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| In [[numerical analysis]], '''von Neumann stability analysis''' (also known as Fourier stability analysis) is a procedure used to check the [[Numerical stability|stability]] of [[finite difference scheme]]s as applied to linear [[partial differential equation]]s.<ref>[http://books.google.co.in/books?id=y77n2ySMJHUC&pg=PA523&dq=von+Neumann+stability+analysis#PPA523,M1 Analysis of Numerical Methods by E. Isaacson, H. B. Keller]</ref> The analysis is based on the [[Fourier decomposition]] of [[numerical error]] and was developed at [[Los Alamos National Laboratory]] after having been briefly described in a 1947 article by [[British people|British]] researchers [[John Crank|Crank]] and [[Phyllis Nicolson|Nicolson]].<ref>
| | If you are looking for a specific plugin, then you can just search for the name of the plugin. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. In the most current edition you can customize your retailer layout and display hues and fonts similar to your site or blog. <br><br>Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. The higher your blog ranks on search engines, the more likely people will find your online marketing site. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>ve labored so hard to publish and put up on their website. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Are you considering getting your website redesigned. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>A built-in widget which allows you to embed quickly video from popular websites. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Should you loved this information and you want to receive more information regarding [http://qrm.li/backupplugin436956 wordpress backup plugin] please visit our web page. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. If you choose a blog then people will be able to post articles on your site and people will be able to make comments on your posts (unless you turn comments off). It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route. <br><br>You will know which of your Word - Press blog posts are attracting more unique visitors which in turn will help you develop better products and services for your customers. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. And, it is better that you leave it on for the duration you are writing plugin code. Verify whether your company has a team of developers or programmers having hands-on experience and knowledge about all Word - Press concepts. |
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| {{Citation
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| | last = Crank | first = J. | last2 = Nicolson | first2 = P.
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| | title = A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of Heat Conduction Type
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| | journal = Proc. Camb. Phil. Soc.
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| | volume = 43 | year = 1947 | pages = 50–67
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| | doi = 10.1007/BF02127704
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| }}</ref>
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| This method is an example of [[temporal discretization|explicit time integration]] where the function that defines governing equation is evaluated at the current time.
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| Later, the method was given a more rigorous treatment in an article<ref>
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| {{Citation
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| | last = Charney | first = J. G. | last2 = Fjørtoft | first2 = R.
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| | last3 = von Neumann | first3 = J.
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| | title = Numerical Integration of the Barotropic Vorticity Equation
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| | journal = Tellus | volume = 2 | year = 1950 | pages = 237–254
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| | doi=10.1111/j.2153-3490.1950.tb00336.x
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| }}</ref> co-authored by [[John von Neumann]].
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| ==Numerical stability==
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| The [[Numerical stability|stability of numerical schemes]] is closely associated with [[numerical error]]. A finite difference scheme is stable if the errors made at one time step of the calculation do not cause the errors to increase as the computations are continued. A ''neutrally stable scheme'' is one in which errors remain constant as the computations are carried forward. If the errors decay and eventually damp out, the numerical scheme is said to be stable. If, on the contrary, the errors grow with time the numerical scheme is said to be unstable. The stability of numerical schemes can be investigated by performing von Neumann stability analysis. For time-dependent problems, stability guarantees that the numerical method produces a bounded solution whenever the solution of the exact differential equation is bounded. Stability, in general, can be difficult to investigate, especially when the equation under consideration is [[List of nonlinear partial differential equations|nonlinear]].
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| In certain cases, von Neumann stability is necessary and sufficient for stability in the sense of Lax–Richtmyer (as used in the [[Lax equivalence theorem]]): The PDE and the finite difference scheme models are linear; the PDE is constant-coefficient with [[periodic boundary conditions]] and have only two independent variables; and the scheme uses no more than two time levels.<ref>
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| {{Citation
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| | last = Smith | first = G. D.
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| | title = Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed.
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| | year = 1985
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| | pages = 67–68
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| }}</ref> Von Neumann stability is necessary in a much wider variety of cases. It is often used in place of a more detailed stability analysis to provide a good guess at the restrictions (if any) on the step sizes used in the scheme because of its relative simplicity.
