% BabyWave.m
% MST 383/683 
% Fall 2020
%
% Forward Euler, Central Difference for u_t = u_x
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear,clc

h = 0.001;
delta = 0.01;
% delta = 0.04;
% delta = 0.1;
% delta = 0.001;

t = 0:h:1;
x = 0:delta:1;

u = NaN(length(x),length(t));
u(:,1) = cos(2*pi*x);
v = u;

% Update matrix for Forward Euler, Central Difference
M = 1*diag(ones(1,length(x)),0) + (h/(2*delta))*diag(ones(1,length(x)-1),1) ...
    + (-h/(2*delta))*diag(ones(1,length(x)-1),-1);
M(end,1) = h/(2*delta);
M(1,end) = -h/(2*delta);

% Update matrix for Backward Euler, Central Difference
A = 1*diag(ones(1,length(x)),0) + (-h/(2*delta))*diag(ones(1,length(x)-1),1) ...
    + (h/(2*delta))*diag(ones(1,length(x)-1),-1);
A(end,1) = -h/(2*delta);
A(1,end) = h/(2*delta);

for i = 1:length(t)-1
    u(:,i+1) = M*u(:,i);
    
    v(:,i+1) = A\v(:,i);
end

% Plotting
[T,X] = meshgrid(t,x);

figure
mesh(T,X,u)
xlabel('t')
ylabel('x')
zlabel('u')

figure
mesh(T,X,v)