clc, clear all, close all

U = 1;              % vecloity of fluid m/s
Lmin = 0;           % 0 m
Lmax = 1;           % 1 m
n = 100;            % nodes/points
dx = Lmax/n;        % step size [m]
r = 0.5;            % Courant number for stability
t = 0;              % initial time
tmax = 1;           % time [s]

% for lax wendroff/friedrich to be stable, r < 1, we have now r = 0.5
dt = (r*dx)/U;      % time step, r = U*dt/dx
kdiff = 0.5*(U^2)*dt;
pausetime = dt;     % Pause time between animation

for k = 1:((tmax/dt)+1)
    t(k) = (k-1)*dt;                    % saving time steps, starting with t=0
    phi(1,k) = -sin(2*pi*t(k));         % left boundary condition phi(x=0,t)
    phi_LW(1,k) = phi(1,k);             % lax-wendroff
end

for l = 1:((Lmax/dx)+1)
    x(l) = (l-1)*dx;                    % saving spatial steps, starting with x = 0
    phi(l,1) = sin(2*pi*x(l));          % initial solution phi(x,t=0)
    phi_LW(l,1) = phi(l,1);             % lax-wendroff
end

for k = 1:(tmax/dt)
    for l = 2:(Lmax/dx)                 % start from 2 since 1 is boundary cond.
        phi(l,k+1) = 0.5*(phi(l+1,k)+phi(l-1,k)) - (r/2)*(phi(l+1,k)-phi(l-1,k));
        phi_LW(l,k+1) = phi_LW(l,k) - (r/2)*(phi_LW(l+1,k)-phi_LW(l-1,k))+ ((kdiff*dt)/(dx^2))*(phi_LW(l-1,k)-2*phi_LW(l,k)+phi_LW(l+1,k));
    end
    pause(pausetime)   % Shows results for each timestep
    %plot(x(1:end),phi(1:end,k),'r');    % lax-friedrich, NEED TO UNCOMMENT FOR PLOT
    plot(x(1:end),phi_LW(1:end,k),'r'); % lax-wendroff, NEED TO UNCOMMENT FOR PLOT
    xlabel('x [m]')
    ylabel('U [m/s]')
    title('The temperature distribution in real time')
end

for k = 1:101
exact_solution(k) = sin(2*pi*k*dx);         % exact solution at t = 1 s
end

hold on
xlabel('x [m]')
ylabel('U [m/s]')
title('Temperature distribution at t = 1s')
plot(x,phi(1:end,end),'r')                  % plot lax-fried. against x for t = 1 s
plot(x,phi_LW(1:end,end),'g')
plot(x,exact_solution(1:end),'b')       % plot exact solution fo t = 1 s
legend('Lax-Friedrich','Law-Wendroff','Exact solution')