ABSTRACT
This chapter deals with finite-difference schemes for partial differential equations. Definitions of convergence, consistency, and stability, as well as the theorem that establishes the relation between them, are quite general. This chapter discusses the construction of finite-difference schemes for partial differential equations, and considers approaches to the analysis of their stability. Moreover, it shows how to prove the theorem that consistent and stable schemes converge. For a wide class of linear operator equations, the theory of their approximate solution (not necessarily numerical) can be developed, and the concepts of consistency, stability, and convergence can be introduced and studied in a unified general framework. In doing so, the exact equation and the approximating equation should basically be considered in different spaces of functions ̶ the original space and the approximating space (like, for example, the space of continuous functions and the space of grid functions).
