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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Partial Differential Equations
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Overview of methods Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values. Method of lines The method of lines (MOL, NMOL, NUMOLHamdi, S., W. E. Schiesser and G. W. Griffiths (2007) Method of lines ''Scholarpedia'', 2(7):2859.) is a technique for solving partial differential equations (PDEs) in which all dimensions except one are discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of ordinary differential equations (ODEs) and differential algebraic equations (DAEs), to be used. A large number of integration routines have been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...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] |
Crank–Nicolson Method
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century. For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally stable. However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step \Delta t times the thermal diffusivity to the square of space step, \Delta x^2, is large (typically, larger than 1/2 per Von Neumann stability analysis). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is often used, which is both stable and immune to oscillations. The method ... [...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] |
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
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, to create a pattern or image on a surface, by allowing the pigment to reach only some parts of the surface. 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 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 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 remain int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Lagrange Polynomial
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree of a polynomial, degree that polynomial interpolation, interpolates a given set of data. Given a data set of graph of a function, 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 formulas, Newton–Cotes method of numerical integration and Shamir's Secret Sharing, Shamir's secret sharing scheme in cryptography. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
