Runge-Kutta-Nystrom Research Articles

Relaxation RKN-type integrators that preserve two invariants for second-order (oscillatory) systems

Recently, the relaxation technique has been widely used to impose conservation of invariants while retaining the full accuracy of the original method. So far, only a single invariant of a system has been considered. In this work, by a mild generalization of the relaxation technique, the Runge–Kutta–Nyström (RKN) integrators are modified to preserve two invariants for second-order system of Ordinary Differential Equations (ODEs). The proposed integrators can be explicit and of arbitrarily high order. The accuracy of the relaxation RKN integrators and the existence of valid relaxation parameters have been proved. The construction of the new integrators is under the framework of adapted RKN (ARKN) integrators which are specially designed for numerical solving second-order oscillatory systems. Therefore, the proposed integrators could be oscillation-preserving in the sense that they exactly integrate homogeneous oscillatory system q′′+Kq=0. Some numerical experiments are conducted to show the advantage and efficiency of the proposed integrators in comparison with the standard (A)RKN integrators.

Embedded exponential Runge–Kutta–Nyström methods for highly oscillatory Hamiltonian systems

Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from O(N2) to O(Nlog⁡N). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs.

Nine-Stage Runge–Kutta–Nyström Pairs Sharing Orders Eight and Six

We explore second-order systems of non-stiff initial-value problems (IVPs), particularly those cases where the first derivatives are absent. These types of problems are of significant interest and have applications in various domains, such as astronomy and physics. Runge–Kutta–Nyström (RKN) pairs stand out as highly effective methods of addressing these IVPs. In order to create a pair with eighth and sixth orders, we need to address a certain known set of equations concerning the coefficients. When constructing such pairs for use in double-precision arithmetic, we often need to meet various conditions. Primarily, we aim to maintain small coefficient magnitudes to prevent a loss of accuracy. Nevertheless, in the context of quadruple precision, we can tolerate larger coefficients. This flexibility enables us to establish pairs with eighth and sixth orders that exhibit significantly reduced truncation errors. Traditionally, these pairs are constructed to go through eight stages per step. Here, we propose using nine stages per step. Then we have available more coefficients in order to further reduce truncation errors. As a result, we construct a novel pair that, as anticipated, achieves superior performance compared to equivalent-order pairs in various significant problem scenarios.

Development of high-order adaptive multi-step Runge–Kutta–Nyström method for solving special second-order ODEs

Runge–Kutta–Nyström (RKN) methods are extensively used to obtain approximate solutions of ordinary differential equations (ODEs). Specifically, they are widely used to directly solve second-order ODEs of the special form. Although the derivation of new higher-order methods with fewer numbers of function evaluations is of great importance in increasing the precision and effectiveness of the methods, however, this is rarely done due to the difficulty or complexity of some derivations. This study focuses on constructing a 7(5) pair of embedded multi-step Runge–Kutta–Nyström (EMSN) method with lower stages for the numerical solutions of special second-order ODEs. An adaptive step size formulation using an embedded procedure is considered, and the numerical findings reveal that the new embedded pair outperforms existing Runge–Kutta (RK) pairs in terms of the minimum number of functions evaluations.

Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs

Runge–Kutta–Nyström (RKN) methods for the numerical solution of inhomogeneous linear initial value problems with constant coefficients are considered.A general procedure to construct explicit s-stage RKN methods with maximal order p=s+1, similar to the developed by the authors (Montijano et al., 2023) for the class of second order IVP under consideration, depending on the nodes ci,i=1,…,s is presented. This procedure requires only the solution of successive linear equations in the elements aij, 1≤j<i≤s of the matrix of coefficients A of the RKN method and avoids the solution of non linear equations.The remarkable fact is that using as free parameters the nodes ci,i=1,…,s with a quadrature relation, the s(s−1)/2 elements of matrix A can be computed by solving successively linear systems with coefficients depending on the nodes, so that if they are non-singular we get a unique s-stage method with maximal order s+1.We obtain an optimized six-stage seventh-order RKN method in the sense that the nodes are chosen so that minimize the leading term of the local error. Finally, some numerical experiments are presented to test the behaviour of the optimized RKN method with others with Radau and Lobatto nodes.

