% Implicit, backward Euler for solving the heat equation: created by the AMATH 301 Spring 2009 class, in class.

clear all; close all;
n = 100;
L = 20;
x2= linspace(-L/2,L/2,n+1);
x=x2(1:n);
dx = x(2)-x(1);

dt = 1;
CFL = dt/dx^2;

Time = 4;
time_steps=Time/dt;
t = 0:dt:Time;

u0 = exp(-x.^2)';
usol(:,1) = u0;

e1 = ones(n,1);
A = spdiags([-CFL*e1 (1+2*CFL)*e1 -CFL*e1],[-1 0 1],n,n);

A(1,n) = -CFL;
A(n,1) = -CFL;

for j = 1:time_steps,
 u1 = A\u0;
 u0 = u1;   
 usol(:,j+1) = u1;   
end

waterfall(x,t,usol');
map=[0 0 0]
colormap(map);