P-stable high-order super-implicit and Obrechkoff methods for periodic initial value problems

This paper discusses the numerical solution of periodic initial value problems. Two classes of methods are discussed, super-implicit and Obrechkoff. The advantage of Obrechkoff methods is that they are high-order one-step methods and thus will not require additional starting values. On the other hand they will require higher derivatives of the right-hand side. In cases when the right-hand side is very complex, we may prefer super-implicit methods. We develop a super-implicit P-stable method of order 12 and Obrechkoff method of order 18.