlabel u, grid off
%================================================
function dvdt = f(t,v)
global c_h

dvdt = c_h .* [0; diff(v)];
% WWX: For periodic BC's:
dvdt(1) = c_h(1)*(v(1)-v(end));