% [u,err,x,t] = lax_friedrichs(@f,t_0,t_f,M,N)
%
% solves the heat equation u_t + u_x = 0 [0,2*pi] 
%
% If I'm going to use the FFT to analyse the solution then I'd like to  take
% N of the form 2^k.  This will also have the advantage of making sure there's
% always a meshpoint at x = pi
%
% f.m is the function that contains the initial data

function [u,u_exact,x,t] = lax_friedrichs(f,t_0,t_f,M,N)

% define the mesh in space
dx = 2*pi/N;
x = 0:dx:2*pi;
x = x';

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

c = 1/2;
%display('this number should be less than 1 for stability:')
mu = c*dt/dx;

% choose the wave number of the initial data and give its decay rate
u = zeros(N+1,M+1);
u(:,1) = f(x);
u_exact(:,1) = u(:,1);

% I want to do the Lax Friedrichs scheme:
%
% u_new(j) = (u_old(j-1)+u_old(j+1))/2 - (mu/2)*(u_old(j+1)-u_old(j-1))

for j=1:M
    for k=2:N
        u(k,j+1) = (u(k+1,j)+u(k-1,j))/2-(mu/2)*(u(k+1,j)-u(k-1,j));
    end
    % I code in the exact values at the endpoints.
    u(1,j+1)=1;
    u(N+1,j+1)=0;
    X = x-c*t(j+1);
    u_exact(:,j+1) = f(X);
end