What’s the name of this phenomenon occurring when numerically computing the explicit Euler method for this PDE IVP and why does it occur? I’ve been asked to calculate numerically the solution of this IVP using Euler’s explicit method on Mathlab:
$$u_t-2u_{xx}=0$$
$$u(x,0)=\sin (2\pi x), \quad x\in(0,1)$$
$$u(0,t)=u(1,t)=0, \quad t\in(0,0.05)$$
However, I find the approximation is really bad for $h = 0.04, k = 4 · 10^{-3}$ (here the exact solution is shown in blue and the red dots correspond to the approximated values):

...then it gets considerably better for a slightly inferior value of $k$: here the same plot is shown for $h = 0.04, k = 4 · 10^{-5}$:

I’d like to know how is this phenomenon called and why does it occur for this particular IVP. Is there any conditions a numerical method has to meet in order not to present this kind of problems?
 A: You run afoul of the limited A-stability of the explicit Euler method. If you write the discretized PDE as ODE system $\dot u=F(u)$, then $L=\frac8{h^2}$ is a Lipschitz constant, one can probably reduce that by some percent. A rule-of-thump says that for $Lk>2$ the method is unstable, for $Lk\approx 1$ mostly looking sensible and with $Lk\le 0.5$ starting to give quantitatively correct results.
In your failed experiment you have $L=5000$ and $Lk=20$, thus instability. In the second case you get $Lk=0.2$ which corresponds to sensible results with the errors still visible in the plot.
You can get an even more detailed picture by applying the discrete Fourier-transform to the space discretization. Then the equations for the different frequencies are decoupled from each other. Note that the Euler iteration in space mode provides a continuing source of floating point noise to the amplitude equations in frequency mode. So even amplitudes starting at zero can be perturbed and then amplified by the instability effects.
