Narrow Results

Fourth order exponential and trigonometric fitted Runge–Kutta methods are developed in this paper. They are applied to problems involving the Schrödinger equation and to other related problems. Numerical results show the superiority of these methods over conventional fourth order Runge–Kutta methods. Based on the methods developed in this paper, a variable-step algorithm is proposed. Numerical experiments show the efficiency of the new algorithm.

In this paper we describe procedures for the construction of efficient methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions. Based on the described procedures we develop two simple and efficient multistep methods for the solution of the above problems. The first method is exponentially-fitted and trigonometrically-fitted and the second has a minimal phase-lag. Both methods are symmetric. Numerical results obtained for several well known problems show the efficiency of the new methods when they are compared with known methods in the literature.

In this paper a dissipative trigonometrically-fitted two-step explicit hybrid method is developed. This method is based on a dissipative explicit two-step method developed recently by Papageorgiou, Tsitouras and Famelis.⁶ Numerical examples show that the procedure of trigonometrical fitting is the only way in one to produce efficient dissipative methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions.

In this paper we present a family of explicit Runge–Kutta methods of 5th algebraic order, one of which has variable coefficients, for the efficient solution of problems with oscillating solutions. Emphasis is placed on the phase-lag property in order to show its importance with regards to problems with oscillating solutions. Basic theory of Runge–Kutta methods, phase-lag analysis and construction of the new methods are described. Numerical results obtained for known problems show the efficiency of the new methods when they are compared with known methods in the literature. Furthermore we note that the method with variable coefficients appears to have much higher accuracy, which gets close to double precision, when the product of the frequency with the step-length approaches certain values. These values are constant and independent of the problem solved and depend only on the method used and more specifically on the expressions used to achieve higher algebraic order.

Exponentially and trigonometrically fitted third algebraic order Runge–Kutta methods for the numerical integration of the Schrödinger equation are developed in this paper. Numerical results obtained for several well known problems show the efficiency of the new methods.

We introduce Runge–Kutta (RK) methods for Retarded Functional Differential Equations (RFDEs). With respect to RK methods (A, b, c) for Ordinary Differential Equations the weights vector b ∈ ℝ^s and the coefficients matrix A ∈ ℝ^s×s are replaced by ℝ^s-valued and ℝ^s×s-valued polynomial functions b(·) and A(·) respectively. Such methods for RFDEs are different from Continuous RK (CRK) methods where only the weights vector is replaced by a polynomial function. We develop order conditions and construct explicit methods up to the convergence order four.