Eighth Algebraic Order Research Articles

Eighth order, Numerov-like schemes with coefficients tailored for superior performance on ODE systems with oscillatory solutions

<p>Second order Ordinary Differential Equations (ODE) were considered. Numerov-like techniques employing effectively seven stages per step and sharing eighth algebraic order were under examination for numerically solving them. The coefficients of these methods were contingent on four independent parameters. To tackle issues with oscillatory solutions, we typically aimed to fulfill specific criteria such as minimizing phase-lag, expanding the periodicity interval, or even neutralizing amplification errors. These latter attributes stemmed from a test problem mimicking an ideal trigonometric trajectory. Here, we suggested training the coefficients of the chosen method family across a broad spectrum of pertinent problems. Following this training using the differential evolution method, we identified a particular method that surpassed others in this category across an even broader array of oscillatory problems.</p>

A new family of 7 stages, eighth‐order explicit Numerov‐type methods

In this paper, we consider the integration of the special second‐order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge‐Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase‐lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.

An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems

In this paper we develop a new implicit eighth algebraic order symmetric six-step method. For this method we request for the first time in the literature vanishing of the phase-lag and its first, second, third and fourth derivatives. The investigation of the new method consists of the following: the development of the method, i.e. the production of the coefficients of the method in order the phase-lag and the derivatives of the phase-lag to be vanished, the computation of the formula of the Local Truncation Error, the application of the new obtained method to a test problem (the radial Schrodinger equation) and the production of the asymptotic form of the local truncation for this test problem the comparison of the asymptotic forms of the local truncation error of known similar methods with the asymptotic form local truncation error of of the new developed method (comparative local truncation error analysis) the stability investigation of the new produced method. This investigation is taken place by using a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the produced results i.e. by studying the interval of periodicity of the developed method. Finally we will apply the new developed method to the approximate solution of the resonance problem of the radial Schrodinger equation. The above mentioned application will help us on the study of the efficiency of the new obtained method. We will test the efficiency of the produced method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.

Family of symmetric linear six-step methods with vanished phase-lag and its derivatives and their application to the radial Schrödinger equation and related problems

A new family of implicit eighth algebraic order symmetric six-step methods is obtained in this paper. For this family, we request for the first time in the literature vanishing of the phase-lag and its derivatives. For this new family of methods, we will investigate the following: (1) the construction of the methods of the family , i.e. the definition of the coefficients of each method of the family in order the phase-lag and its derivatives to be vanished, (2) the calculation of the formula of the Local Truncation Error, (3) the comparative local truncation error analysis (i.e. the application methods of the family on a test problem and the investigation of their behaviors), (4) the stability analysis of the methods of the new proposed family, by applying the methods of the family to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by investigating the obtained results i.e. by studying the interval of periodicity of the obtained methods of the family. We finally investigate the behavior the methods of the new family by applying the new methods to the numerical solution of the resonance problem of the radial Schrodinger equation. We prove the efficiency of the obtained methods of the new family by comparing it with (1) well known methods in the literature and (2) very recently obtained methods.

Trigonometrically fitted two-step obrechkoff methods for the numerical solution of periodic initial value problems

In this paper, we present a new two-step trigonometrically fitted symmetric Obrechkoff method. The method is based on the symmetric two-step Obrechkoff method, with eighth algebraic order, high phase-lag order and is constructed to solve IVPs with periodic solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. The numerical results obtained by the new method for some problems show its superiority in efficiency, accuracy and stability.

An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

Our new linear symmetric semi-embedded predictor–corrector method (SEPCM) presented here is based on the multistep symmetric method of Quinlan and Tremaine (1990), with eight steps and eighth algebraic order and constructed to solve numerically the two-dimensional Kepler problem. It can also be used to integrate related IVPs with oscillatory solutions for which the frequency is unknown. Firstly we present a SEPCM (see Panopoulos and Simos, 2013 [36,37]) in pair form. This form has the advantage that reduces the computational expense. From this form we construct a new symmetric eight-step method. The new scheme has constant coefficients and algebraic order ten. We tested the efficiency of our newly developed scheme against some well known methods from the literature. We measure the efficiency of the methods and conclude that the new scheme is the most efficient of all the compared methods and for all the problems solved.

A high algebraic order predictor–corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems

In this paper an eighth algebraic order predictor–corrector explicit four-step method is studied. The main scope of this paper is to study the consequences of (1) the vanishing of the phase-lag and its first, second, third and fourth derivatives and (2) the high algebraic order on the efficiency of the new developed method. A theoretical and computational study of the obtained method is also presented. More specifically, the theoretical study of the new predictor–corrector method consists of: Finally, the computational study of the new predictor–corrector method consists of the application of the new produced predictor–corrector explicit four-step method to the numerical solution of the resonance problem of the radial time independent Schrodinger equation.

A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation

A low computational cost eighth algebraic order hybrid two-step method with vanished phase-lag and its first, second, third and fourth derivatives is developed in this paper. We also investigate the local truncation error, the stability and the result of the elimination of the phase-lag and its derivatives on the effectiveness of the produced method.

A New Optimized Symmetric Embedded Predictor- Corrector Method (EPCM) for Initial-Value Problems with Oscillatory Solutions

In this work a new optimized symmetric eight-step embedded predictor-corrector method (EPCM) with minimal phase-lag and algebraic order ten is presented. The method is based on the symmetric multistep method of Quinlan-Tremaine (1), with eight steps and eighth algebraic order and is constructed to solve numerically IVPs with oscillatory solutions. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new optimized method with minimal phase-lag is noticeably most efficient of all the compared methods and for all the problems solved including the two-dimensional Kepler problem and the radial Schr ¨ odinger equation.

