I have the 1D heat equation $u_t= \alpha u_{xx}$ on $x \in [0,1]$.
It has homogeneous Dirichlet boundary conditions.
I intend to use the forward Euler numerical scheme. $$\frac{u^{n+1}-u^{n}}{k} = \alpha u^{n}_{xx} $$
The discrete energy of the heat equation is $$ E^{n} = \int_0^1 (u^{n})^{2} dx $$
Now I need to prove energy stability of the above numerical scheme, i.e., $$ E^{n+1} \leq E^{n} $$
Can you please help me how to prove? I took the expression $E^{n+1}-E^{n}$ and used the original discretized equation in it but no use.