A new modified RungeKuttaNyström method of fourth algebraic order is developed. The new modified RKN method is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives. Numerical results indicate that the new modified method is much more efficient than other methods derived for solving numerically the Schrödinger equation.
Among the most commonly used methods in the numerical integration of second order differential equations, in which the first derivative terms are omitted, are RungeKutta (Nyström) methods. These methods have been used widely due to their simplicity and their accuracy and mainly because they are onestep methods and thus they require no additional starting values.
In the last decades many researchers developed optimized RungeKutta (Nyström) methods, based mostly on the exponential fitting and the phase lag properties [
In the last years, a new methodology has been developed, which is based on the phase lag derivatives. Researchers that have been using the mentioned methodology achieve higher accuracy at their methods [
In the present paper a new modified RungeKuttaNyström method is constructed. The new method contains four additional variable coefficients (in comparison with the classical RKN method), which depend on
The new modified RKN method that has been obtained will be used for the numerical solution of second order differential equations and more specifically for the numerical integration of the radial Schrödinger equation.
The
For the classical method
Consider the problem
By applying a numerical method for the solution of (
The difference
The difference
Based on the above definitions, it is easy for one to see that we have two new errors, the dissipation and the dispersion errors. These errors can be expressed via Taylor series around an initial value
In order to develop the new method we use the test equation (
The eigenvalues of the amplification matrix
In phase analysis one compares the phases of
Apply the RKN method (
From Definition
If at a point
We can also put forward an alternative definition for the case of infinite order of phase lag.
To obtain phase lag of order infinity the relation
From Definition
If one has phase lag of order infinity and at a point
For the construction of a method with nullification of phase lag, amplification error, and their derivatives, one must satisfy the conditions
The new method that we are going to develop in this section, is a fourstage explicit RungeKuttaNyström method with the FSAL technique (first stage as last), so the method actually uses three stages at each step for the function evaluations. From (
By substituting the coefficients that have been used by the DEP algorithm in [
By applying numerical method (
For small values of
In this section we study the algebraic order of the new modified RKN method. We require that
By expanding in Taylor series the corresponding numerical solution
By equating (
Order 2:
order 3:
order 4:
By following the same procedure for
Order 1:
order 2:
order 3:
order 4:
For the classical RKN method [
In order to verify the algebraic order of the the new modified explicit RungeKuttaNyström method, first we will produce the algebraic order conditions. To do that, we apply the above methodology for the new method. At first, we want to extract the conditions for
By equating (
Order 2:
order 3:
order 4:
By following the same procedure for the approximate solution of
Order 1:
order 2:
order 3:
order 4:
As it is proved, in order to construct an explicit modified RungeKuttaNyström method of fourth algebraic order, (
For the proposed modified RKN method (
For
So it is proved that the new RKN method is of fourth algebraic order. Moreover the local truncation error in
At this point, we have already compute the Taylor expansions of
the exact solution
the first derivative
The LTE verifies the fourth algebraic order of the new modified method. From the above procedure the local truncation error in
Respectively, the local truncation error in
From (
The onedimensional or radial Schrödinger equation has the form
We call the term
For the purpose of our numerical illustration we will take the domain of integration as
In the case of positive energies
(
For positive energies and for
We use the following eigenenergies:
In the case of negative energies
In order to solve this problem numerically, by a chosen eigenvalue, we integrate forward from the point
For this problem we use the following eigenenergies:
In order to investigate the case of
In this section four RungeKuttaNyström methods are compared (including the new method). These methods have four algebraic orders with four stages; also all of them are using the FSAL properties. The methods used in the comparison have been denoted by
MRKNDPAF4: the new fourthorder RKN method with four stages (three effective stages with FSAL property), derived in Section
RKNPAF4: the fourthorder RKN method with four stages (three effective stages with FSAL property), phase lag, and amplification error of order infinity of Papadopoulos et al. [
DEPRKN4: the highorder method of pair RKN
EFRKN4: the fourthorder exponential fitted RKN method with four stages (three effective stages with FSAL property) of Franco [
One way to measure the efficiency of the method is to compute the accuracy in the decimal digits, that is,
In the case of LennardJones potential, we compare the phase shifts with the values found in [
LennardJones potential with

Kobeissi et al. [ 
MRKNDPAF4  EFRKN4  DEPRKN4  RKNPAF4 



































































LennardJones potential with

Kobeissi et al. [ 
MRKNDPAF4  EFRKN4  DEPRKN4  RKNPAF4 



































































The frequency is given by the suggestion of Ixaru and Rizea [
The numerical results were obtained by using the highlevel language MATLAB. In Figures
Efficiency for the Schrödinger equation using
Efficiency for the Schrödinger equation using
Efficiency for the Schrödinger equation using
Efficiency for the Schrödinger equation using
Efficiency for the Schrödinger equation using
Efficiency for the Schrödinger equation using
Efficiency for the Schrödinger equation using
Efficiency for the Schrödinger equation using
Numerical results indicate that the new method derived in Section
More specifically in the case of WoodsSaxon potential with positive energies, the new method remains more accurate than the RKNPAF4 and EFRKN4 methods up to two decimals at all eigenvalues. Also our method is more accurate than the classical DEPRKN4 method, by two decimals for the eigenvalue
In the case of WoodsSaxon potential with negative energies, the new method has almost the same accuracy with the RKNPAF4 and EFRKN4 methods, but it remains (even a little) more accurate at the majority of the negative eigenvalues. In comparison with the DEPRKN4 method, the new method is much more accurate and specifically by one digit for the eigenvalue
At last, for the LennardJones potential the new method is more accurate than all the other methods for eigenvalues
The modified RKN method, developed in this paper, is much more efficient than the classical one, in any case. The new method remained more efficient for all the eigenvalues and in some cases was more accurate than the other methods up to two decimals. Moreover we observe that the accuracy difference between the new method and the other methods increased as the eigenvalue increased (for details about the original proof see [
T. E. Simos is an active member of the European Academy of Sciences and the European Academy of Sciences and Arts and a corresponding member of the European Academy of Arts, Sciences and Humanities.