Different variations of the “energy method” can be used to show that PDEs are well posed, to show that discrete approximations are stable, and to establish (global) convergence rates under mesh refinements. The energy approach is very broadly applicable, and can handle many cases that include: boundary conditions; variable coefficients (and nonlinearities); and nonperiodic PS methods.

However, this flexibility and power comes at a price of often significant technical difficulty. This appendix is intended to provide only a first flavor of this rich subject to readers who are unfamiliar with it. For this purpose, we here consider five examples, all relating to the heat equation on the interval [–1, 1]. More systematic descriptions can be found in Richtmyer and Morton (1967), Gustafsson et al. (1995), and (for PS methods) Canuto et al. (1988).