I'm trying to apply the theta method to the 2D heat transfer equation: $$\partial_t u -\Delta u = f$$
With a second order discretization in space.
Now, I was able to show 2nd order in space by keeping $\Delta t$ fixed, decreasing $\Delta x =\Delta y $ and plotting in loglog the error against $\Delta x $ obtaining 2 parallel lines.
I would also like to show that the theta method is of 1st order in time. My instinct was to do the same thing by decreasing only $\Delta t$. However, the error soon stops decreasing and remains constant (probably because the space discretization is fixed so I can't reduce the error past some amount?).
Was this supposed to work and give me a straight line?
I also tried, for $\theta < 0.5$, to use the stability condition to decrease both time and space steps and I obtain the expected graph, but it does not feel right since this could only the space grid refinement doing the work.