Intervals of periodicity and absolute stability of explicit nyström methods fory″=f(x,y)

Abstract

We examine intervals of periodicity and absolute stability of explicit Nystrom methods fory″=f(x,y) by applying these methods to the test equationy″=−λ2y,λ>0. We consider in detail general families of fourth-order explicit Nystrom methods; necessary and sufficient conditions are given to characterize methods which possess non-vanishing intervals of periodicity and absolute stability. We establish closed-form expressions giving intervals of periodicity and/or absolute stability, in case these exist, for any fourth-order method. We then show that the methodM4 (1/6, 5/6) has the largest interval of periodicity out of all fourth-order methods; we also obtain the fourth-order method with the largest interval of absolute stability. The corresponding results for second and third-order explicit Nystrom methods are also included.