Consistency, Stability, and Convergence

The three central themes of this book are presented: consistency, stability, and convergence. The three concepts are interrelated by the Lax-Richtmyer Equivalence Theorem. Derivative approximations provide a simple example of discretization error, while an introduction to IEEE floating point motivates the study of roundoff error and stability. The three types of numerical error intrinsic to digital computing are discussed: (i) roundoff error, (ii) truncation or discretization error, and (iii) numerical instability. A-stability and L-stability are analyzed, as well as the relationship between local truncation error and local error.