Numerical Methods for Systems of Initial Value Problems The State of the Art

State of the art software for initial value problems will be surveyed and recent developments and the current and future implications of these developments will be identified. Software libraries such as IMSL, NAG and SLATEC are now available at most major computing sites and, with a little guidance, the initial value software that is provided can prove invaluable to practitioners. Using routines from these libraries as examples we will identify the important problem characteristics and details of the programming environment that will determine the most appropriate method. In particular characteristics such as the form of the differential equations, whether it is stiff, the size of the system and the accuracy desired will be shown to be important. The extent to which numerical integration methods can be considered as interchangeable modules or ‘black boxes’ will also be discussed and future developments which should make this approach more feasible will be identified. On the other hand, the importance of special purpose methods which exploit the special structure of particular classes of problems will be acknowledged and examples of systems where this can be critical will be presented.