Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations (original) (raw)

A New Optimized Runge-Kutta Method for Solving Oscillatory Problems

International Journal of Pure and Apllied Mathematics, 2016

A new explicit Runge-Kutta method of fifth algebraic order is developed in this paper, for solving second-order ordinary differential equations with oscillatory solutions. The new method has zero phase-lag, zero amplification error and zero first derivative of the phaselag. Numerical results show that the new proposed method is more efficient as compared with other Runge-Kutta methods in the scientific literature, for the numerical integration of oscillatory problems.

Solving Oscillatory Problems Using an Optimized Runge–Kutta Method

2018

New explicit Runge–Kutta method with zero phase-lag, zero first derivative of the phase-lag and zero amplification error is derived for the effective numerical integration of second-order initial-value problems with oscillatory solutions in this paper. The new method is based on the sixth-stage fifth-order Runge–Kutta method. Numerical illustrations show that the new proposed method is much efficient as compared with other Runge–Kutta methods in the scientific literature, for the numerical integration of oscillatory problems.

New Phase-Fitted and Amplification-Fitted Modified Runge-Kutta Method for Solving Oscillatory Problems

Global Journal of Pure and Applied Mathematics, 2016

In this paper, a new phase-fitted and amplification-fitted modified Runge-Kutta (MRK) method is constructed to solve first-order ordinary differential equations with oscillatory solutions. This new method is based on the Runge-Kutta Zonn-eveld method with fourth algebraic order. The numerical results for the new method have been compared with other existing methods. Findings have shown that the new method is more efficient than the other existing methods.

A New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution of Oscillatory Problems

Applied Mathematical Sciences, 2016

In this work, a new way for constructing an efficiently modified Runge-Kutta (RK) method to solve first-order ordinary differential equations with oscillatory solutions is provided. The proposed method solves the first-order ODEs by first converting the second order ODEs to an equivalent first-order ODEs. The method of the embedded has algebraic orders five and four. The numerical results of the new method have been compared with those of existing methods and showed that the new method is more efficient.

P A NEW EFFICIENT PHASE-FITTED AND AMPLIFICATION-FITTED RUNGE-KUTTA METHOD FOR OSCILLATORY PROBLEMS

International Journal of Pure and Applied Mathematics, 2016

A new Runge-Kutta (RK) method is constructed to solve first-order differential equations with oscillatory solutions. This new method is based on the Runge-Kutta method of order four with seven-stage. Numerical tests are performed, and the results of the new method is compared with the existing methods. The numerical results show that the new method is more efficient.

Embedded 5(4) Pair Trigonometrically-Fitted Two Derivative Runge-Kutta Method with FSAL Property for Numerical Solution of Oscillatory Problems

2017

Based on First Same As Last (FSAL) technique, an embedded trigonometrically-fitted Two Derivative Runge-Kutta method (TDRK) for the numerical solution of first order Initial Value Problems (IVPs) is developed. Using the trigonometrically-fitting technique, an embedded 5(4) pair explicit fifth-order TDRK method with a “small” principal local truncation error coefficient is derived. The numerical experiments are carried out and showed that our new method is more accurate and efficient when compared with other existing Runge-Kutta (RK) and TDRK methods of the same order.

New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems

Abstract and Applied Analysis, 2013

Two new Runge-Kutta-Nyström (RKN) methods are constructed for solving second-order differential equations with oscillatory solutions. These two new methods are constructed based on two existing RKN methods. Firstly, a three-stage fourth-order Garcia's RKN method. Another method is Hairer's RKN method of four-stage fifth-order. Both new derived methods have two variable coefficients with phase-lag of order infinity and zero amplification error (zero dissipative). Numerical tests are performed and the results show that the new methods are more accurate than the other methods in the literature.

Explicit Runge Kutta method with trigonometrically fitted for solving first order ODEs

AIP Conference Proceedings, 2016

A new fourth-order explicit Runge-Kutta method for solving first order ordinary differential equations AIP Conf. Abstract. In this note, an explicit trigonometrically-fitted (RK) method is developed to determine the approximate solution of the first-order IVPs with oscillatory solution. The proposed method solves first order ODEs by first converting the second order ODEs to an equivalent first order; which is based on the RK method of order four. The numerical experiment performed shows the efficacy of our newly developed method.

Two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems

Applied Mathematics and Computation, 2015

The construction of new embedded pairs of explicit Runge-Kutta methods specially adapted to the numerical solution of oscillatory problems is analyzed. Based on the order conditions for this class of methods, two new embedded pairs of orders 4(3) and 6(4) which require five and seven stages per step, respectively, are constructed. The derivation of the new embedded pairs is carried out paying special attention to the minimization of the principal term of the local truncation error as well as the dispersion and dissipation errors of the higher order formula. Several numerical experiments are carried out to show the efficiency of the new embedded pairs when they are compared with some standard and specially adapted pairs proposed in the scientific literature for solving oscillatory problems.

Explicit Runge-Kutta methods for initial value problems with oscillating solutions

Journal of Computational and Applied Mathematics, 1996

New pairs of embedded Runge-Kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained. These pairs have been derived taking into account not only the usual properties of accuracy, stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by . Three nine-stage embedded pairs of Runge-Kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and/or dissipation are selected among the members of a family of pairs depending on several free parameters. Some numerical results are presented to show the efficiency of the new methods.