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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 to ... References Numerical differential equations {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stencil (numerical Analysis)
In mathematics, especially the areas of numerical analysis concentrating on the 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 and 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 Graphical representations of node arrangements and their coefficients arose early in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification). Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, ''discretization'' may also refer to modification of variable or category ''granularity'', as when multiple discrete variables are aggregated or multiple discrete categories fused. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level conside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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's.Communications 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] |
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 ƒ of a real variable at a point ''x'' can be approximated using a five-point stencil as: :f'(x) \approx \frac Notice that the center point ƒ(''x'') itself is not involved, only the four neighboring points. Derivation This formula can be obtained by writing out the four Taylor series of ƒ(''x'' ± ''h'') and ƒ(''x'' ± 2''h'') up to terms of ''h''3 (or up to terms of ''h''5 to get an error estimation as well) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
