Neumann stability analysis

An intuitive way of investigating the stability properties of a finite-difference scheme is the von Neumann stability analysis, which we briefly outline as follows. The von Neumann analysis is local, being based on the assumption that the coefficients of the difference equation are so slowly varying in space and time as to be considered constant. Under such assumptions, the eigenmodes (the independent solutions) of the difference equation may be written in the general form  

The time propagation of the solution is considered to be stable if the amplification factor satisfies the condition  

In order to express the amplification factor for the forward-difference representation of the one-dimensional diffusion equation, one has to replace the general form (8-21) of the eigenmodes into the difference equation (8-11)  

By combining the exponentials and employing the trigonometric identity 1 - c.osx = 2sinz(x/2), one obtains for the amplification factor  

The significance of this result is that the timestep At insuring the stability of the algorithm is limited by an upper bound, which is proportional to the diffusion time across a cell of width h. This makes the explicit scheme, characterized by forward time differencing, conditionally stable and proves that the value A - 1/2 is indeed critical. 

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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. 

If the Neumann stability analysis is applied to the three calculation schemes that we have discussed in the previous section, the following stability conditions are obtained in the assumption of a linear isotherm ... 

Analogous stabihty analyses can be executed for the other time-discretization schemes as well. It is important to note here that although the von Neumann stability analysis yields a limiting time-step estimate to keep the roundoff errors bounded, it does not preclude the occurrence of bounded but unphysical solutions. A classical example is the Crank-Nicolson scheme, which from the von Neumann viewpoint is unconditionally stable but can give rise to bounded unphysical solutirms in case all the coefficients of the discretization equation do not happen to be of the same sign [2]. 

The von Neumann stability analysis apphes ordy to difference equations with constant coefficients and periodic boundary conditions. Instability may arise from specification of boundary conditions or nonlinear terms in differential equations. Instability caused by nonlinear terms is called nonlinear instability and was first noticed in a numerical solution of the nonhnear barotropic vorticity equationinthe early days of munerical weatherprediction. 

The von Neumann stability analysis can be used to examine the long term time stability of the different finite difference approximations. This technique only applies to linear partial differential equations with constant coefficients, but much can be learned from such simple cases. This analysis begins by assuming that the solution of the finite difference system can be expressed as a superposition of Fourier modes having the form... 

Here C is defined by the boundary value in the case of the Dirichlet conditions (3.1.3b), (3.1.3d) at one of the end points or by the space averages of the initial concentrations in the case of the Neumann conditions (3.1.3a), (3.1.3c) at both ends. In the spirit of a standard linear stability analysis consider a small perturbation of the equilibrium of the form... 

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). 

To obtain a rough physical understanding of the mechanism of the instability, attention may be focused first on a planar detonation subjected to a one-dimensional, time-dependent perturbation. Since the instability depends on the wave structure, a model for the steady detonation structure is needed to proceed with a stability analysis. As the simplest structure model, assume that properties remain constant at their Neumann-spike values for an induction distance after which all of the heat of combustion is released instantaneously. If v is the gas velocity with respect to the shock at the Neumann condition, then may be expressed approximately in terms of the explosion time given by equation (B-57) as Z = vt. From normal-shock relations for an ideal gas with constant specific heats in the strong-shock limit, the Neumann-state conditions are expressible by v/vq = po/p —... 

Study the finite difference literature on parabolic equations and transient modeling, and summarize the von Neumann stability criterion for explicit and implicit schemes. What are its strengths and limitations What new stability tests are available to study nonlinearities and heterogeneities Comment on group velocity and wave-based stability analysis. 

When p = 0.5, the method is the Crank-Nicholson implicit method. The expansion point should be taken at (i+l/2,j). The truncation error is of the order (Ax)2 plus order (Ay)2. No stability criterion comes out of the von Neumann analysis, but difficulties can come about if diagonal dominance is not kept for the tridiagonal algorithm. 

Concerning the stability of the previous schemes, von Neumann analysis leads to the Courant criterion... 

Once the pseudoemulsion films break, then the possible configurations of the bridging drop in a Plateau border are determined by Neumann s triangle and three surface tensions—0 0, Oqw, and a w There has been no analysis of the stability of the resulting configurations. It is, however, possible to speculate. We show a... 

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