% [u,x,t,kk,amp] = burgers_ab(u0,D,t_0,t_f,M,N)
%
% solves the periodic Burgers equation with viscosity:
%                u_t + u u_x = D u_xx
% with initial data u_0 with periodic boundary conditions using
% spectral derivatives in
% space and second-order Adams-Bashforth time-stepping.  t_0 is the initial time, t_f is the
% final time, N is the number of mesh-points, and M is the number of
% time steps.  err is the error. it also returns the power spectrum of the
% solution, where kk is the collection of wave numbers and amp = log(f_k^2)
% is the magnitude of the kth Fourier mode.
%
% N SHOULD BE A POWER OF 2!
%
% (It will work no matter what, but will go faster if N is a power
% of 2.)
%
% Note: for burgers equation with initial data cos(x), we know that the
% solution blows up at time: t = 1
%
%
% Also, the initial data u0 should be given at x=0,x=dx,....x=2*pi-dx only.
function [u,x,t,kk,amp] = burgers_ab(u0,D,t_0,t_f,M,N)

% define the mesh in space
dx = 2*pi/N;
x = 0:dx:2*pi-dx;
x = x';
kk = 0:1:(N/2-1);

% define the mesh in time
dt = (t_f-t_0)/M;
t = t_0:dt:t_f;

u(:,1) = u0;
[kk,amp(:,1)] = find_spec(u0);

% I first take one time step.  I do this using the first-order accurate
% integrating factor method (see burgers_if.m)
U_now = fft(u(:,1));
G = convolve(U_now);
% the zero mode:
U_new(1) = U_now(1) - dt*G(1);
for k = 1:floor(N/2)-1
    U_new(k+1) = exp(-D*k^2*dt)*(U_now(k+1) - dt*G(k+1));
    U_new(N-k+1) = exp(-D*k^2*dt)*(U_now(N-k+1) - dt*G(N-k+1));
end
% update so I'm ready for another time step.
U_old = U_now;
G_old = G;
U_now = U_new;

% now that I have U_old and U_now, I can begin Adams-Bashforth.

% reverse transform to get u.  Note that in a real computation I might
% take many timesteps before "printing out" a solution.  And I need to
% transform back to real space only when I want to print out a
% solution.
j = 1;
u(:,j+1) = real(ifft(U_new));
% note: for a linear system, I wouldn't have to take the real part. If
% I've coded everything correctly then I'll have complex conjugates
% where I need them and the ifft will return a real-valued function.
% But when I have a nonlinearity things get messed up and I have to
% take the real part of the ifft
[kk,amp(:,j+1)] = find_spec(u(:,j+1));


% I now start the second-order Adams-Bashforth method
for j=2:M
    G_now = convolve(U_now);
    % At time t_n, I use convolve.m to compute the nonlinearity u^n u^n_x.  It
    % gives me the fourier transform of u^n u^n_x.  Let's call that g^n.  And
    %
    %     d/dt hat{u}_k + hat{g}_k = - D k^2 hat{u}_k
    %
    %     d/dt hat{u}_k = - D k^2 hat{u}_k - hat{g}_k = F(u)_k
    %
    % leads to second-order Adams-Bashforth
    %
    %     u^{n+1}_k = u^n_k + .5*dt*(3*F(u^n)_k - F(u^{n-1})_k)
    %               = U_new + .5*dt*(3*F(U_now) - F(U_old))

    % the zero mode:
    U_new(1) = U_now(1) + .5*dt*(-3*G_now(1)+G_old(1));
    for k = 1:floor(N/2)-1
        U_new(k+1) = U_now(k+1)+.5*dt*(-D*k^2*(3*U_now(k+1)-U_old(k+1))-3*G_now(k+1)+G_old(k+1));
        U_new(N-k+1) = U_now(N-k+1)+.5*dt*(-D*(-k)^2*(3*U_now(N-k+1)-U_old(N-k+1)) - 3*G_now(N-k+1) + G_old(N-k+1) );
    end
    % update so I'm ready for another time step.
    U_old = U_now;
    G_old = G_now;
    U_now = U_new;
    % reverse transform to get u.  Note that in a real computation I might
    % take many timesteps before "printing out" a solution.  And I need to
    % transform back to real space only when I want to print out a
    % solution.
    u(:,j+1) = real(ifft(U_new));
    % note: for a linear system, I wouldn't have to take the real part. If
    % I've coded everything correctly then I'll have complex conjugates
    % where I need them and the ifft will return a real-valued function.
    % But when I have a nonlinearity things get messed up and I have to
    % take the real part of the ifft
    [kk,amp(:,j+1)] = find_spec(u(:,j+1));
end