Numerov Type Research Articles

Explicit, ninth order, two step methods for solving inhomogeneous linear problems x″(t)=Λx(t)+f(t)

Here we consider the second order Inhomogeneous Linear Initial Value Problems with constant coefficients. Two–step hybrid methods (i.e. of Numerov–type) are considered for addressing this problem. A special set of order conditions is given and solved for derivation a low cost new method. Actually, we manage to save one stage (function evaluation) per step for this type of problems in comparison with the best existed methods. This is crucial as demonstrated in various numerical tests where our new proposal outperforms standard methods in the relevant literature.

A new eighth order exponentially fitted explicit Numerov-type method for solving oscillatory problems

In this work we derive a new family of seven stages, eighth order, exponential fitted, Numerov (i.e. two-step) type methods. A particular variable coefficient method is constructed. The numerical results show that it outperforms exponential fitted two step hybrid method found in the literature sharing similar characteristics. Moreover, the presented method outperforms Numerov type methods with minimized phase-lag for problems in consideration share a non dominant linear part.

Numerov type variable mesh approximations for 1D unsteady quasi-linear biharmonic problem: application to Kuramoto-Sivashinsky equation

We present two new two-level compact implicit variable mesh numerical methods of order two in time and two in space, and of order two in time and three in space for the solution of 1D unsteady quasi-linear biharmonic problem subject to suitable initial and boundary conditions. The simplicity of the proposed methods lies in their three-point discretization without requiring any fictitious points for incorporating the boundary conditions. The derived methods are shown to be unconditionally stable for a model linear problem for uniform mesh. We also discuss how our formulation is able to handle linear singular problem and ensure that the developed numerical methods retain their orders and accuracy everywhere in the solution region. The proposed difference methods successfully works for the highly nonlinear Kuramoto-Sivashinsky equation. Many physical problems are solved to demonstrate the accuracy and efficiency of the proposed methods. The numerical results reveal that the obtained solutions not only approximate the exact solutions very well but are also much better than those available in earlier research studies.

A Novel Numerical Algorithm of Numerov Type for 2D Quasi-linear Elliptic Boundary Value Problems

In this article, using three function evaluations, we discuss a nine-point compact scheme of O(Δ y2 + Δ x4) based on Numerov-type discretization for the solution of 2D quasi-linear elliptic equations with given Dirichlet boundary conditions, where Δy > 0 and Δx > 0 are grid sizes in y- and x-directions, respectively. Iterative methods for diffusion-convection equation are discussed in detail. We use block iterative methods to solve the system of algebraic linear and nonlinear difference equations. Comparative results of some physical problems are given to illustrate the usefulness of the proposed method.

Trigonometrically fitted block Numerov type method for y′′ = f(x, y, y′)

A trigonometrically fitted block Numerov type method (TBNM), is proposed for solving y???=?f(x, y, y?) directly without reducing it to an equivalent first order system. This is achieved by constructing a continuous representation of the trigonometrically fitted Numerov method (CTNM) and using it to generate the well known trigonometrically fitted Numerov method (TNUM) and three new additional methods, which are combined and applied in block form as simultaneous numerical integrators. The stability property of the TBNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

Stability of a Numerov type finite–difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis

We consider an initial-boundary value problem for the one-dimensional nonstationary Schrodinger equation on the half-axis and study a two-level symmetric finite-difference scheme of Numerov type with higher approximation order. This scheme is constructed on a finite mesh, which is uniform with respect to space, with a nonlocal approximate transparent boundary condition of a general form (of Dirichlet-to-Neumann type). We obtain assertions about the stability of the finite-difference scheme in two norms with respect to the initial data and free terms in the equation and in the approximate transparent boundary condition under suitable conditions in the form of inequalities on the operator of approximate transparent boundary condition. Bibliography: 12 titles.

On the use of AGE algorithm with a high accuracy Numerov type variable mesh discretization for 1D non-linear parabolic equations

In this paper, the use of N-AGE and Newton-N-AGE iterative methods on a variable mesh for the solution of one dimensional parabolic initial boundary value problems is considered. Using three spatial grid points, a two level implicit formula based on Numerov type discretization is discussed. The local truncation error of the method is of \(O({k^2h_l^{-1} +kh_l +h_l^3})\), where h l > 0 and k > 0 are the step lengths in space and time directions, respectively. We use a special technique to handle singular parabolic equations. The advantage of using these algorithms is highlighted computationally.

Trigonometrically fitted explicit Numerov-type method for periodic IVPs with two frequencies

In this paper, new trigonometrically fitted Numerov type methods for the periodic initial problems are proposed. These methods are based on the original Numerov-type sixth order method with fifth internal stages motivated by Tsitouras (see [Ch. Tsitouras, Explicit Numerov type methods with reduced number of stages, Comput. Math. Appl. 45 (2003) 37–42]). Some parameters are added to these methods so that they can integrate exactly the combination of trigonometrically functions with two frequencies. 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.

An explicit Numerov-type method for second-order differential equations with oscillating solutions

In this paper a new explicit Numerov-type method is introduced. The construction is based on a modification of a sixth-order explicit Numerov-type method recently developed by Tsitouras [Ch. Tsitouras, Explicit Numerov type methods with reduced number of stages, Comput. Math. Appl. 45 (2003) 37–42]. Two free parameters are added in order to nullify the phase-lag and the amplification. The method is useful only when a good estimate of the frequency of the problem is known in advance. The parameters depend on the product of the estimated frequency and the stepsize. Numerical results obtained for well-known test problems show the efficiency of the new method.

