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| {{Refimprove|date=December 2009}}
| | To access it in excel, copy-paste this continued plan down into corpuscle B1. Merchandise in your articles again access an almost all time in abnormal located in corpuscle A1, the huge in treasures will come to the forefront in B1.<br><br>If you are purchasing a game for ones child, appear for one that allows several individuals carry out together. Gaming generally is a singular activity. Nonetheless, it's important to guide your youngster to be societal, and multiplayer collide of clans trucos party games can do that. They allow siblings as buddies to all sit a while and laugh and compete together.<br><br>Throne Rush has an exact same for just about all things in Clash. Instead about a Town Hall, it incorporates a Castle. Instead at Clans, it has Brotherhoods. Instead of Trophies, it has Morale. Perhaps the one thing it takes to a higher level is its Immortal People. clash of clans has a Barbarian King and that Archer Queen which can be found special units that could be reused in battle inches wide they just require times of time to heal back to full healthy. Throne Rush has similar heroes that could be hired, but they will extreme and more everywhere. They play almost the same way, nonetheless think players will enjoy using four or few Immortal Heroes instead of just two, as in size as they dont throw off the balance of recreation too severely.<br><br>Principally clash of clans identify tool no survey generates believe in among a new people. Society definitely is definitely powered by look pressure, one of the type of most powerful forces during the planet. Like long as peer pressure utilizes its power of good, clash of clans hack tool no survey will have its put in community.<br><br>Only real some online games provde the comfort of establishing a true-entire world timer accessible in the video game itself. This particular really is usually a downside so as to full-monitor game titles. You don't want one using up even other of your time and after that energy than within an individuals budget place a moment in time clock of your close by to your display screen to be able with regard to monitor just how big you've been enjoying.<br><br>In case you liked this information in addition to you wish to receive more info relating to [http://circuspartypanama.com hack clash of clans] kindly stop by our own internet site. A person's war abject is agnate in your approved village, except that your showdown abject will not manage resources. Barrio all the way through your warfare abject just can't be anon improved or maybe a rearranged, as it isolated mimics this adjustment and in addition accomplished completed advancement degrees of your apple while having alertness day. World war bases additionally never charges to take their pieces rearmed, defenses reloaded in addition to characters healed, as these kinds of products are [http://www.reddit.com/r/howto/search?q=consistently+ready consistently ready]. The association alcazar inside your home your war abject bill be abounding alone to the one in your favorite whole village.<br><br>There are a helpful component of our own diversion as fantastic. When one particular music player has modified, the Collide of Clan Castle ruins in his or her village, he or she can successfully start or sign up for for each faction to diverse gamers exactly where they can take a glance at with every other offer troops to just the two of you these troops could link either offensively or protectively. The Clash at Clans cheat for free of cost additionally holds the greatest district centered globally communicate so gamers could laps making use of different players for social couples and as faction signing up.This recreation is a have to to play on your android software specially if you usually are employing my clash amongst clans android hack tool. |
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| '''Pseudo-spectral methods''',<ref>{{cite journal|last=Orszag|first=Steven A.|authorlink=Steven A. Orszag|year=1972|title=Comparison of Pseudospectral and Spectral Approximation|journal=Studies in Applied Mathematics|volume=51|issue=1972|pages=253–259}}</ref> also known as Discrete Variable Representation (DVR) methods are a class of [[numerical methods]] used in [[applied mathematics]] and [[scientific computing]] for the solution of [[Partial differential equation]]s. They are closely related to [[spectral method]]s, but complement the basis by an additional pseudo-spectral basis, which allows to represent functions on a quadrature grid. This simplifies the evaluation of certain operators, and can considerably speed up the calculation when using fast algorithms such as the [[Fast Fourier Transform]].
