29 - Finite Difference Methods for Hyperbolic Problems

Summary

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.