High-accuracy non-dissipative extended one-step methods for oscillatory problems

Abstract

Recently, Chawla and Al-Zanaidi [7] examined stability of the class of extended one-step methods introduced in [6] for the numerical integration of first-order initial value problems y′ = f(t y); y(t ₀) = n which possess oscillating solutions. They obtained non-dissipative methods of orders two to five, while the maximum possible order of an extended one-step method is seven [10]. In the present paper we investigate the existence of non-dissipative extended one-step methods of orders six and seven. We show that there exists a uniquely determined non-dissipative method of order six. We derive a two-parameter family of extended one-step methods of order seven; while there exists no non-dissipative method, we obtain a method of order seven which is “almost” non-dissipative. The present methods are computationally tested by an example.

Keywords:

• Initial-value problem
• ordinary differential equations
• oscillating solutions
• extended one-step methods
• non-dissipative methods

C.R. Categories:

• G.1.7