I have the following problem: $$ -u''(x)+5u'(x)=f(x), x \in (0,1) $$ $$ u(0)=u(1)=0 $$ Now I have to find a discretization for Finite Differences, so my Matrix $A_h$ is strictly diagonally dominant (Where $A_h \cdot u=b_h$ ).
How do I know which discretization I have to choose without just trying every discretization?
The answer to what is "best" depends on the metric you employ. Do you want easily parallelizable solutions? Pick small stencils and small grids and solve them on parallel machines. Do you have only one core and know that your solutions are very regular? Pick large stencils with moderate grid sizes, giving you high order of accuracy.
I don't think there are finite difference schemes which are strictly diagonal dominant since you always have the condition that, in order to approximate a derivate, the actual point value should vanish. Thus for a scheme of the form with order $k \in \mathbb{N} \geq 1$ $$ u^{(k)}(x) \approx \sum_{i=1}^{N>1} w_i u(x + h_i) $$ The Taylor expansion reads $$\sum_{i=1}^{N>1} w_i u(x) \cdot (h_i)^0 + \mathcal{O}(\max_i \{h_i\})= \sum_{i=1}^{N>1} w_i u(x) \overset{!}{=} 0 \: \forall \: u(x) $$ This can only be achieved if for any scheme $$\sum_i w_i = 0$$ with renders a strictly diagonal-dominant matrix impossible.
