%% Example 6.8
% Burgers Equations


%% Initialization
close all;
clear N f dX X dt t x D l j u


N  = 32;
f  = @(x) 10*sin(x);
dX = 2*pi/N;
X  = [0:dX:2*pi-dX]';
dt = 0.005;
t  = 0;

x  = [0:0.01:2*pi];
plot(x,f(x),'k','LineWidth',2);
hold on;


%% Create Differentiation matrix :

D = zeros(N,N);
for l = [1:N]
    for j = [1:N]
        if l~=j
            D(l,j) = 0.5*(-1)^(l-j)/tan(pi*(l-j)/N);
        end;
    end;
end;



%% Iterate and plot :

u   = f(X);

for t = [dt:dt:1]
    u = u + dt*(D*D*u - diag(u)*D*u);
    
    switch t
        case .1
            plot(X,u,'ko');
        case .2
            plot(X,u,'kx');
        case .4
            disp('here')
            plot(X,u,'k*');
        case .6
            plot(X,u,'ks');
    end;
end;
    
legend('t=0','t=0.1','t=0.2','t=0.4','t=0.6')