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Stencil (numerical Analysis)
In mathematics, especially the areas of numerical analysis concentrating on the numerical partial differential equations, numerical solution of partial differential equations, a stencil is a geometric arrangement of a nodal group that relate to the point of interest by using a numerical approximation routine. Stencils are the basis for many algorithms to numerically solve partial differential equations (PDE). Two examples of stencils are the five-point stencil and the Crank–Nicolson method stencil. Stencils are classified into two categories: Compact stencil, compact and Non-compact stencil, non-compact, the difference being the layers from the point of interest that are also used for calculation. In the notation used for one-dimensional stencils n-1, n, n+1 indicate the time steps where timestep n and n-1 have known solutions and time step n+1 is to be calculated. The spatial location of finite volumes used in the calculation are indicated by j-1, j and j+1. Etymology Graphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finite Difference Coefficient
In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward. Central finite difference This table contains the coefficients of the central differences, for several orders of accuracy and with uniform grid spacing:. For example, the third derivative with a second-order accuracy is : f(x_) \approx \frac + O\left(h_x^2 \right), where h_x represents a uniform grid spacing between each finite difference interval, and x_n = x_0 + n h_x. For the m-th derivative with accuracy n, there are 2p + 1 = 2 \left\lfloor \frac \right\rfloor - 1 + n central coefficients a_, a_, ..., a_, a_p. These are given by the solution of the linear equation system : \begin 1 & 1 & ... & 1 & 1 \\ -p & -p+1 & ... & p-1 & p \\ (-p)^2 & (-p+1)^2 &... & (p-1)^2 & p^2 \\ ... & ... &...&...&... \\ ... & ... &...&...&... \\ ... & ... &...&...&... \\ (-p)^ & (-p+1)^ & ... & (p-1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nine-point Stencil
In numerical analysis, given a square grid in two dimensions, the nine-point stencil of a point in the grid is a Stencil (numerical analysis), stencil made up of the point itself together with its eight "neighbors". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiation. This stencil is often used to approximate the Laplacian of a function of two variables. Motivation If we discretize the 2D Laplace operator, Laplacian by using Finite difference method, central-difference methods, we obtain the commonly used five-point stencil, represented by the following convolution kernel: D_=\begin 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0\end Even though it is simple to obtain and computationally lighter, the central difference kernel possess an undesired intrinsic Anisotropy, anisotropic property, since it doesn't take into account the diagonal neighbours. This intrinsic anisotropy poses a problem when applied on c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Five-point Stencil
In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiation. In one dimension In one dimension, if the spacing between points in the grid is ''h'', then the five-point stencil of a point ''x'' in the grid is : \. 1D first derivative The first derivative of a function ''f'' of a real variable at a point ''x'' can be approximated using a five-point stencil as: :f'(x) \approx \frac The center point ''f''(''x'') itself is not involved, only the four neighboring points. Derivation This formula can be obtained by writing out the four Taylor series of f(x \pm h) and f(x \pm 2h) at the point a, up to terms of ''h''3 (or up to terms of ''h''5 to get an error estimation as well), evaluating each series at a = x \mp h and a = x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since a Padé approximant is a rational function, an artificial sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a primitive monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A primitive monomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Taylor Expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lagrange Polynomial
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration, Shamir's secret sharing scheme in cryptography, and Reed–Solomon error correction in coding theory. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stencil
Stencilling produces an image or pattern on a surface by applying pigment to a surface through an intermediate object, with designed holes in the intermediate object. The holes allow the pigment to reach only some parts of the surface creating the design. The stencil is both the resulting image or pattern and the intermediate object; the context in which ''stencil'' is used makes clear which meaning is intended. In practice, the (object) stencil is usually a thin sheet of material, such as paper, plastic, wood or metal, with lettering, letters or a design cut from it, used to produce the letters or design on an underlying surface by applying pigment through the cut-out holes in the material. The key advantage of a stencil is that it can be reused to repeatedly and rapidly produce the same letters or design. Although aerosol paint, aerosol or painting stencils can be made for one-time use, typically they are made with the intention of being reused. To be reusable, they must rem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Non-compact Stencil
In numerical mathematics, a non-compact stencil is a type of discretization method, where any node surrounding the node of interest may be used in the calculation. Its computational time grows with an increase of layers of nodes used. Non-compact stencils may be compared to compact stencils.Communications in Numerical Methods in Engineering, Copyright © 2008 John Wiley & Sons, Ltd. See also *Nine-point stencil *Five-point stencil In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors". It is used to write finite difference approximations t ... References Numerical differential equations {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Compact Stencil
In mathematics, especially in the areas of numerical analysis called numerical partial differential equations, a compact stencil is a type of stencil that uses only nine nodes for its discretization method in two dimensions. It uses only the center node and the adjacent nodes. For any structured grid utilizing a compact stencil in 1, 2, or 3 dimensions the maximum number of nodes is 3, 9, or 27 respectively. Compact stencils may be compared to non-compact stencils. Compact stencils are currently implemented in many partial differential equation solvers, including several in the topics of CFD, FEA, and other mathematical solvers relating to PDE'sCommunications in Numerical Methods in Engineering, Copyright © 2008 John Wiley & Sons, Ltd. Two Point Stencil Example The two point stencil for the ''first derivative'' of a function is given by: f'(x_0)=\frac + O\left(h^2\right) . This is obtained from the Taylor series expansion of the first derivative of the function given by: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
