Behaviour of a new type of Runge–Kutta methods when integrating satellite orbits (original) (raw)
An approach is treated for numerical integration of ordinary differential equations systems of the first order with choice of a computation scheme, ensuring the required local precision. The treatment is made on the basis of schemes of Runge-Kutta-Fehlberg type. Criteria are proposed as well as a method for the realization of the choice of an 'optimum' scheme. The effectiveness of the presented approach to problems in the field of satellite dynamics is illustrated by results from a numerical experiment. These results refer to a case when a satisfactory global stability of the solution for all treated cases is available. The effectiveness has been evaluated as good, especially when solving multi-variable problems in the sphere of simulation modelling. Comment: 8 pages, 2 figures
Hacettepe Journal of Mathematics and Statistics
In this paper, a trigonometrically-fitted Two Derivative Runge-Kutta method (TFTDRK) of high algebraic order for the numerical integration of first order Initial Value Problems (IVPs) which possesses oscillatory solutions is constructed. Using the trigonometrically-fitted property, a sixth order four stage Two Derivative Runge-Kutta (TDRK) method is designed. The numerical experiments are carried out with the comparison with other existing Runge-Kutta methods (RK) to show the accuracy and efficiency of the derived methods.
A precise Runge–Kutta integration and its application for solving nonlinear dynamical systems
Applied Mathematics and Computation, 2007
In this paper, precise integration is compounded with Runge-Kutta method and a new effective integration method is presented for solving nonlinear dynamical system. Arbitrary dynamical system can be expressed as nonhomogeneous linear differential equation problem. Precise integration is used to solve its corresponding homogeneous equation and Runge-Kutta methods can be used to calculate the nonhomogeneous, nonlinear items. The precise integration may have large time step and the time step of the Runge-Kutta methods can be adjusted to improve the computational precision. The handling technique in this paper not only avoids the matrix inversion but also improves the stability of the numerical method. Finally, the numerical examples are given to demonstrate the validity and effectiveness of the proposed method.
Sixth-order symmetric and symplectic exponentially fitted modified Runge–Kutta methods of Gauss type
Computer Physics Communications, 2008
The construction of exponentially fitted Runge-Kutta (EFRK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is considered. Based on the symplecticness, symmetry, and exponential fitting properties, two new threestage RK integrators of the Gauss type with fixed or variable nodes, are obtained. The new exponentially fitted RK Gauss type methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(λt), exp(−λt)}, λ ∈ C, and in particular {sin(ωt), cos(ωt)} when λ = iω, ω ∈ R. The algebraic order of the new integrators is also analyzed, obtaining that they are of sixth-order like the classical three-stage RK Gauss method. Some numerical experiments show that the new methods are more efficient than the symplectic RK Gauss methods (either standard or else exponentially fitted) proposed in the scientific literature.
A New Optimized Runge-Kutta-Nyström Method to Solve Oscillation Problems
2015
In this article, a new Runge-Kutta-Nyström method is derived. The new RKN method has zero phase-lag, zero amplification error and zero first derivative of phase-lag. This method is basically based on the sixth algebraic order Runge-Kutta-Nyström method, which has proposed by Dormand, El-Mikkawy and Prince. Numerical illustrations show that the new proposed method is much efficient as compared with other Runge-Kutta-Nyström methods in the scientific literature, for the numerical integration of oscillatory problems.
Obrechkoff versus super-implicit methods for the integration of Keplerian orbits
Astrodynamics Specialist Conference, 2000
This paper discusses the numerical solution of rst order initial value problems and a special class of second order ones (those not containing rst derivative). Two classes of methods are discussed, super-implicit and Obrechko . We will show equivalence of super-implicit and Obrechko schemes. The advantage of Obrechko methods is that they are high order one-step methods and thus will not require additional starting values. On the other hand they will require higher derivatives of the right hand side. In case the right hand side is complex, we m a y prefer super-implicit methods. The super-implicit methods may in general have a larger error constant, but one can get the same error constant for the cost of an extra future value.
Symmetric and symplectic exponentially fitted Runge-Kutta methods of high order
2010
The construction of high order symmetric, symplectic and exponentially fitted Runge-Kutta (RK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is analyzed. Based on the symplecticness, symmetry, and exponential fitting properties, three new four-stage RK integrators, either with fixed-or variable-nodes, are constructed. The algebraic order of the new integrators is also studied, showing that they possess eighth-order of accuracy as the classical four-stage RK Gauss method. Numerical experiments with some oscillatory test problems are presented to show that the new methods are more efficient than other symplectic four-stage eighth-order RK Gauss codes proposed in the scientific literature.
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.
Acceleration of Runge-Kutta integration schemes
Discrete Dynamics in Nature and Society, 2004
A simple accelerated third-order Runge-Kutta-type, fixed time step, integration scheme that uses just two function evaluations per step is developed. Because of the lower number of function evaluations, the scheme proposed herein has a lower computational cost than the standard third-order Runge-Kutta scheme while maintaining the same order of local accuracy. Numerical examples illustrating the computational efficiency and accuracy are presented and the actual speedup when the accelerated algorithm is implemented is also provided.
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.