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| ==Illustration of the method==
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| The von Neumann method is based on the decomposition of the errors into [[Fourier series]]. To illustrate the procedure, consider the one-dimensional [[heat equation]]
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| :<math>
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| \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}
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| </math>
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| defined on the spatial interval <math>L</math>, which can be discretized<ref>in this case, using the [[FTCS|FTCS discretization scheme]]</ref> as
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| :<math>
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| \quad (1) \qquad u_j^{n + 1} = u_j^{n} + r \left(u_{j + 1}^n - 2 u_j^n + u_{j - 1}^n \right)
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| </math>
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| where
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| :<math>r = \frac{\alpha\, \Delta t}{\Delta x^2}</math>
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| and the solution <math>u_j^{n}</math> of the discrete equation approximates the analytical solution <math>u(x,t)</math> of the PDE on the grid. | |
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| Define the [[round-off error]] <math>\epsilon_j^n</math> as
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| :<math>
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| \epsilon_j^n = N_j^n - u_j^n
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| </math>
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| where <math>u_j^n</math> is the solution of the discretized equation (1) that would be computed in the absence of round-off error, and <math>N_j^n</math> is the numerical solution obtained in [[Floating point|finite precision arithmetic]]. Since the exact solution <math>u_j^n</math> must satisfy the discretized equation exactly, the error <math>\epsilon_j^n</math> must also satisfy the discretized equation.<ref>{{cite book | title = Computational Fluid Dynamics: The Basics with Applications | author = Anderson, J. D., Jr. | authorlink=John D. Anderson|edition = | publisher = [[McGraw Hill]] | year = 1994 | isbn = }}</ref>
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| Thus
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| :<math>
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| \quad (2) \qquad \epsilon_j^{n + 1} = \epsilon_j^n + r \left(\epsilon_{j + 1}^n - 2 \epsilon_j^n + \epsilon_{j - 1}^n \right)
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| </math>
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| is a recurrence relation for the error. Equations (1) and (2) show that both the error and the numerical solution have the same growth or decay behavior with respect to time. For linear differential equations with periodic boundary condition, the spatial variation of error may be expanded in a finite Fourier series, in the interval <math>L</math>, as
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| :<math>
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| \quad (3) \qquad \epsilon(x) = \sum_{m=1}^{M} A_m e^{ik_m x}
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| </math>
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| where the [[wavenumber]] <math>k_m = \frac{\pi m}{L}</math> with <math>m = 1,2,\ldots,M</math> and <math>M = L/\Delta x</math>. The time dependence of the error is included by assuming that the amplitude of error <math>A_m</math> is a function of time. Since the error tends to grow or decay exponentially with time, it is reasonable to assume that the amplitude varies exponentially with time; hence
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| :<math> | |
| \quad (4) \qquad \epsilon(x,t) = \sum_{m=1}^{M} e^{at} e^{ik_m x}
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| </math>
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| where <math>a</math> is a constant.
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| Since the difference equation for error is linear (the behavior of each term of the series is the same as series itself), it is enough to consider the growth of error of a typical term:
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| :<math>
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| \quad (5) \qquad \epsilon_m(x,t) = e^{at} e^{ik_m x}
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| </math>
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| The stability characteristics can be studied using just this form for the error with no loss in generality. To find out how error varies in steps of time, substitute equation (5) into equation (2), after noting that
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| *:<math>
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| \begin{align}
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| \epsilon_j^n & = e^{at} e^{ik_m x} \\
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| \epsilon_j^{n+1} & = e^{a(t+\Delta t)} e^{ik_m x} \\
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| \epsilon_{j+1}^n & = e^{at} e^{ik_m (x+\Delta x)} \\
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| \epsilon_{j-1}^n & = e^{at} e^{ik_m (x-\Delta x)},
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| \end{align}
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| </math>
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| to yield (after simplification) | |
| :<math>
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| \quad (6) \qquad e^{a\Delta t} = 1 + \frac{\alpha \Delta t}{\Delta x^2} \left(e^{ik_m \Delta x} + e^{-ik_m \Delta x} - 2\right).
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| </math> | |
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| Using the identities
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| :<math>
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| \qquad \cos(k_m \Delta x) = \frac{e^{ik_m \Delta x} + e^{-ik_m \Delta x}}{2} \qquad \text{and} \qquad \sin^2\frac{k_m \Delta x}{2} = \frac{1 - \cos(k_m \Delta x)}{2}
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| </math>
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| equation (6) may be written as
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| :<math>
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| \quad (7) \qquad e^{a\Delta t} = 1 - \frac{4\alpha \Delta t}{\Delta x^2} \sin^2 (k_m \Delta x/2)
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| </math>
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| Define the amplification factor
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| :<math>
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| G \equiv \frac{\epsilon_j^{n+1}}{\epsilon_j^n}
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| </math>
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| The necessary and sufficient condition for the error to remain bounded is that <math>\vert G \vert \leq 1.</math>
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| However,
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| :<math>
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| \quad (8) \qquad G = \frac{e^{a(t+\Delta t)} e^{ik_m x}}{e^{at} e^{ik_m x}} = e^{a\Delta t}
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| </math>
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| Thus, from equations (7) and (8), the condition for stability is given by
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| :<math>
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| \quad (9) \qquad \left\vert 1 - \frac{4\alpha \Delta t}{\Delta x^2} \sin^2 (k_m \Delta x/2) \right\vert \leq 1
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| </math>
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| For the above condition to hold at all <math>\sin^2 (k_m \Delta x/2)</math>, we have
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| :<math>
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| \quad (10) \qquad \frac{\alpha \Delta t}{\Delta x^2} \leq \frac{1}{2}
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| </math>
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| Equation (10) gives the stability requirement for the FTCS scheme as applied to one-dimensional heat equation. It says that for a given <math>\Delta x</math>, the allowed value of <math>\Delta t</math> must be small enough to satisfy equation (10).
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| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Numerical analysis]]
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| [[Category:Fourier analysis]]
| |
If you are looking for a specific plugin, then you can just search for the name of the plugin. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. In the most current edition you can customize your retailer layout and display hues and fonts similar to your site or blog.
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ve labored so hard to publish and put up on their website. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Are you considering getting your website redesigned. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration.
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