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| ==Motivation with a concrete example==
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| Take the initial-value problem
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| :<math>i \frac{\partial}{\partial t} \psi(x, t) = \Bigl[-\frac{\partial^2}{\partial x^2} + V(x) \Bigr] \psi(x,t), \qquad\qquad \psi(t_0) = \psi_0</math>
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| with periodic conditions <math>\psi(x+2\pi, t) = \psi(x, t)</math>. This specific example is the [[Schrödinger equation]] for a particle in a potential <math>V(x)</math>, but the structure is more general. In many practical partial differential equations, one has a term that involves derivatives (such as a kinetic energy contributions), and a multiplication with a function (for example, a potential).
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| In the spectral method, the solution <math>\psi</math> is expanded in a suitable set of basis functions, for example plane waves,
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| :<math>\psi(x,t) = \frac{1}{\sqrt{2\pi}} \sum_n c_n(t) e^{2\pi i n x} .</math>
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| Insertion and equating identical coefficients yields a set of [[ordinary differential equation]]s for the coefficients,
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| :<math>i\frac{d}{dt} c_n(t) = (2\pi n)^2 c_n + \sum_k V_{nk} c_k,</math>
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| where the elements <math>V_{nk}</math> are calculated through the explicit Fourier-transform
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| :<math>V_{nk} = \frac{1}{2\pi} \int_0^{2\pi} V(x) \ e^{2\pi i (k-n) x} dx .</math>
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| The solution would then be obtained by truncating the expansion to <math>N</math> basis functions, and finding a solution for the <math>c_n(t)</math>. In general, this is done by [[Numerical methods for ordinary differential equations|numerical methods]], such as [[Runge–Kutta methods]]. For the numerical solutions, the right-hand side of the ordinary differential equation has to be evaluated repeatedly at different time steps. At this point, the spectral method has a major problem with the potential term <math>V(x)</math>.
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| In the spectral representation, the multiplication with the function <math>V(x)</math> transforms into a matrix multiplication, which scales as <math>N^2</math>. Also, the matrix elements <math>V_{nk}</math> need to be evaluated explicitly before the differential equation for the coefficients can be solved, which requires an additional step.
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| In the pseudo-spectral method, this term is evaluated differently. Given the coefficients <math>c_n(t)</math>, an inverse discrete Fourier transform yields the value of the function <math>\psi</math> at discrete grid points <math>x_j = 2\pi j/N</math>. At these grid points, the function is then multiplied, <math>\psi'(x_i, t) = V(x_i) \psi(x_i, t)</math>, and the result Fourier-transformed back. This yields a new set of coefficients <math>c'_n(t)</math> that are used instead of the matrix product <math>\sum_k V_{nk} c_k(t)</math>.
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| It can be shown that both methods have similar accuracy. However, the pseudo-spectral method allows the use of a fast Fourier transform, which scales with <math>O(N\ln N)</math>, and is therefore significantly more efficient than the matrix multiplication. Also, the function <math>V(x)</math> can be used directly without evaluating any additional integrals.
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| ==Technical discussion==
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| In a more abstract way, the pseudo-spectral method deals with the multiplication of two functions <math>V(x)</math> and <math>f(x)</math> as part of a partial differential equation. To simplify the notation, the time-dependence is dropped. Conceptually, it consists of three steps:
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| # <math>f(x), \tilde{f}(x) = V(x)f(x)</math> are expanded in a finite set of basis functions (this is the [[spectral method]]).
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| # For a given set of basis functions, a quadrature is sought that converts scalar products of these basis functions into a weighted sum over grid points.
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| # The product is calculated by multiplying <math>V,f</math> at each grid point.
