  In this paper we study the stability of explicit finite difference
discretizations of linear advection-diffusion equations (ADE) with arbitrary
order of accuracy in the context of method of lines. The analysis first focuses
on the stability of the system of ordinary differential equations (ODE) that is
obtained by discretizing the ADE in space and then extends to fully discretized
methods where explicit Runge-Kutta methods are used for integrating the ODE
system. In particular, it is proved that all stable semi-discretization of the
ADE gives rise to a conditionally stable fully discretized method if the
time-integrator is at least first-order accurate, whereas high-order spatial
discretization of the advection equation cannot yield a stable method if the
temporal order is too low. In the second half of this paper, we extend the
analysis to a partially dissipative wave system and obtain the stability
results for both semi-discretized and fully-discretized methods. Finally, the
major theoretical predictions are verified numerically.
