Backward Error Analysis and Structure Preservation

Abstract

The origin of backward error analysis dates back to the work of Wilkinson (1960) in numerical linear algebra. For the study of integration methods for ordinary differential equations, its importance was seen much later. The present chapter is devoted to this theory. It is very useful, when the qualitative behaviour of numerical methods is of interest, and when statements over very long time intervals are needed. The formal analysis (construction of the modified equation, study of its properties) gives already a lot of insight into numerical methods. For a rigorous treatment, the modified equation, which is a formal series in powers of the step size, has to be truncated. The error, induced by such a truncation, can be made exponentially small, and the results remain valid on exponentially long time intervals.

One of the greatest virtues of backward analysis ... is that when it is the appropriate form of analysis it tends to be very markedly superior to forward analysis. Invariably in such cases it has remarkable formal simplicity and gives deep insight into the stability (or lack of it) of the algorithm.

(J.H. Wilkinson, IMA Bulletin 1986)