STABILITY AND CONVERGENCE OF FINITE DIFFERENCE SCHEMES WITH SINGULAR COEFFICIENTS.
Abstract
A technique is developed for the numerical analysis of well posed initial value problems containing singular coefficients. To do this, the single Banach space utilized in the Lax-Richtmyer theory has been replaced by a sequence of finite-dimensional Banach spaces. For each of these spaces introduces the mean p-th power norm and define convergence with respect to this sequence as an increment of t approaches zero. One finds that strong stability of the finite difference operators implies convergence while weak stability of order 2/p is a necessary condition for convergence. It is shown that every scalar first order initial value problem can be approximated by a difference scheme possessing a certain invariant subspace of a fixed dimension. For such schemes a useful sufficient condition has been developed for strong stability and the results are applied to the m-dimensional, spherically symmetric diffusion equation u sub t equals u sub rr + (m-1)r/10 u sub r. It is found that for three-point centered space differences, strong stability in the maximum norm can be established if m is even and lambda equals in increment of t over an increment of r squared is less than 1/2. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1965
- Accession Number
- AD0475271
Entities
People
- Dennis Eisen
Organizations
- Adelphi University