Neumann method

As an extension of Problem 11, integrate a second time to obtain the equation for the meniscus profile in the Neumann method. Plot this profile as y/a versus x/a, where y is the vertical elevation of a point on the meniscus (above the flat liquid surface), x is the distance of the point from the slide, and a is the capillary constant. (All meniscus profiles, regardless of contact angle, can be located on this plot.)... 

Abstract. We review the recent development of quantum dynamics for nonequilibrium phase transitions. To describe the detailed dynamical processes of nonequilibrium phase transitions, the Liouville-von Neumann method is applied to quenched second order phase transitions. Domain growth and topological defect formation is discussed in the second order phase transitions. Thermofield dynamics is extended to nonequilibrium phase transitions. Finally, we discuss the physical implications of nonequilibrium processes such as decoherence of order parameter and thermalization. 

The von Neumann method described above usually works well, and is reasonably easy to apply. One reason it works well, despite the fact that it totally ignores conditions at the boundaries, is that errors that often arise at interior points away from the boundaries and spread from there [private communication with O. Osterby 1996], However, boundary conditions can affect stability, especially if derivative (or mixed) boundary conditions hold [116,117,118,119,334], It might be safer to consider all points in space in some way. The following somewhat brief treatment is described in greater mathematical detail in such texts as Smith [514] or Lapidus and Pinder [350],... 

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. 

Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [135, 174, 136]. This method is used to study the effects of numerical diffusion on the solution. 

Moreover, through the von Neumann method along with the use of discrete Fourier modes [1], the stability condition for any member of the (2, M) family is extracted. For instance, the (2, 4) case has... 

The stability criterion of the technique is extracted by means of the ordinary von Neumann method as... 

J. A. Pereda, L. A. Vielva, A. Vegas, and A. Prieto, Analyzing the stability of the FDTD technique by combining the von Neumann method with the Routh-Hurwitz... 

Concerning the stability of the higher order curvilinear ADI-FDTD algorithm, the von Neumann method is applied and the two subiterations are expressed in matrix form as... 

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. 

Theoretical interpretations of the Neumann equations and semi-predictive methods to estimate the beta parameter have been proposed (Kwok et al., 2000) and validated with good results (Kwok and Neumann, 2000a) but as the results with the original Neumann method are overall similar these modified approaches have not been used in subsequent publications. 

We have already mentioned that theories for interfacial tension constimte a controversial topic and the advent of the van Oss-Good and Neumann methods has not made the topic less controversial. Actually, these two are possibly the most widely discussed theories today and for this reason, and to pay justice to all opinions, we will divide the discussion into different sections the opinions of the developers on their own and on each other s theories and some of the independent studies carried out by others before expressing our own views. 

They believe that these deviations are not due to experimental error or problems of the theory but aU have a clear explanation. In most cases, they attribute the problems to either interaction between the solid and the liquid and/or the presence of a non-zero spreading pressure due to vapour adsorption from the liquid. They improve the smoothness of their plots, in these cases, by eliminating several (in some cases many of the) liquids, sometimes even alkanes, used in the analysis in order to maintain the maximum possible inertness of the test liquids used to estimate the solid surface tension using the Neumann method. They mention that with this approach, i.e. careful selection of test liquids (bulky molecules are often useful), some of the experimental contact angle data, even with the goniometer method used in the extensive studies of Zisman, can be used in the context of the Neumann method (Kwok and Nemnann, 2000b). 

Despite the criticism, the Neumann method is useful in many practical applications. Table 15.5 and other similar results from the literature (see references in the table) provide Neumann values for several polymers obtained from the method. Clearly, the Neumann method has been applied to a wide range of polymers and other solids (over 50 different solid surfaces) with surface tensions from around or below 10 mN m up to more than 40 mN m This list includes thus hydrophobic, non-polar and polar polymers. This wide applicability is a positive characteristic of the method. 

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [254]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [157, 158, 215]. This method is used to study the effects of numerical diffusion on the solution. 

To solve Eqs. 26 and 27, the initial values at n = 0 and n= must be known. Additionally, storing the velocities and stresses from both the current and the previous time step increases memory requirements. Approximating the spatial derivatives with the central difference formula and the temporal derivatives with the forward difference formula may seem a convenient alternative. However, such a combination leads to a FD scheme that is unconditionally unstable, i.e., it will not converge regardless of the value of the Courant number C. The stability of a numerical scheme is often analyzed using the von Neumann method, which is based on Fourier decomposition of the numerical solution. 

In this section, we discuss the stability of finite difference approximations using the well-known von Neumann procedure. This method introduces an initial error represented by a finite Fourier series and examines how this error propagates during the solution. The von Neumann method applies to initial-value problems for this reason it is used to analyze the stability of the explicit method for parabolic equations developed in Sec. 6.4.2 and the explicit method for hyperbolic equations developed in Sec. 6.4.3. 

The stability of the explicit solution (6.91) of the hyperbolic equation (6.88) can be similarly analyzed using the von Neumann method. The homogeneous equation for the error propagation of that solution is... 

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