Economical handling of Runge–Kutta–Nyström step rejection

Runge–Kutta–Nyström (RKN) pairs are amongst the most popular methods for solving numerically Initial Value Problems of second order. Here we consider the special case with first derivative absent from the right hand side of the classical definition of this problem. While using a RKN pair, we may experience rejection of some steps through the interval of integration. Traditionally, the whole amount of evaluations are then dropped and we proceed with a completely new round of computations. In this work we propose avoiding this waste and continue by using again the rejected RKN stages. We especially focus on a RKN pair of orders six and four. After a step rejection we use again all the previously evaluated stages and add only one new stage. We proceed by evaluating the output in a smaller step. By this technique we manage to reduce significantly the cost in a set of problems that are known to pose difficulties in RKN step size changing algorithms.

Error analysis of modified Runge–Kutta–Nyström methods for nonlinear second-order delay boundary value problems

This paper is concerned with the numerical solutions of nonlinear second-order boundary value problems with time-variable delay. By adapting Runge–Kutta–Nyström (RKN) methods and combining Lagrange interpolation, a class of modified RKN (MRKN) methods are suggested for solving the problems. Under some suitable conditions, MRKN methods are proved to be convergent of order min{p,q}, where p,q are the local orders of MRKN methods and Lagrange interpolation, respectively. Numerical experiments further confirm the computational effectiveness and accuracy of MRKN methods.

Runge–Kutta–Nyström Pairs of Orders 8(6) for Use in Quadruple Precision Computations

The second-order system of non-stiff Initial Value Problems (IVP) is considered and, in particular, the case where the first derivatives are absent. This kind of problem is interesting since since it arises in many significant problems, e.g., in Celestial mechanics. Runge–Kutta–Nyström (RKN) pairs are perhaps the most successful approaches for solving such type of IVPs. To achieve a pair attaining orders eight and six, we have to solve a well-defined set of equations with respect to the coefficients. Here, we provide a simplified form of these equations in a robust algorithm. When creating such pairings for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients’ magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders eight and six with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems.

Runge–Kutta–Nyström methods of eighth order for addressing Linear Inhomogeneous problems

Second order Linear Inhomogeneous Initial Value Problems with constant coefficients are considered here. Runge–Kutta–Nyström (RKN) methods are amongst the most effective for addressing these problems. For attaining a certain algebraic order of accuracy, we need to solve a reduced set of order conditions (in comparison with conventional RKN) with respect to the coefficients. Here, using a seven stages method we manage to solve these equations up to eighth order. Differential Evolution technique was used for achieving this. Thus, our main contribution is that we saved one stage since conventional eight order RKN methods use eight stages for advancing a step. This new method is combined at no cost with a fifth and third order companion formulas. As consequence, a triplet of orders 8(5)3 is given for estimating local errors and adjust effectively the step lengths. The new triplet of methods is applied to various relevant problems and is found to outperform standard RKN pairs from the literature.

Runge–Kutta–Nyström symplectic splitting methods of order 8

Different families of Runge–Kutta–Nyström (RKN) symplectic splitting methods of order 8 are presented for second-order systems of ordinary differential equations and are tested on numerical examples. They show a better efficiency than state-of-the-art symmetric compositions of 2nd-order symmetric schemes and RKN splitting methods of orders 4 and 6 for medium to high accuracy. For some particular examples, they are even more efficient than extrapolation methods for high accuracies and integrations over relatively short time intervals.

A trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström pair to solve oscillating systems

In this study, a trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström (RKN) pair with six stages is formulated, considering a previous method developed by El‐Mikkawy and Rahmo. The obtained adapted pair integrates exactly the usual test equation: . The local truncation error of the new method is presented, showing that the algebraic order of the original method is maintained. The periodicity interval of the new method is computed, showing that the developed method is “almost” P‐stable. The numerical examples considered clearly show the superiority of the new developed embedded pair over other RKN methods of algebraic orders 6(4) with six stages appeared in the literature.