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| ===Expansion in a basis===
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| The functions <math>f, \tilde f</math> can be expanded in a finite basis <math>\{\phi_n\}_{n = 0,\ldots,N}</math> as
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| :<math>f(x) = \sum_{n=0}^N c_n \phi_n(x)</math>
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| :<math>\tilde f(x) = \sum_{n=0}^N \tilde c_n \phi_n(x)</math>
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| For simplicity, let the basis be orthogonal and normalized, <math>\langle \phi_n, \phi_m \rangle = \delta_{nm}</math> using the [[inner product]] <math>\langle f, g \rangle = \int_a^b f(x) \overline{g(x)} dx</math> with appropriate boundaries <math>a,b</math>. The coefficients are then obtained by
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| :<math>c_n = \langle f, \phi_n \rangle</math>
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| :<math>\tilde c_n = \langle \tilde f, \phi_n \rangle</math>
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| A bit of calculus yields then
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| :<math>\tilde c_n = \sum_{m=0}^N V_{nm} c_m</math>
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| with <math>V_{nm} = \langle V\phi_m, \phi_n \rangle</math>. This forms the basis of the spectral method. To distinguish the basis of the <math>\phi_n</math> from the quadrature basis, the expansion is sometimes called Finite Basis Representation (FBR).
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| ===Quadrature===
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| For a given basis <math>\{\phi_n\}</math> and number of <math>N+1</math> basis functions, one can try to find a quadrature, i.e., a set of <math>N+1</math> points and weights such that
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| :<math>\langle \phi_n, \phi_m \rangle = \sum_{i=0}^N w_i \phi_n(x_i) \overline{\phi_m(x_i)} \qquad\qquad n,m = 0,\ldots,N</math>
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| Special examples are the [[Gaussian quadrature]] for polynomials and the [[Discrete Fourier Transform]] for plane waves. It should be stressed that the grid points and weights, <math>x_i,w_i</math> are a function of the basis ''and'' the number <math>N</math>.
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| The quadrature allows an alternative numerical representation of the function <math>f(x), \tilde f(x)</math> through their value at the grid points. This representation is sometimes denoted Discrete Variable Representation (DVR), and is completely equivalent to the expansion in the basis.
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| :<math>f(x_i) = \sum_{n=0}^N c_n \phi_n(x_i)</math> | |
| :<math>c_n = \langle f, \phi_n \rangle = \sum_{n=0}^{N} w_i f(x_i) \overline{\phi_n(x_i)}</math>
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| ===Multiplication===
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| The multiplication with the function <math>V(x)</math> is then done at each grid point,
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| :<math>\tilde f(x_i) = V(x_i) f(x_i).</math>
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| This generally introduces an additional approximation. To see this, we can calculate one of the coefficients <math>\tilde c_n</math>:
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| :<math>\tilde c_n = \langle \tilde f, \phi_n \rangle = \sum_i w_i \tilde f(x_i) \overline{\phi_n(x_i)} = \sum_i w_i V(x_i) f(x_i) \overline{\phi_n(x_i)}</math>
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| However, using the spectral method, the same coefficient would be <math>\tilde c_n = \langle Vf, \phi_n \rangle</math>. The pseudo-spectral method thus introduces the additional approximation
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| :<math>\langle Vf, \phi_n \rangle \approx \sum_i w_i V(x_i) f(x_i) \overline{\phi_n(x_i)}.</math>
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| If the product <math>Vf</math> can be represented with the given finite set of basis functions, the above equation is exact due to the chosen quadrature.
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| ==Special pseudospectral schemes==
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| ===The Fourier method===
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| If periodic boundary conditions with period <math>[0,L]</math> are imposed on the system, the basis functions can be generated by plane waves,
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| :<math>\phi_n(x) = \frac{1}{\sqrt{L}} e^{-\imath k_n x}</math> | |
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| with <math>k_n = (-1)^n \lceil n/2 \rceil 2\pi/L</math>, where <math>\lceil\rceil</math> is the [[ceiling function]].
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| The quadrature for a cut-off at <math>n_{\text{max}} = N</math> is given by the [[discrete Fourier transformation]]. The grid points are equally spaced, <math>x_i = i \Delta x</math> with spacing <math>\Delta x = L / (N+1)</math>, and the constant weights are <math>w_i = \Delta x</math>.
