Optimally stable fifth-order integration methods: a numerical approach

In this paper, the results of the first detailed and systematic study of the family of fifth order implicit linear multistep methods requiring function evaluations at four backpoints are given. Also described is a practical and efficient nonlinear optimization procedure which made it possible to locate in a precise manner the methods of this type which possess optimal ranges of relative stability in the sense-that the corresponding relative stability regions contain maximal disks centered at the origin.