Numerov method

We first follow the flow chart for the simple case of elastic scattering of structureless atoms. The number of internal states, Nc, is one, quantum scattering calculations are feasible and recommended, for even the smallest modem computer. The Numerov method has often been used for such calculations (41), but the recent method based on analytic approximations by Airy functions (2) obtains the same results with many fewer evaluations of the potential function. The WKB approximation also requires a relatively small number of function evaluations, but its accuracy is limited, whereas the piecewise analytic method (2) can obtain results to any preset, desired accuracy. 

Many of the properties of Rydberg atoms depend on the wavefunction in the r > r0 region where the potential is simply a coulomb potential. In this region we can very easily calculate the wavefunction numerically using the Numerov method, which can be applied to an equation of the form7... 

If x increases in steps of size h, then the basic equation of the Numerov method... 

In 1924, the Russian astronomer Numerov (transliterating his own name as Noumerov), published a paper [421] in which he described some improvements in approximations to derivatives, to help with numerical simulations of the movement of bodies in the solar system. His device has been adapted to the solution of pdes, and was introduced to electrochemistry by Bieniasz in 2003 [108]. The method described by Bieniasz is also called the Douglas equation in some texts such as that of Smith [514], where a rather clear description of the method is found. With the help of the Numerov method, it is possible to attain fourth order accuracy in the spatial second derivative, while using only the usual three points. The first paper by Bieniasz on this method treated equally spaced grids, and was followed by another on unequally spaced grids [107], The method makes it practical to use higher-order time derivative approximations without the complications of, say, the (6,5)-point scheme described above, which makes the solution of the system of equations a little complicated (and computer time consuming). 

Note also that, if there are homogeneous chemical reaction terms on the right-hand side of (9.31), they can be accommodated without problems they will lead to some additional terms operated on by What must not be present are convection terms, since these are spatial first derivatives, making the Numerov method, in this form, impossible to use. However, Bieniasz has devised an improved version, called the extended Numerov method [110], which indeed can handle first spatial derivatives and thus convective systems. 

Higher-order methods Chap. 9, Sect. 9.2.2 for multipoint discretisations. The four-point variant with unequal intervals is probably optimal the system can be solved using an extended Thomas algorithm without difficulty. Numerov methods (Sect. 9.2.7) can achieve higher orders with only three-point approximations to the spatial second derivative. They are not trivial to program. 

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. 

In 35 the numerical solution of the two-dimensional time-independent Schrodinger equation is studied using the method of partial discretization. The discretized problem is treated as a problem of the numerical solution of a system of ordinary differential equations and Numerov type methods are used to solve it. More specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe el al. and the minimum phase-lag method of Rao et al. are applied to this problem. The methods are applied for the calculation 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 discretization. Conclusions are presented. 

In 48 multiderivative methods are developed for the numerical integration of the one-dimensional Schrodinger equation. The method is called multiderivative since uses derivatives of order two and four. Application of these methods to the resonance problem of the radial Schrodinger equation indicates that the new developed method is more efficient than the Numerov method and other well known methods of the literature. 

They were determined numerically by the Cooley-Numerov method(31.32). with a spline representation of V (t). potential supports three bound states. 

Numerov Discretization. - Flack and Vanden Berghe101 have used the well known Numerov method of discretization which leads to a tridiagonal (not necessarily symmetric) form for A and B. The method is of algebraic order 4, phase-lag order 4. The interval of periodicity is (0,6). 

The generalized Numerov method produced in this paper has the form... 

We note here that one of the most used methods for the numerical solution of the coupled differential equations arising from the Schrodinger equation is the Iterative Numerov method of Allison.113... 

For the propagation of the multichannel wavefunction 4>(R), in real or complex-scaled coordinates, an efficient algorithm is furnished by fhe Fox-Goodwin-Numerov method [8, 44], which results from a discrefizafion of the differential operator appearing in Eqs. (39). Given adjacent points R — h, R, and R + h on the grid, we define an inward mafrix (labeled /) and an outward matrix (labeled o) as ... 

FIGURE 4.7 The number of nodes in a Numerov-method solution as a function of the energy .,... 

Dimensionless Variables. The Numerov method requires that we guess values of . What should be the order of magnitude of our guesses 10 J, 10 J,... To answer this question, we reformulate the Schrbdinger equation using dimensionless variables, taking the harmonic oscillator as the example. 

For reasonable accuracy, one usually needs a minimum of 100 points, so we shall take Sr = 0.1 to give us 100 points. As is evident from the derivation of the Numerov method, Sr must be small. A reasonable rule might be to have 5, no greater than 0.1. 

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. 

An alternative to a Numerov-method computer program is a spreadsheet. Most colleges and universities have computer labs that make a spreadsheet program available. 

FIGURE 4.B Spreadsheet for Numerov-method solution of the harmonic oscillator. 

The Numerov method is a numerical method that allows one to find bound-state energies and wave functions for the one-particle, one-dimensional Schrodinger equation. 

Use the Numerov method to find the lowest three stationary-state energies for a particle in a one-dimensional box of length / with walls of infinite height. 

Use (4.75) to show that if one multiplies i/ri in the Numerov method by a constant c, then ij/2, ijfi, are all multiplied by c, so the entire wave function is multiplied by c, which does not affect the eigenvalues we find. 

Use the normalized Numerov-method harmonic-oscillator wave functions found by going from —5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the v = 0 and v = l states. 

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