The stability of linear operators - Colburn, Hugh Edwin Geoffrey (Department of Mathematics and Applied Mathematics, 2016)
In the approximation and solution of both ordinary and partial differential equations by finite difference equations, it is well-known that for different ratios of the time interval to the spatial intervals widely differing solutions are obtained. This problem was first attacked by John von Neumann using Fourier analysis. It has also been studied in the context of the theory of semi-groups of operators. It seemed that the problem could be studied with profit if set in a more abstract structure. The concepts of the stability of a linear operator on a (complex) Banach space and the stability of a Banach sub-algebra of operators were formed in an attempt to generalize the matrix 2 theorems of H.O. Kreiss as applied to the L² stability problem. Chapter 1 deals with the stability and strict stability of linear operators. The equivalence of stability and convergence is discussed in Chapter 2 and special cases of the Equivalence Theorem are considered in Chapters 3 and 4. In Chapter 5 a brief account of the theory of discretizations is given and used to predict instability in non-linear algorithms.