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.

Numerical solution of the two-dimensional time independent Schr�dinger equation with Numerov-type methods

The solution of the two-dimensional time-independent Schrodinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and Numerov type methods are used to solve it. Specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe et al. [Int. J. Comp. Math 32 (1990) 233–242], and the minimum phase-lag method of Rao et al. [Int. J. Comp. Math 37 (1990) 63–77] are applied to this problem. All methods are applied for the computation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon–Heils potential. The results are compared with the results produced by full discterization.

Zero Dissipative, Explicit Numerov-Type Methods for Second Order IVPs with Oscillating Solutions

New explicit, zero dissipative, hybrid Numerov type methods are presented in this paper. We derive these methods using an alternative which avoids the use of costly high accuracy interpolatory nodes. We only need the Taylor expansion at some internal points then. The method is of sixth algebraic order at a cost of seven stages per step while their phase lag order is fourteen. The zero dissipation condition is satisfied, so the methods possess an non empty interval of periodicity. Numerical results over some well known problems in physics and mechanics indicate the superiority of the new method.

Explicit Numerov type methods with reduced number of stages

We present in this paper a new approach for the derivation of hybrid explicit Numerov type methods. The new methodology does not require the intermediate use of high accuracy interpolatory nodes, since we only need the Taylor expansion of the internal points. As a consequence, a sixth-order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now. Numerical results over some well-known problems in physics and mechanics indicate the superiority of the new method.

Explicit two-step methods for second-order linear IVPs

We present a new type of method for the integration of systems of linear inhomogeneous initial value problems with constant coefficients. Our methods are of hybrid explicit Numerov type. The methods are constructed without the intermediate use of high accuracy interpolatory nodes, since only the Taylor expansion at the internal points is needed. Then we derive the order conditions taking advantage of the special structure of the problem considered. We present a method with algebraic order seven at a cost of only four stages per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation, indicate the superiority of the new method.

EXPLICIT NUMEROV TYPE METHODS FOR SECOND ORDER IVPs WITH OSCILLATING SOLUTIONS

New explicit hybrid Numerov type methods are presented in this paper. These efficient methods are constructed using a new approach, where we do not need the use of the intermediate high accuracy interpolatory nodes, since only the Taylor expansion of the internal points is needed. The methods share sixth algebraic order at a cost of five stages per step while their phase-lag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phase-lag is more important than the nonempty interval in constructing numerical methods for the solution of Schrödinger equation and related problems.1–3 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.

Dissipative high phase-lag order methods

New explicit hybrid Numerov type methods are presented in this paper. They share eighth algebraic order while their phase-lag order varies between 18 and 22. Their main characteristic is that they are dissipative so they possess an empty interval of periodicity. Numerical illustrations indicate that this choice was successful since the new methods outperforms the older ones which were scheduled to integrate effectively periodic problems.

Getting started with Numerov’s method

Second-order ordinary differential equations of the Numerov type (no first derivative and the given function linear in the solution) are common in physics, but little discussion is devoted to the special first step that is needed before one can apply the general algorithm. We give an explicit algorithm to calculate the first point of the solution with an accuracy appropriate to that obtained with the general algorithm. © 1997 American Institute of Physics.

Bessel and Neumann-fitted methods for the numerical solution of the radial Schrödinger equation

Two two-step methods for the numerical solution of some problems related to the Schrödinger equation are developed in this paper. One is of the Numerov type and of algebraic order 4 and the other is of the Runge-Kutta type and of algebraic order 5. Each of these have free parameters that are defined such that the methods are fitted to spherical Bessel and Neumann functions. From these methods a variable-step technique is devised. This variable-step technique is applied to the phase-shift and resonance problems of the radial Schrödinger equation. Results indicate that the new approach is more efficient than several other well-known methods.

An inverse iteration method using multigrid for quantum chemistry

We present a method to compute the lowest eigenpairs of a generalized eigenvalue problem resulting from the discretization of a stationary Schrödinger equation by a fourth order finite difference scheme of Numerov type. We propose to use an inverse iteration method combined with a Rayleigh-Ritz procedure to correct several eigenvectors at the same time. The linear systems in the inverse iteration scheme are regularized by projections on lower dimensional spaces and approximately solved by a multigrid algorithm.We apply the method to the electronic structure calculation in quantum chemistry.

An explicit hybrid method of Numerov type for second-order periodic initial-value problems

We consider a two-parameter family of explicit hybrid methods of Numerov type for the numerical integration of second-order intial-value problems. When these methods are applied to the linear equation: y″ + ω 2 y = 0, ω > 0, we determine the parameters α, β so that the phase lag (frequency distortion) of the method is minimal. The resulting method has (algebraic) order 4 and a small frequency distortion of size ( 1 3628800 )v 8 ( v = ωh, h being the step size) and in addition it possesses an interval of periodicity of size 4.63, which is larger than the interval of periodicity corresponding to the explicit method of Chawla and Rao (1986). The application of this method to equations describing free and weakly forced oscillations reveals its superiority over other methods.