An eighth-order exponentially fitted two-step hybrid method of explicit type for solving orbital and oscillatory problems

The construction of an eighth-order exponentially fitted (EF) two-step hybrid method for the numerical integration of oscillatory second-order initial value problems (IVPs) is considered. The EF two-step hybrid methods integrate exactly differential systems whose solutions can be expressed as linear combinations of exponential or trigonometric functions and have variable coefficients depending on the frequency of each problem. Based on the order conditions and the EF conditions for this class of methods, we construct an explicit EF two-step hybrid method with symmetric nodes and algebraic order eight which only uses seven function evaluations per step. This new method has the highest algebraic order we know for the case of explicit EF two-step hybrid methods. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order EF scheme is more efficient than other standard and EF two-step hybrid codes recently proposed in the scientific literature

A class of explicit high-order exponentially-fitted two-step methods for solving oscillatory IVPs

The derivation of new exponentially fitted (EF) modified two-step hybrid (MTSH) methods for the numerical integration of oscillatory second-order IVPs is analyzed. These methods are modifications of classical two-step hybrid methods so that they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(¿t), exp(-¿t)}, ¿¿C, or equivalently {sin(¿t), cos(¿t)} when ¿=i¿, ¿¿R, where ¿ represents an approximation of the main frequency of the problem. The EF conditions and the conditions for this class of EF schemes to have algebraic order p (with p=8) are derived. With the help of these conditions we construct explicit EFMTSH methods with algebraic orders seven and eight which require five and six function evaluation per step, respectively. These new EFMTSH schemes are optimal among the two-step hybrid methods in the sense that they reach a certain order of accuracy with minimal computational cost per step. In order to show the efficiency of the new high order explicit EFMTSH methods in comparison to other EF and standard two-step hybrid codes from the literature some numerical experiments with several orbital and oscillatory problems are presented

Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods

A class of explicit two-step hybrid methods for second-order IVPs

AbstractA class of explicit two-step hybrid methods for the numerical solution of second-order IVPs is presented. These methods require a reduced number of stages per step in comparison with other hybrid methods proposed in the scientific literature. New explicit hybrid methods which reach up to order five and six with only three and four stages per step, respectively, and which have optimized the error constants, are constructed. The numerical experiments carried out show the efficiency of our explicit hybrid methods when they are compared with classical Runge–Kutta–Nyström methods and other explicit hybrid codes proposed in the scientific literature

Exponentially fitted fifth-order two-step peer explicit methods

The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differential systems were introduced by R. Weiner et al [6, 7, 11, 12, 13] for solving different types of problems either in sequential or parallel computers. In this work, we study exponentially fitted three-stage peer schemes that are able to fit functional spaces with dimension six. Finally, some numerical experiments are presented to show the behaviour of the new peer schemes for some periodic problems

A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages

A two-step trigonometrically fitted semi-implicit hybrid method for solving special second order oscillatory differential equation

In this paper, we derived a semi-implicit hybrid method (SIHM) which is a two-step method to solve special second order ordinary differential equations (ODEs). The SIHM which is three-stage and fourth-order is then trigonometrically fitted and denoted by TF-SIHM3(4). The method is constructed using trigonometrically fitted properties instead of using phase-lag and amplification properties. Numerical integration show that TF-SIHM3(4) is more accurate in term of accuracy compared to the existing explicit and implicit methods of the same order

Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs

We consider time-averaging methods based on the Magnus series expansion jointly with exponential integrators for the numerical integration of general linear non-homogeneous differential equations. The schemes can be considered as averaged methods which transform, for one time step, a non-autonomous problem into an autonomous one whose flows agree up to a given order of accuracy at the end of the time step. The problem is reformulated as a particular case of a matrix Riccati differential equation and the Möbius transformation is considered, leading to a homogeneous linear problem. The methods proposed can be used both for initial value problems (IVPs) as well as for two-point boundary value problems (BVPs). In addition, they allow to use different approximations for different parts of the equation, e.g. the homogeneous and non-homogeneous parts, or to use adaptive time steps. The particular case of separated boundary conditions using the imbedding formulation is also considered. This formulation allows us to transform a stiff and badly conditioned BVP into a set of well conditioned IVPs which can be integrated using some of the previous methods. The performance of the methods is illustrated on some numerical examples. © 2012 IMACS. Published by Elsevier B.V. All rights reserved.We would like to thank the referees for their suggestions and comments that helped us to improve the presentation of the work as well as to clarify the main results. The authors acknowledge the support of the Generalitat Valenciana through the project GV/2009/032. The work of S.B. has also been partially supported by Ministerio de Ciencia e Innovacion (Spain) under the coordinated project MTM2010-18246-C03 (co-financed by the ERDF of the European Union) and the work of E.P. has also been partially supported by Ministerio de Ciencia e Innvacion of Spain, by the project MTM2009-08587.Blanes Zamora, S.; Ponsoda Miralles, E. (2012). Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs. Applied Numerical Mathematics. 62(8):875-894. doi:10.1016/j.apnum.2012.02.00187589462

On the Construction and Comparison of an Explicit Iterative Algorithm with Nonstandard Finite Difference Schemes

An explicit iterative algorithm to solve both linear and nonlinear problems of ordinary differential equations with initial conditions is formulated with main focus given on its comparison with some non-standard finite difference schemes. Two first order linear initial value problems (IVPs) with periodic behavior are used to analyze the performance of the proposed algorithm with respect to maximum absolute error and computational effort where proposed algorithm performs better in both cases. The proposed algorithm efficiently follows the oscillatory behavior of models like Lotka-Volterra predator-prey and mass-spring system (damped case) in comparison to the nonstandard schemes. All necessary computations have been carried out through MATLAB version 8.1 (R2013a) in double precision arithmetic. Numerical results obtained by the proposed algorithm are found to be computationally reliable and practical in comparison with two nonstandard finite difference schemes discussed in literature. Keywords: Iterative algorithm, nonstandard finite difference scheme, Initial conditions, Maximum absolute error

Two point block multistep methods with trigonometric−fitting for solving oscillatory problems

In this paper, we present the absolute stability of the existing 2-point implicit block multistep step methods of step number k = 3 and k = 5 and solving special second order ordinary differential equations (ODEs). The methods are then trigonometrically fitted so that they are suitable for solving highly oscillatory problems arising from the special second order ODEs. Their explicit counterparts are also trigonometrically fitted so that in the implementation the methods can act as a predictor-corrector pairs. The numerical results based on the integration over a large interval are given to show the performance of the proposed methods. From the numerical results we can conclude that the new trigonometrically-fitted methods are superior in terms of accuracy and execution time, compared to the existing methods in the scientific literature when used for solving problems which are oscillatory in nature