New family for Runge‐Kutta‐Nyström pairs of orders 6(4) with coefficients trained to address oscillatory problems

In this paper, we consider the integration of the special second‐order initial value problem (IVP) by Runge‐Kutta‐Nyström (RKN) pairs of algebraic orders 6 and 4. For achieving this, we have to solve a certain set of order conditions. The main contribution of the present work is that we introduce a new methodology of solving them that possesses nine free parameters instead of four used by similar pairs until now. This family uses the FSAL (first stage as last) devise. Selecting these extra coefficients properly we may construct methods with better properties. Here, we exploit the free parameters in order to derive a pair with coefficients trained to furnish better results on problems with periodic solutions. Extensive numerical tests justify our approach.

Higher-order symplectic integration techniques for molecular dynamics problems

The explicit symplectic difference schemes with a number of stages from 1 to 5 are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using Gröbner bases is proposed. It is shown that it is possible to significantly increase the accuracy of symplectic schemes without increasing the number of stages by obtaining all possible real schemes with the aid of Gröbner bases. For the numbers of stages 2, 3, and 4, the parameters of Runge–Kutta–Nyström (RKN) schemes and partitioned Runge–Kutta (PRK) schemes are obtained using the Gröbner basis technique. It is shown that there exists only one explicit invertible (symmetric) two-stage RKN scheme. In the cases of the vanishing Vandermonde determinant, 20 new real four-stage RKN schemes and 18 new PRK schemes of Forest–Ruth and Yoshida were found in the analytic form with the aid of Gröbner bases. In the cases of the nonzero Vandermonde determinant, four symmetric four-stage RKN schemes were derived in the analytic form with the aid of the Gröbner basis technique. In addition, for the number of stages 4 and 5, new nonsymmetric schemes were found using the Nelder–Mead numerical optimization method. In particular, four new schemes were obtained for the number of stages 4. For the number of stages 5, three new schemes were obtained in addition to four schemes known earlier. For each specific number of stages, a scheme has been found that is the best in terms of the minimum of the leading term of the approximation error. The relations have been established between the polynomials entering the expansions of the errors of each scheme for the momentum and for the particle coordinate. Verification of the schemes was carried out on a problem that has an exact solution. It is shown that the symplectic five-stage RKN scheme provides a more accurate conservation of the total energy balance of the particle system than schemes of lower orders of accuracy. The stability studies of the schemes were performed using the Mathematica software package.

Collocation Runge–Kutta–Nyström methods for solving second-order initial value problems

ABSTRACT A class of modified collocation Runge–Kutta–Nyström (RKN) methods for solving second-order initial value problem , , has been formulated and studied in the paper. The new methods are applicable to a larger class of second-order initial value problems compared to the classical collocation RKN methods and they reduce to the classical methods when is absent from the equation. Superconvergence for the new methods is attained when the set of collocation points satisfies orthogonality conditions. We proved that an s-stage modified collocation RKN method is of accuracy order of at least s for any set of collocation parameters and at most when are Gauss points. The stability function and the stability of some modified RKN methods have also been investigated. Numerical experiments are included to demonstrate the advantage of the new methods.

A DIAGONALLY IMPLICIT RUNGE-KUTTA-NYSTROM (RKN) METHOD FOR SOLVING SECOND ORDER ODES ON PARALLEL COMPUTERS

In this paper, diagonally implicit Runge-Kutta-Nystrom (RKN) method of high-order for the numerical solution of second order ordinary differential equations (ODE) possessing oscillatory solutions to be used on parallel computers is constructed. The method has the properties of minimized local truncation error coefficients as well as possessing non-empty interval of periodicity, thus suitable for oscillatory problems. The method was tested with standard test problems from the literature and numerical results compared with the analytical solution to show the advantage of the algorithm

Oscillation-preserving algorithms for efficiently solving highly oscillatory second-order ODEs

