Numerov s method

The well known Numerov s method (which is indicated as Method I)... 

The Method of Raptis and Allison" is more efficient than the Numerov s method and the P-stable method of Chawla ... 

Finally the method of Chawla has similar efficiency with the Numerov s method. 

It is instructive to note that Numerov s method integrates exactly... 

The phase-lag of the constructed method is calculated based on the direct formula for the calculation of the phase-lag developed in the previous paper of Raptis and Simos.33 The free parameter Co of the method is computed based on the requirement that the phase-lag is of order infinity. The produced new method has an interval of periodicity equal to [0,19.43328], which is much larger than the interval of periodicity of Numerov s method. 

Method MI Numerov s method Method Mil Derived by Ixaru and Rizea6 Method Mill Derived by Raptis and Cash15 Method MIV Derived by Cash, Raptis and Simos16 Method MV Derived by Simos7 Method MVI Derived by Simos8... 

Explicit Methods. - 4.2.1 Fourth Algebraic Order Methods. - Simos82 has derived an explicit version of Numerov s method. This version has three extra layers of the form ... 

Recently, Konguetsof, Avdelas and Simos108 have developed a generalization of Numerov s method for the numerical solution of the problem... 

The Exponentially-fitted four-step method developed by Raptis (Method II) is more efficient than the Numerov s Method (Method I). 

The exponentially-fitted four-step method developed by Raptis (Method V) is better than the Numerov s Method (Method I), the Exponentially-fitted four-step method developed by Raptis (Method II), the two-step Numerov-type Method with minimum phase-lag produced by Chawla and Rao (Method III) and the two-step method developed by Raptis and Allis on (Method IV). 

The new two-step Numerov-Type method with phase-lag and its first derivative equal to zero (Method VII) is better than the Numerov s Method... 

The new two-step Numerov-Type method with phase-lag and its first, second and third derivatives equal to zero (Method IX) is better than the Numerov s Method (Method I), the Exponentially-fitted four-step method developed by Raptis (Method II), the two-step Numerov-type Method with minimum phase-lag produced by Chawla and Rao (Method III), the two-step method developed by Raptis and Allison (Method IV), the exponentially-fitted four-step method developed by Raptis (Method V), the new two-step Numerov-Type method with phase-lag and its first derivative equal to zero (Method VII), the new two-step Numerov-Type method with phase-lag and its first and second derivatives equal to zero (Method VIII). 

M. M. Chawla and P. S. Rao, A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II Explicit method, J. Comput. Appl. Math., 1986, 15, 329—337. 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

The explicit version of the Numerov method which has better stability properties than Niunerov s method since it has an interval of periodicity equal to (0, 12) while Niunerov s method has an interval of periodicity equal to (0, 6). This method has been produced by Chawla and is indicated as method MI. (2) The method developed by Tsitouras, which is indicated as method MIL (3) The method developed by Simos and Williams, which is indicated as method Mill. (4) The method developed by Papageorgiou et al.f which is indicated as method MIV (5) The method developed by Tsitouras, which is indicated as method MV (6) The method developed by Simos (Case IX), which is indicated as method MVI. (7) The exponentially fitted method developed by Simos,which is indicated as method MVII. (8) The exponentially fitted method developed by Simos and Aguiar (see Appendix C), which is indicated as method MVIII (Case I). (9) The exponentially fitted method developed by Simos and Aguiar, which is indicated as method MIX (Case II). (10) The exponentially fitted method developed in this review (Sections 3.4.1 and 3.4.2), which is indicated as method... 

In order to illustrate the efficiency of the new produced methods, the author applied them to the well-known undamped Duffing equation with Dooren s parameters. The numerical results show that the Numerov method fitted with the Fourier components is much more stable, accurate and efficient than the one with no Fourier component. The accuracy of the fitted method with the first three Fourier components can attain 10 for a remarkable range of step sizes, which is much higher than the one of the traditional Numerov method, with eight orders for step size of tc/2.011. 

T. E. Simos and P. S. Williams, A family of Numerov-type exponentially-fitted methods for the numerical integration of the Schrodinger equation, Comput. Chem., 1997, 21,403-417. 

The relevant equations for the derivative Numerov-Cooley (DNC) method closely follow Cooley s [111] presentation. Let R be the radial coordinate or bond displacement coordinate, P R) a radial eigenfunction, and U R) the potential function. The one-dimensional Schrodinger equation is then... 

Computer Program for the Numerov Method. Table 4.1 contains a BASIC computer program that applies the Numerov method to the harmonic-oscillator Schrodinger equation. The character in the names of variables makes these variables double precision. M is the number of intervals between and and equals (x,niax r,o)/ S- Lines 55 and 75 contain two times the potential-energy function, which must be modified if the problem is not the harmonic-oscillator. If there is a node between two successive values of x then the values at these two points will have opposite signs (see Problem 4.43) and statement 90 will increase the nodes counter NN by 1. 

FIGURE 13.2 The v = S Morse vibrational wave function for H2 as found by the Numerov method. 

Use the Morse function and the Numerov method (Section 4.4) to (a) find the lowest six vibrational energy levels of the H2 molecule in its ground electronic state, which has Djhc = 38297 cm Vg/c = 4403.2 cm and Rg = 0.741 A, where h and c are Planck s constant and the speed of light (b) find (/ ) for each of these vibrational states. 

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