Introduction

The last chapter concerned linear stability. This chapter concerns nonlinear stability. Linear stability theory is classic, dating to the late 1940s. The study of nonlinear stability is far newer. Starting with papers by Boris and Book (1973) and Van Leer (1974), nonlinear stability theory developed over roughly the next fifteen years. Although nonlinear stability theory may someday undergo major revision, no significant new developments have appeared in the literature since the late 1980s. Thus after a period of intensive development, nonlinear stability theory has plateaued, at least temporarily.

To keep the discussion within reasonable bounds, this chapter concerns only explicit forward-time finite-difference approximations. Furthermore, this chapter concerns mainly one-dimensional scalar conservation laws on infinite spatial domains. As far as more realistic scenarios go, the Euler equations are discussed briefly in Section 16.12, multidimensions are discussed briefly at the end of this introduction, and solid and far-field boundaries are discussed briefly in Section 19.1. Unlike solid and far-field boundaries, the periodic boundaries in Chapter 15 and the infinite boundaries in this chapter do not pose any difficult stability issues.

The last chapter began with a general introduction to stability, both linear and nonlinear. Any impatient readers who skipped the last chapter should go back and read Section 15.0. While linear and nonlinear stability share the same broad philosophical principles, especially the emphasis on spurious oscillations, the details are completely different. Thus, except for its introduction, the last chapter is not a prerequisite for this chapter. One of the more important nonlinear stability conditions relies heavily on the wave speed split form described in Section 13.5. Readers who skipped this section should go back and read it.