In the last few decades, Runge-Kutta-Nystrom (RKN) methods have made significant progress and the study of RKN-type methods for solving highly oscillatory differential equations has received a great deal of attention. In this paper, from the point of view of geometric integration, the oscillation-preserving behaviour of the existing RKN-type methods in the literature are analysed in detail. To this end, it is convenient to introduce the concept of oscillation preservation for RKN-type methods. It turns out that if both the internal stages and the updates of an RKN-type method respect the qualitative and global features of the highly oscillatory solution, then the method is oscillation preserving. Since the internal stages of standard RKN and adapted RKN (ARKN) methods are inimical to the oscillation-preserving structure, neither ARKN methods nor the symplectic and symmetric RKN methods, and standard RKN methods are oscillation preserving. Other concerns relating to oscillation preservation are also considered. In particular, we are concerned with the computational issues for efficiently solving semi-discrete wave equations such as semi-discrete Klein-Gordon (KG) equations and damped sine-Gordon equations. The results of numerical experiments show the importance of the oscillation-preserving property for an RKN-type method and the remarkable superiority of oscillation-preserving integrators when applied to nonlinear multi-frequency highly oscillatory systems.

Volume-preserving exponential integrators and their applications

It is well known that various dynamical systems preserve volume in phase space such as all Hamiltonian systems. This qualitative geometrical property of the analytical solution should be respected in the sense of Geometric Integration. This paper studies the volume-preserving property of exponential integrators for different vector fields. For exponential integrators, we first derive a necessary and sufficient condition of volume preservation. Then based on this condition, volume-preserving exponential integrators are discussed in detail for four kinds of vector fields. It is shown that symplectic exponential integrators can be volume preserving for a much larger class of vector fields than Hamiltonian systems. On the basis of the analysis, some applications of volume-preserving exponential integrators are discussed. For solving highly oscillatory second-order systems, novel volume-preserving exponential integrators are derived, and for separable partitioned systems, extended Runge–Kutta–Nyström (ERKN) integrators of volume preservation are presented. Moreover, the volume preservation of Runge–Kutta–Nyström (RKN) methods is also discussed. Five illustrative numerical experiments are carried out to demonstrate the notable superiority of volume-preserving exponential integrators in comparison with volume-preserving Runge-Kutta methods.

The performance of the N-body integrator SSS

The integrator SSS performs accurate N-body simulations of the Solar System when there is a mix of massive bodies and test particles. The orbital motion of all bodies at all times is integrated using a 12-10 explicit Runge-Kutta Nystrom (RKN) pair. The test particles are divided into sets and each set integrated on a different processor. The explicit RKN pair uses an order 12 interpolant for the position and velocity when checking for collisions. We report on two significant improvements to SSS. The first improvement reduced the local round-off error in interpolated values by approximately four orders of magnitude, permitting more accurate modelling of collisions. The technique used to reduce the round-off error can be applied to other high-order interpolants. The second improvement is hand optimization of the implementation of SSS. This optimization increased the speed of SSS by approximately 60%, permitting more accurate modelling through the use of more test particles. We also present a summary of the numerical performance of SSS on a simulation of the Sun, the planets Earth to Neptune, and 500,000 test particles over 100 million years.

Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations y′′(x)=f(x,y,y′)

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y′′(x)=f(x,y,y′) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.

Approximate solutions for the nonlinear duffing oscillator with an external force

This paper is concerned with a finding of an approximate solution to the Duffing oscillator with a linear external force using a modified differential transform method (MDTM) and Runge-Kutta-Nystrom (RKN) method. The results obtained from the two methods are then numerically and graphically compared. We found that both results are close to one another when an amplitude and frequency of external forces are very small. However, both results are quite different when the amplitude and frequency of external forces are large. Their results are in agreement with the phase plane diagram which demonstrates a stable equilibrium point for both methods in case of small amplitude and frequency of external force. However, for large amplitude and frequency of external forces, only MDTM shows the stable equilibrium point.