Obrechkoff Method Research Articles

Exponentially-Fitted Fourth-Derivative Single-Step Obrechkoff Method for Oscillatory/Periodic Problems

The quest for accurate and more efficient methods for solving periodic/oscillatory problems is gaining more attention in recent time. This paper presents the construction and implementation of a family of exponentially-fitted Obrechkoff methods using a six-step flowchart discussed in the literature. A single-step Obrechkoff method involving terms up to the fourth derivative was used as the base method. We also present the stability and convergence properties of the constructed family of methods. Two numerical examples were used to illustrate the performance of the constructed methods.

Obrechkoff two-step method fitted with Fourier spectrum for undamped Duffing equation

A new kind of Obrechkoff two-step method with Fourier spectrum for the numerical solution of undamped Duffing equation is constructed in this paper. This kind of Obrechkoff two-step method is based on a new family of Obrechkoff two-step method which incorporates odd order derivatives to improve the accuracy, and is derived by the Fourier-fitting technique. The stability analysis and error analysis are carried out. The numerical results confirm our analyses and demonstrate that the new kind of method has great advantages in accuracy, efficiency and stability.

An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

In this paper, we present an explicit six-step singularly P-stable Obrechkoff method of tenth algebraic order for solving second-order linear periodic and oscillatory initial value problems of ordinary differential equations. The advantage of this new singularly P-stable Obrechkoff method is that it is a high-order explicit method, and thus does not require additional predictor stages. The numerical stability and phase properties of the new method is analyzed. Four numerical examples show that the new explicit method is more accurate than Obrechkoff schemes of the same order when applied to the numerical solution of second-order initial value problems with highly oscillatory solutions.

A New Symmetric p-stable Obrechkoff Method with Optimal Phase lag for Oscillatory Problems

In this paper, we derive a class of symmetric p-stable Obrechkoff methods via Padé approximation approach (PAA) for the numerical solution of special second order initial value problems (IVPs) in ordinary differential equations (ODEs). We investigate periodicity analysis on the proposed scheme to verify p-stability property. The new algorithms possess minimum phase-lag error which shows that they can accurately solve oscillatory problems. Reports on several numerical experiments are provided to illustrate the accuracy of the method.

A new four-step P-stable Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation

In this paper, we present a new four-step implicit P-stable Obrechkoff method of tenth algebraic order for solving one-dimensional second-order linear periodic and oscillatory initial value problems of ordinary differential equations such as reasonable Schrödinger equation. Numerical stability and phase properties of the new methods are analyzed. Numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature.

Stability and Convergence of Two-Step Obrechkoff Scheme For Second-Order Two-Point Boundary Value Problem

Mathematical modeling of scientific and engineering processes often yield Boundary Value Problems (BVPs). One of the broad categories of numerical methods for solving BVPs is the finite difference methods, in which the differential equation is replaced by a set of difference equations which are solved by direct or iterative methods. In this paper, we use some properties of matrices to analyze the stability and convergence of the prominent finite difference methods - two-step Obrechkoff method - for solving the boundary value problem $u^{\prime \prime} = f(t,u)$, $a < x < b$, $u(a) = \eta_1$, $u(b) = \eta_2$. Conditions for the stability and convergence of the two-step Obrechkoff method method were established.

A New Two-step Obrechkoff Method with Vanished Phase-lag and Some of its Derivatives for the Numerical Solution of Radial Schrodinger Equation and Related IVPs with Oscillating Solutions

A new two-step implicit linear Obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schrodinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

Numerical Solution of First Order Initial Value Problems Using a Self-Starting Implicit Two-Step Obrechkoff-Type Block Method

The conventional two-step implicit Obrechkoff method is a discrete scheme that requires additional starting values when implemented for the numerical solution of first order initial value problems. This paper therefore presents a two-step implicit Obrechkoff-type block method which is self-starting for solving first order initial value problems, hence bypassing the rigour of developing and implementing new starting values for the method. Numerical examples are considered to show the new method performing better when com-pared with previously existing methods in literature.

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.

A fifth order implicit method for the numerical solution of differential-algebraic equations

An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.

High phase-lag order trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems

In this paper, we present the two-step trigonometrically fitted symmetric Obrechkoff methods with algebraic order of twelve. The method is based on the symmetric two-step Obrechkoff method, with 12 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.

Contractivity-preserving explicit Hermite–Obrechkoff ODE solver of order 13

A new optimal, explicit, Hermite–Obrechkoff method of order 13, denoted by HO(13), that is contractivity-preserving (CP) and has nonnegative coefficients is constructed for solving nonstiff first-order initial value problems. Based on the CP conditions, the new 9-derivative HO(13) has maximum order 13. The new method usually requires significantly fewer function evaluations and significantly less CPU time than the Taylor method of order 13 and the Runge–Kutta method DP(8,7)13M to achieve the same global error when solving standard \(N\)-body problems.

