Hello I am trying to use an explicit Euler method and to show the solution is always unstable no matter the time step.

𝜕𝑇/𝜕𝑡 + 𝑎 𝜕𝑇/𝜕𝑥 = 0 for 0 <= x <= 1

With the initial condition and boundary conditions of,

𝑇(𝑥, 0) = 𝑒 −200(𝑥−0.25) 2 , 𝑇(0,𝑡) = 0, 𝑇(1,𝑡): outflow condition

𝑎 = 1, ∆𝑥 = 0.01