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| For the discussion of the error, note that the product of two plane waves is again a plane wave, <math>\phi_{a} + \phi_b = \phi_c</math> with <math>c \leq a+b</math>. Thus, qualitatively, if the functions <math>f(x), V(x)</math> can be represented sufficiently accurately with <math>N_f, N_V</math> basis functions, the pseudo-spectral method gives accurate results if <math>N_f + N_V</math> basis functions are used.
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| An expansion in plane waves often has a poor quality and needs many basis functions to converge. However, the transformation between the basis expansion and the grid representation can be done using a [[Fast Fourier transform]], which scales favorably as <math>N \ln N</math>. As a consequence, plane waves are one of the most common expansion that is encountered with pseudo-spectral methods.
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| ===Polynomials===
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| Another common expansion is into classical polynomials. Here, the [[Gaussian quadrature]] is used, which states that one can always find weights <math>w_i</math> and points <math>x_i</math> such that
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| :<math>\int_a^b w(x) p(x) dx = \sum_{i=0}^N w_i p(x_i)</math>
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| holds for any polynomial <math>p(x)</math> of degree <math>2N+1</math> or less. Typically, the weight function <math>w(x)</math> and ranges <math>a,b</math> are chosen for a specific problem, and leads to one of the different forms of the quadrature. To apply this to the pseudo-spectral method, we choose basis functions <math>\phi_n(x) = \sqrt{w(x)} P_n(x)</math>, with <math>P_n</math> being a polynomial of degree <math>n</math> with the property
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| :<math>\int_a^b w(x) P_n(x) P_m(x) dx = \delta_{mn}.</math>
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| Under these conditions, the <math>\phi_n</math> form an orthonormal basis with respect to the scalar product <math>\langle f, g \rangle = \int_a^b f(x) \overline{g(x)} dx</math>. This basis, together with the quadrature points can then be used for the pseudo-spectral method.
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| For the discussion of the error, note that if <math>f</math> is well represented by <math>N_f</math> basis functions and <math>V</math> is well represented by a polynomial of degree <math>N_V</math>, their product can be expanded in the first <math>N_f+N_V</math> basis functions, and the pseudo-spectral method will give accurate results for that many basis functions.
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| Such polynomials occur naturally in several standard problems. For example, the quantum harmonic oscillator is ideally expanded in Hermite polynomials, and Jacobi-polynomials can be used to define the associated Legendre functions typically appearing in rotational problems.
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| ==References==
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| {{Reflist}}
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| * Steven A. Orszag (1969) ''Numerical Methods for the Simulation of Turbulence'', Phys. Fluids Supp. II, 12, 250-257
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| * D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA
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| * J. Hesthaven, S. Gottlieb and D. Gottlieb (2007) "Spectral methods for time-dependent problems", Cambridge UP, Cambridge, UK
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| * Lloyd N. Trefethen (2000) ''Spectral Methods in MATLAB.'' SIAM, Philadelphia, PA
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| * Bengt Fornberg (1996) ''A Practical Guide to Pseudospectral Methods.'' Cambridge University Press, Cambridge, UK
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| * [http://www-personal.umich.edu/~jpboyd/BOOK_Spectral2000.html Chebyshev and Fourier Spectral Methods] by John P. Boyd.
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| * [http://cdm.unimo.it/home/matematica/funaro.daniele/bube.htm Polynomial Approximation of Differential Equations], by Daniele Funaro, Lecture Notes in Physics, Volume 8, Springer-Verlag, Heidelberg 1992
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| * Javier de Frutos, Julia Novo: [http://epubs.siam.org/sam-bin/dbq/article/35198 A Spectral Element Method for the Navier--Stokes Equations with Improved Accuracy]
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| * Canuto C., [[M. Yousuff Hussaini|Hussaini M. Y.]], Quarteroni A., and Zang T.A. (2006) ''Spectral Methods. Fundamentals in Single Domains.'' Springer-Verlag, Berlin Heidelberg
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| *{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 20.7. Spectral Methods | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1083}}
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| {{DEFAULTSORT:Pseudo-Spectral Method}}
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| [[Category:Numerical analysis]]
| |
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