The new class of implicit [formula omitted]-stable hybrid Obrechkoff method for the numerical solution of first order initial value problems

In a recent paper, Shokri et al. (2011) [1] introduce a new class of hybrid Obrechkoff methods for the numerical solution of second-order initial value problems. In this work, we will derive L-stable hybrid Obrechkoff methods for the numerical integration of stiff first-order initial value problems. The numerical results obtained by the new method for some problems show its superiority in efficiency, accuracy and stability.

A new two-step P-stable hybrid Obrechkoff method for the numerical integration of second-order IVPs

In this paper, we introduce a class of new P-stable hybrid Obrechkoff methods for the numerical solution of second-order initial value problems. We generate a two-step, symmetric, P-stable hybrid Obrechkoff method of order 8 with two off-step, symmetric, points and phase lag of order at least 2 and for special cases, this order can be increased up to 6. The numerical results obtained by the new method for some problems show its superiority in efficiency, accuracy and stability.

Highly-accurate ground state energies of the He atom and the He-like ions by Hartree SCF calculation with Obrechkoff method

In this paper, we use the Obrechkoff one-step method to improve the performance of the Hartree SCF calculation and obtain the Hartree energies of the two-electron atom and ions for the nuclear charge up to 20 with 14 significant figures, which is more accurate than those reported in the literature by four figures. Our calculation demonstrates that the highly-accurate Obrechkoff one-step method for the SCF calculation has the obvious advantage in the accuracy and efficiency over the previous methods.

Hermite–Birkhoff–Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size

A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y ′ = f ( t , y ) with initial conditions y ( t 0 ) = y 0 . Its formula uses y ′ , y ″ and y ‴ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.

P-stable high-order super-implicit and Obrechkoff methods for periodic initial value problems

This paper discusses the numerical solution of periodic initial value problems. Two classes of methods are discussed, super-implicit and Obrechkoff. The advantage of Obrechkoff 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 cases when the right-hand side is very complex, we may prefer super-implicit methods. We develop a super-implicit P-stable method of order 12 and Obrechkoff method of order 18.

A HIGHLY ACCURATE AND EFFICIENT TRIGONOMETRICALLY-FITTED P-STABLE THREE-STEP METHOD FOR PERIODIC INITIAL-VALUE PROBLEMS

In this paper we present a new three-step method, which is a trigonometrically-fitted P-stable Obrechkoff method with phase-lag (frequency distortion) infinity. In this new method, we make use of higher-even-order derivatives including the eighth-order to increase the accuracy. On the other hand, we adopt a special structure to reduce the computational complexity of high-derivatives. The numerical illustration demonstrate that the new method has advantage in accuracy, periodic stability and efficiency.

A HIGH-ACCURATE AND EFFICIENT OBRECHKOFF FIVE-STEP METHOD FOR UNDAMPED DUFFING'S EQUATION

In this paper, we present a five-step Obrechkoff method to improve the previous two-step one for a second-order initial-value problem with the oscillatory solution. We use a special structure to construct the iterative formula, in which the higher-even-order derivatives are placed at central four nodes, and show there existence of periodic solutions in it with a remarkably wide interval of periodicity, [Formula: see text]. By using a proper first-order derivative (FOD) formula to make this five-step method to have two advantages (a) a very high accuracy since the local truncation error (LTE) of both the main structure and the FOD formula are the same as O (h14); (b) a high efficiency because it avoids solving a polynomial equation with degree-nine by Picard iterative. By applying the new method to the well-known problem, the nonlinear Duffing's equation without damping, we can show that our numerical solution is four to five orders higher than the one by the previous Obrechkoff two-step method and it takes only 25% of CPU time required by the previous method to fulfil the same task. By using the new method, a better "exact" solution is found by fitting, whose error tolerance is below 5×10-15, than the one widely used in the lectures, whose error tolerance is below 10-11.

A new kind of high-efficient and high-accurate P-stable Obrechkoff three-step method for periodic initial-value problems

In order to improve the efficiency and accuracy of the previous Obrechkoff method, in this paper we put forward a new kind of P-stable three-step Obrechkoff method of O ( h 10 ) for periodic initial-value problems. By using a new structure and an embedded high accurate first-order derivative formula, we can avoid time-consuming iterative calculation to obtain the high-order derivatives. By taking advantage of new trigonometrically-fitting scheme we can make both the main structure and the first-order derivative formula to be P-stable. We apply our new method to three periodic problems and compare it with the previous three Obrechkoff methods. Numerical results demonstrate that our new method is superior over the previous ones in accuracy, efficiency and stability.