clear

%%
% We solve the advection equation on a domain with periodic end conditions.
% Our approach is the method of lines.
c = 2;
[x, Dx, ~] = diffper(300, [-4,4]);
f = @(t,u) -c*(Dx*u);

%%
% The following initial condition isn't mathematically periodic, but the
% deviation is less than machine precision. 
u_init = 1 + exp(-3*x.^2);
t = linspace(0, 3, 41);
[t, U] = ode45(f, t, u_init);

%%
% A nice way to visualize solutions with moving features is by a
% waterfall plot.
waterfall(x, t, U)
view(-13, 65)
xlabel('x')
ylabel('t')
zlabel('u(x,t)')
title('Advection equation')

%%
% The bump moves with speed 2 to the right, reentering on the left as it
% exits to the right because of the periodic conditions.