A new phase-fitted eight-step symmetric embedded predictor–corrector method (EPCM) for orbital problems and related IVPs with oscillating solutions

Our new phase-fitted embedded predictor–corrector method (EPCM) presented here is based on the multistep symmetric method of Quinlan–Tremaine (1990), with eight steps and eighth algebraic order and constructed to solve numerically the two-dimensional Kepler problem. It can also be used to integrate other orbital problems and related IVPs with oscillatory solutions. First we present a EPCM (Panopoulos et al. (2011) and Panopoulos et al. (2013)) pair form. From this form we construct a new eight-step method. The new scheme has algebraic order ten and infinite order of phase-lag. We tested the efficiency of our newly developed scheme against to some recently constructed optimized methods and other well known methods from the literature. We measure the efficiency of the methods and conclude that the new scheme is noticeably more efficient of all the compared methods and for all the problems solved, including the radial Schrödinger equation.

A new optimized symmetric 8-step semi-embedded predictor–corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems

In this work we introduce a new predictor-corrector (PC) pair form (SEPCM) for the numerical integration of second-order initial-value problems and a new optimized eight-step symmetric predictor-corrector method with minimal phase- lag and algebraic order ten is constructed. The new method is based on the multistep symmetric method of Quinlan-Tremaine (Astron. J. 100(5):1694-1700, 1990), with eight steps and eighth algebraic order and constructed to solve numerically the radial time-independent Schrodinger equation during the resonance problem with the use of the Woods-Saxon potential. It can also be used to integrate related IVPs with oscilla- tory solutions such as orbital problems. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new optimized method is the most efficient of all the compared methods and for all the problems solved.

A NEW EIGHT-STEP SYMMETRIC EMBEDDED PREDICTOR-CORRECTOR METHOD (EPCM) FOR ORBITAL PROBLEMS AND RELATED IVPs WITH OSCILLATORY SOLUTIONS

A new multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new eight-step symmetric embedded predictor-corrector method is constructed. The new PC method is based on the multistep symmetric method of Quinlan & Tremaine, with eight steps and eighth algebraic order, and is constructed to solve numerically the N-body problem. The new integrator has algebraic order 10 and it can be used to solve problems, for which the frequency is not known. We investigate the behavior of the new algorithm by integrating the five outer-planet problem and the two-body problem with various eccentricities. Regarding the five outer-planet problem, we calculate the error of the integrator in the solution, the Hamiltonian, and the phase after forward and backward integration over various intervals that are multiples of the period of Jupiter.

An Optimized Symmetric 8-Step Semi-Embedded Predictor-Corrector Method for IVPs with Oscillating Solutions

In this paper we present a new optimized symmetric eight-step semi-embedded predictor-corrector method (SEPCM) with minimal phase-lag. The method is based on the symmetric multistep method of Quinlan-Tremaine (1), with eight steps and eighth algebraic order and is constructed to solve IVPs with oscillating solutions. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions

In this paper we present a new optimized symmetric eight-step predictor-corrector method with phase-lag of order infinity (phase-fitted). The method is based on the symmetric multistep method of Quinlan–Tremaine, with eight steps and eighth algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation during the resonance problem with the use of the Woods–Saxon potential. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.

EXPLICIT EIGHTH ORDER TWO-STEP METHODS WITH NINE STAGES FOR INTEGRATING OSCILLATORY PROBLEMS

We present a new explicit hybrid two step method for the solution of second order initial value problem. It costs only nine function evaluations per step and attains eighth algebraic order so it is the cheapest in the literature. Its coefficients are chosen to reduce amplification and phase errors. Thus the method is well suited for facing problems with oscillatory solutions. After implementing a MATLAB program, we proceed with numerical tests that justify our effort.

Stage reduction on P-stable Numerov type methods of eighth order

We present an implicit hybrid two step method for the solution of second order initial value problem. It costs only six function evaluations per step and attains eighth algebraic order. The method satisfy the P-stability property requiring one stage less. We conclude dealing with implementation issues for the methods of this type and give some first pleasant results from numerical tests.

Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation

Multiderivative methods with minimal phase-lag are introduced in this paper, for the numerical solution of the one-dimensional Schrödinger equation. The methods are called multiderivative since they use derivatives of order two, four or six. Numerical application of the newly introduced method to the resonance problem of the one-dimensional Schrödinger equation shows its efficiency compared with other similar well-known methods of the literature.

A generator of dissipative methods for the numerical solution of the Schrödinger equation

In this paper a generator of hybrid methods with minimal phase-lag is developed for the numerical solution of the Schrodinger equation and related problems. The generator's methods are dissipative and are of eighth algebraic order. In order to have minimal phase-lag with the new methods, their coefficients are determined automatically. Numerical results obtained by their application to some well known problems with periodic or oscillating solutions and to the coupled differential equations of the Schrodinger type indicate the efficiency of these new methods.

Symmetric Eighth Algebraic Order Methods with Minimal Phase-Lag for the Numerical Solution of the Schrödinger Equation

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrodinger equation indicate that the new method is more efficient than older ones.

New P-Stable Eighth Algebraic Order Exponentially-Fitted Methods for the Numerical Integration of the Schrödinger Equation

A family of P-stable high algebraic order exponentially-fitted methods for the numerical solution of the Schrodinger equation is developed in this paper. Numerical illustration to the resonance problem of the radial Schrodinger equation indicates that the new proposed methods are generally more efficient than the previously developed exponentially-fitted methods of the same kind.