Neumann series method

The method of solution described in this subsection will be referred to as the Neumann series method, or the method of successive generations. The term successive approximation is reserved for the solution method described below. 

A more analytical method of stability analysis is the method of von Neumann [424, 565] (note that [424] is mostly incorrectly cited as being of the year 1951 [139]). The method focusses on an interior point along X in the grid and looks at the propagation of an error at that point, making certain reasonable assumptions, using Fourier series (which is why the method on occasion is also called the Fourier series method). 

The Neumann series or the successive-approximation methods provide formulations that, theoretically, should yield the particular solution of Eq. (252). Practical implementation of those methods calls for special care in order to avoid a contribution from the fundamental mode [the term of Eq. (254)]. A fundamental-mode contamination can result, for example, if the source condition of Eq. (253) is not satisfied exactly. Numerical round-... 

A typical regularization method for the Fredholm integral equation of the first kind is to add a term Q(f> y) (with q a fixed number) to the left-hand side of Eq. (66), thus obtaining a Fredholm integral equation of the second kind, whose solution is stable. The difiieulty of this method is related to the fact that the formula expressing such a solution (the Neumann series ) does converge only for q > A, where A is the norm of the operator A [45]. 

This chapter has been used to illustrate the method discussed in Chap. 2, which may be used to solve thermoviscoelastic boundary value problems involving temperature fields simultaneously varying with position and time. The method relies upon the solution of integral equations in terms of Neumann series expansions. 

Once the random field involved in the stochastic boundary value problem has been discretized, a solution method has to be adopted in order to solve the boundary value problem numerically. The choice of the solution method depends on the required statistical information of the solution. If only the first two statistical moments of the solution are of interest second moment analysis), the perturbafion method can be applied. However, if a. full probabilistic analysis is necessary, Galerkin schemes can be utilized or one has to resort to Monte Carlo simulations eventually in combination with a von Neumann series expansion. 

D. Marx and J. Hiitter, in Modern Methods and Algorithms of Quantum Chemistry, J. GrotendorsC, ed., John von Neumann Institute for Computing, Julich, Germany, 2000, pp. 301-449. See http //www.fz-juelich,de/nic-series. 

For a review see M. Dolg, in Modern methods and algorithms of quantum chemistry, J. Grotendorst (Ed.), NIC Series Vol. 1, John von Neumann-Institute for Computing, Forschungszentrum Jiilich, Germany (2000). 

P. Knowles, M. Schiitz, H.J. Werner, Ab initio Methods for Electron Correlation in Molecules in Modern Methods and Algorithms of Quantum Chemistry, Edited by Grotendorst, John von Neumann, Institute for computing, NIC series 88, 69 (2002). J.F. McCann Thesis, Queen s University Belfast (1984). 

Marx D, Flutter J, Ab Initio Molecular Dynamics Theory and Implementation, In Modern Methods and Algorithms of Quantum Chemistry, edited by J Grotrndorst, NIC Series, Vol 1 (John von Neumann Institute of Computing Jiilich, 2000), pp 301—449 and refs therein... 

Ahlrichs R, Elliot S, Huniar U (2000) In J. Grotendorst J (ed) Ab Initio Treatment of Large Molecules in Modem Methods and Algorithms of Quantum Chemistry. Proceedings, 2nd edn. John von Neumann Institute for Computing, Jiilich, NIC Series 3, pp 7—25... 

P. Sherwood, in Hybrid quantum mechanicslmolecular mechanics approaches in Modern Methods and Algorithms of Quantum Chemistry, Proceedings, ed. J. Grotendorst, John van Neumann Institute of Computing, Jiilich, NIC series, second edn., 2000, vol. 3, pp. 285-305 available on-line. 

D.Y. Kwok, A.W. Neumann, Contact Angle Measurements, in Surface Characterization Methods. Principles, Techniques and Applications. Surfactant Series no. 87, A.J. Milting, Ekl., Marcel Dekker (1999) chapter 2, p. 37-86. (Review, many practical hints, 191 refs.)... 

We use here the Neumann stability analysis [57], which is the most widely used procedure for the determination of the stabihty of a calculation scheme using a finite difference approximation. In this stability analysis, an initial error is introduced as a finite Fourier series and one studies the growth or decay of this error during the calculation. The Neumann method applies only to initial value problems with a periodical initial condition it neglects the influence of the bormd-ary condition, and it is applied only to linear finite difference approximations with constant coefficients, i.e., to linear equations. This method gives only a necessary condition for the stability of a munerical procedure. It turns out, however, that this condition is sufficient in many cases. 

The stability attributes of the aforementioned method are explored via the combined von Neumann and Routh-Hurwitz technique [2]. According to this approach, the error that appears during the computation of any field component is described by a single term of a Fourier series expansion ... 

Worth, G.A. and Robb, M.A., Adv. Chem. Phys., 124, 355-341, 2002 Marx, D. and Hutter, J., Modern Methods and Algorithms of Quantum Chemistry, Proceedings, second Edition, J. Groten-dorst (Ed.), John Von Neumann Institute for Computing, Jtilich, NIC Series, Vol. 3, pp.329-477, 2000. 

Fromm has shown that the difference equations are conditionally stable to infinitesimal fluctuations, using the von Neumann method, in which the perturbation is expanded in a Fourier series and a given Fourier component is examined. 

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