General Question: If I have an IVP with periodic and continuous initial condition, which rules the accuracy of the scheme - the manner in which we approximate spatial derivative or the acuuracy of the ODE solving scheme.

Specific Case: I'm solving

$u_t = u_{xx} + u$ ,

$x \in \mathbb{R} $, $t>0$, $u(t,x) = u(t, x + \pi )\forall x \in \mathbb{R} $

$u(t =0, x) =cos(x) + sin(2x)$

With the following approach - I discretisize $x$ to a grid $ x_0 ... x_N$, and then solve numerically a system of $N$ coupled ODE's.

It turns out that no matter how I approximate $u_{xx}$, the global error of the solution depends only on the scheme with which I solved the ODE - Forward Euler, Adam Beshfort, Runge Kutta etc.

I looked at this question in MO, and even when I took the spectral approach, it hasn't changed the global error.