In this chapter we study finite difference schemes for parabolic partial differential equations. The notions of conditional and unconditional stability, and CFL condition are introduced to analyze the classical schemes for the heat equation. Different techniques, like maximum principles and energy arguments are presented to obtain stability in different norms. Then, we turn to the study of the pure initial value problem, the grand goal being to discuss the von Neumann stability analysis. To accomplish this we introduce the notions of Fourier-Z transform of grid functions and the symbol of a finite difference scheme. This allows us to state the von Neumann stability condition and prove that it is necessary and sufficient for stability. These notions are also used to present a covergence analysis that is somewhat different than the one presented in previous sections.

We present finite difference schemes for hyperbolic problems. We begin with the transport equation and show the necessity of a certain type of upwinding and/or numerical diffusion for stability. This is illustrated by analyzing all the classical schemes: upwind, downwind, centered, Lax–Friedrichs, Lax–Wendroff. Beam–Warming, and Crank–Nicolson. We then focus on the topic of positivity and max-norm dissipativity of finite difference schemes. We present and sketch the proof of Godunov’s theorem. A brief discussion of dispersion relations is then carried out. Next, we study schemes for the wave equation, and show how to properly choose the discrete initial velocity to attain the desired consistency. To show stability we employ energy and negative norm arguments. The last section is dedicated to developing schemes for symmetric hyperbolic systems. The most well-known finite difference schemes are presented, and their matrix valued symbols are introduced. The symbol is then used to develop the von Neumann stability analysis for this case.