Two Topics in Nonlinear Stability
Jesús María Sanz-Serna · Advances in Numerical Analysis · 1991
Abstract There are several ways in which the word stability may be understood in a numerical analysis context. First of all, stability is used in expressions like ‘stability and consistency imply convergence’. In this first sense, stability refers to dependence of a numerical result on the data, and, as such, applies to most numerical computations, including numerical linear algebra, quadrature, differential equations, etc. This first notion is akin to the idea of ‘well posedness’. In fact, in many cases, a numerical procedure is said to be stable in this sense if it is well posed uniformly with respect to the relevant parameters, such as the dimension of the problem, grid-size, etc. In a second use, the term stability applies in connection with the long time behaviour of discretizations of time-dependent problems in ordinary or partial differential equations. The stability of discretizations of nonlinear differential equations, ordinary or partial, is the unifying theme of the present work. In Sections 4.2 to 4.8, we deal with the question of how best to define ‘stability’ of a nonlinear discretization so that the familiar ‘stability and consistency imply convergence’ holds.