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We use a methodology of optimization of the efficiency of a hybrid two-step method for the numerical solution of the radial Schrödinger equation and related problems with periodic or oscillating solutions. More specifically, we study how the vanishing of the phase-lag and its derivatives optimizes the efficiency of the hybrid two-step method.

In this paper, we investigate the numerical solution of systems of second order differential equations of the form

Problems for which their models are expressed with the above system of equations can be found in different fields of applied sciences such as astronomy, astrophysics, quantum mechanics, quantum chemistry, celestial mechanics, electronics physical chemistry, and chemical physics (see [

The optimization of the efficiency of a numerical method for the numerical solution of the radial Schrödinger equation and related problems with periodic or oscillating solutions is the subject of this paper. More specifically, in this paper, we will investigate how the procedure of vanishing of the phase-lag and its first derivative optimizes, the efficiency of a numerical method. As a result the produced methods via the above procedure, they are very efficient on any problem with periodic or oscillating solutions or on any problem with solution which contains the functions

The purpose of this paper is the computation of the coefficients of the proposed hybrid two-step method in order:

to have the highest algebraic order,

to have the phase-lag vanished,

and finally, to have the first derivative of the phase-lag vanished as well.

The procedure of vanishing of the phase lag and its first derivative is based on the direct formula for the determination of the phase-lag for

We will investigate the efficiency of the new methodology based on the error analysis and stability analysis of the new proposed method. We will also apply the studied methods to the numerical solution of the radial Schrödinger equation and to related problems.

We will consider a hybrid two-step method of sixth algebraic order. Based on this method, we will develop the new optimized method which is of sixth algebraic order and it has phase-lag and its first derivative equal to zero. We will investigate the stability and the error of the produced method. We will apply the obtained method to the resonance problem of the radial Schrödinger equation. This is one of the most difficult problems arising from the radial Schrödinger equation. The construction of the paper is given below.

In Section

The development of the new optimized method is presented in Section

The error analysis is presented in Section

The stability analysis of the new produced method is presented in Section

The numerical results are presented in Section

Finally, in Section

For the numerical solution of the initial value problem:

If the method is symmetric, then

When a symmetric

The characteristic equation associated with (

The symmetric

The formula mentioned in the above theorem is a direct method for the computation of the phase-lag of any symmetric

Consider the following family of hybrid two-step methods (see [

The above-mentioned method belongs to the families of hybrid (Runge-Kutta type) symmetric two-step methods for the numerical solution of problems of the form

Consider the method (

If we apply the method (

Requiring the above method to have its phase-lag vanished and by using the formulae (

Demanding the method to have the first derivative of the phase-lag vanished as well, we have the equation:

Requiring now the coefficients of the new proposed method to satisfy the equations (

For some values of

The behaviour of the coefficients is given in the following Figure

Behaviour of the coefficients of the new proposed method given by (

The local truncation error of the new proposed method (mentioned as

We will study the following methods.

It holds that

It holds that

The radial time independent Schrödinger equation is of the form:

Based on the paper of Ixaru and Rizea [

We express the derivatives

Finally, we substitute the expressions of the derivatives, produced in the previous step, into the local truncation error formulae.

We use the procedure mentioned above and the formulae:

We consider two cases in terms of the value of

The energy is close to the potential, that is,

Hence, we have the following asymptotic expansions of the Local Truncation Errors.

It holds that

It holds that

For the standard hybrid two-step method, the error increases as the fourth power of

Applying the new method to the scalar test equation:

The corresponding characteristic equation is given by.

A symmetric

A method is called P-stable if its interval of periodicity is equal to

A method is called singularly almost P-stable if its interval of periodicity is equal to

In Figure

For the solution of the Schrödinger equation, the frequency of the exponential fitting is equal to the frequency of the scalar test equation. So, it is necessary to observe the surroundings of the first diagonal of the

In the case that the frequency of the scalar test equation is equal with the frequency of phase fitting, that is, in the case that

From the above analysis, we have the following theorem.

The method developed in Section

Based on the analysis presented above, we studied the interval of periodicity of some well-known methods mentioned in the previous paragraph. The results presented in the Table

Comparative stability analysis for the methods mentioned in Section

Method | Interval of Periodicity |
---|---|

CL | |

NM (see Section |

In order to study the efficiency of the new developed method, we apply it

to the radial time-independent Schrödinger equation, and

to a nonlinear orbital problem.

The radial Schrödinger equation can be presented as

The above equation presents the model for a particle in a central potential field where

In (

The function

The quantity

The quantity

We note here that the models which are given via the radial Schrödinger equation are boundary-value problems. In these cases, the boundary conditions are

In order to apply the new obtained method to the radial Schrödinger equation, the value of parameter

We use as a potential the well-known Woods-Saxon potential which can be written as

The behaviour of Woods-Saxon potential is shown in Figure

The Woods-Saxon potential.

It is well known that for some potentials, such as the Woods-Saxon potential, the definition of parameter

For the purpose of obtaining our numerical results, it is appropriate to choose

For example, in the point of the integration region

We consider the numerical solution of the radial Schrödinger equation (

In the case of positive energies,

The above equation has linearly independent solutions

For positive energies, we have the so-called resonance problem. This problem consists either of finding the phase-shift

The boundary conditions for this problem are

We compute the approximate positive eigenenergies of the Woods-Saxon resonance problem using the following.

The eighth-order multistep method developed by Quinlan and Tremaine [

The tenth-order multistep method developed by Quinlan and Tremaine [

The twelfth-order multistep method developed by Quinlan and Tremaine [

The fourth-algebraic-order method of Chawla and Rao with minimal phase-lag [

The hybrid sixth-algebraic-order method developed by Chawla and Rao with minimal phase-lag [

The standard form of the eighth-algebraic-order method developed in Section

The new developed hybrid two-step method with vanished phase-lag and its first derivative (obtained in Section

The computed eigenenergies are compared with reference values (the reference values are computed using the well-known two-step method of Chawla and Rao [

Accuracy (digits) for several values of CPU Time (in seconds) for the eigenvalue

Accuracy (Digits) for several values of CPU Time (in Seconds) for the eigenvalue

Consider the nonlinear system of equations:

The analytical solution of the problem is the following:

The system of equations (

For this problem, we have

Accuracy (digits) for several values of CPU time (in seconds) for the nonlinear orbital problem. The nonexistence of a value of accuracy (digits) indicates that for this value of CPU time, accuracy (digits) is less than 0.

The purpose of this paper was the optimization of the efficiency of a hybrid two-step method for the approximate solution of the radial Schrödinger equation and related problems. We have described how the methodology of vanishing of the phase-lag and its first derivative optimize the behaviour of the specific numerical method. The results of the application of this methodology was a hybrid two-step method that is very efficient on any problem with oscillating solutions or problems with solutions contain the functions

From the results presented above, we can make the following remarks.

The standard form of the eighth-algebraic-order method developed in Section

The tenth-order multistep method developed by Quinlan and Tremaine [

The twelfth-order multistep method developed by Quinlan and Tremaine [

Finally, the new developed hybrid two-step method with vanished phase-lag and its first derivative (obtained in Section

All computations were carried out on an IBM PC-AT compatible 80486 using double-precision arithmetic with 16 significant digits accuracy (IEEE standard).

The author wishes to thank the anonymous reviewers for their careful reading of the manuscript and their fruitful comments and suggestions.