Consider the system of ODE $$\displaystyle \frac{d \bf u}{dt}=A {\bf u} ~~~~~~~~~~~~~~~~~~~(a)$$ formed as a result of a particular discretization say $D_1$ on a linear PDE $$L u(t,x)=0$$ We see that the numerical integration by a particular discretization say $D_2$ on $(a)$ is stable if the eigen values of a matrix $A$ lies in the stability region of $D_2$.
But I couldn't connect the stability of solution for $(a)$ with the stability region of time integration $D_2$. How the time integration $D_2$ become stable if the eigen values of $A$ lies in the stability region of $D_2$, and how it is related with the stability of solution for the system of linear ODE system $(a)$ ?