Diagonally Implicit Runge–Kutta–Nyström Methods for Oscillatory Problems

Abstract

Listen

Translate article

Implicit Runge–Kutta–Nyström (RKN) methods are constructed for the integration of second-order differential equations possessing an oscillatory solution. Based on a linear homogeneous test model we analyse the phase errors (or dispersion) introduced by these methods and derive so-called dispersion relations. Diagonally implicit RKN methods of relatively low algebraic order are constructed, which have a high order of dispersion (up to 10). Application of these methods to a number of test examples (linear as well as nonlinear) yields a greatly reduced phase error when compared with “conventional” DIRKN methods.