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Pseudo-spectral Cause And Effect (Pt.
II) - Napalm Death - Utopia Banished 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 equations. They are closely related to spectral methodsbut complement the basis by an additional pseudo-spectral basis, which allows representation of 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. In many practical partial differential equations, one has a term that involves derivatives such as a kinetic energy contributionand a multiplication with a function for example, a potential. Insertion and equating identical coefficients yields a set of ordinary differential equations for the coefficients.
In general, this is done by numerical methodssuch 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. In the pseudo-spectral method, this term is evaluated differently. It can be shown that both methods have similar accuracy. To simplify the notation, the time-dependence is dropped. Conceptually, it consists of three steps:. The coefficients are then obtained by.
This forms the basis of the spectral method. Special examples are the Gaussian quadrature for polynomials and the Discrete Fourier Transform for plane waves.
This representation is sometimes denoted Discrete Variable Representation DVRand is completely equivalent to the expansion in the basis.
This generally introduces an additional approximation. The pseudo-spectral method thus introduces the additional approximation. An expansion in plane waves often has a poor quality and needs many basis functions to converge. As a consequence, plane waves are one of the most common expansion that is encountered with Take It Away - The Used - Berth methods.
Another Psuedo Static - Heterodactyl - Fourier expansion is into classical polynomials. This basis, together with the quadrature points can then be used for the pseudo-spectral method. Such polynomials occur naturally in several standard problems.
For example, the Psuedo Static - Heterodactyl - Fourier harmonic oscillator is ideally expanded in Hermite polynomials, and Jacobi-polynomials can be used to define the associated Legendre functions typically appearing in rotational Psuedo Static - Heterodactyl - Fourier . From Wikipedia, the free encyclopedia.
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