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Operator Splitting
This is a list of operator splitting topics. General *Alternating direction implicit method — finite difference method for parabolic, hyperbolic, and elliptic partial differential equations *GRADELA — simple gradient elasticity model *Matrix splitting — general method of splitting a matrix operator into a sum or difference of matrices *Paul Tseng — resolved question on convergence of matrix splitting algorithms *PISO algorithm — pressure-velocity calculation for Navier-Stokes equations *Projection method (fluid dynamics) — computational fluid dynamics method *Reactive transport modeling in porous media — modeling of chemical reactions and fluid flow through the Earth's crust *Richard S. Varga — developed matrix splitting *Strang splitting Strang splitting is a numerical method for solving differential equations that are decomposable into a sum of differential operators. It is named after Gilbert Strang. It is used to speed up calculation for problems involving opera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Direction Implicit Method
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions.. It is an example of an operator splitting method. ADI for matrix equations The method The ADI method is a two step iteration process that alternately updates the column and row spaces of an approximate solution to AX - XB = C. One ADI iteration consists of the following steps:1. Solve for X^, where \left( A - \beta_ I\right) X^ = X^\left( B - \beta_ I \right) + C. 2. Solve for X^, where X^\left( B - \alpha_ I \right) = \left( A - \alpha_ I\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GRADELA
GRADELA is a simple gradient elasticity model involving one internal length in addition to the two Lamé parameters. It allows to eliminate elastic singularities and discontinuities and to interpret elastic size effects. This model has been suggested by Elias C. Aifantis. The main advantage of GRADELA over Mindlin's elasticity models (which contains five extra constants) is the fact that solutions of boundary value problems can be found in terms of corresponding solutions of classical elasticity by operator splitting method. In 1992-1993 it has been suggested by Elias C. Aifantis a generalization of the linear elastic constitutive relations by the gradient modification that contains the Laplacian in the form : \sigma_ = \Bigl( \lambda \varepsilon_ \delta_ + 2 \mu \varepsilon_ \Bigr) - l^2_s \, \Delta \, \Bigl( \lambda \varepsilon_ \delta_ + 2 \mu \varepsilon_{ij} \Bigr) , where l_s is the scale parameter. References * E. C. Aifantis"On the role of gradients in the local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Splitting
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960. Regular splittings We seek to solve the matrix equation where A is a given ''n'' × ''n'' non-singular matrix, and k is a given column vector with ''n'' components. We split the matrix A into where B and C are ''n'' × ''n'' matrices. If, for an arbitrary ''n'' × ''n'' matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative entries, we write M1 ≥ M2. Definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Tseng
Paul Tseng () was a Chinese-American (Hakka Taiwanese) and Canadian applied mathematician and a professor at the Department of Mathematics at the University of Washington, in Seattle, Washington. Tseng was recognized by his peers to be one of the leading optimization researchers of his generation. On August 13, 2009, Paul Tseng went missing while kayaking in the Yangtze River in the Yunnan province of China and is presumed dead. Biography Paul Tseng was born September 21, 1959 in Hsinchu, Taiwan. In December 1970, Tseng's family moved to Vancouver, British Columbia. Tseng received his B.Sc. from Queen's University in 1981 and his Ph.D. from Massachusetts Institute of Technology in 1986. In 1990 Tseng moved to the University of Washington's Department of Mathematics. Tseng has conducted research primarily in continuous optimization and secondarily in discrete optimization and distributed computation. Research Tseng made many contributions to mathematical optimization, publ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PISO Algorithm
PISO algorithm (Pressure-Implicit with Splitting of Operators) was proposed by Issa in 1986 without iterations and with large time steps and a lesser computing effort. It is an extension of the SIMPLE algorithm used in computational fluid dynamics to solve the Navier-Stokes equations. PISO is a pressure-velocity calculation procedure for the Navier-Stokes equations developed originally for non-iterative computation of unsteady compressible flow, but it has been adapted successfully to steady-state problems. PISO involves one predictor step and two corrector steps and is designed to satisfy mass conservation using predictor-corrector steps. Algorithm steps The algorithm can be summed up as follows: #Set the boundary conditions. #Solve the discretized momentum equation to compute an intermediate velocity field. #Compute the mass fluxes at the cells faces. #Solve the pressure equation. #Correct the mass fluxes at the cell faces. #Correct the velocities on the basis of the new press ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection Method (fluid Dynamics)
In fluid dynamics, The projection method is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 as an efficient means of solving the incompressible Navier-Stokes equations. The key advantage of the projection method is that the computations of the velocity and the pressure fields are decoupled. The algorithm The algorithm of the projection method is based on the Helmholtz decomposition (sometimes called Helmholtz-Hodge decomposition) of any vector field into a solenoidal part and an irrotational part. Typically, the algorithm consists of two stages. In the first stage, an intermediate velocity that does not satisfy the incompressibility constraint is computed at each time step. In the second, the pressure is used to project the intermediate velocity onto a space of divergence-free velocity field to get the next update of velocity and pressure. Helmholtz–Hodge decomposition Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reactive Transport Modeling In Porous Media
Reactive transport modeling in porous media refers to the creation of computer models integrating chemical reaction with transport of fluids through the Earth's crust. Such models predict the distribution in space and time of the chemical reactions that occur along a flowpath. Reactive transport modeling in general can refer to many other processes, including reactive flow of chemicals through tanks, reactors, or membranes; particles and species in the atmosphere; gases exiting a smokestack; and migrating magma. Overview Reactive transport models are constructed to understand the composition of natural waters; the origin of economic mineral deposits; the formation and dissolution of rocks and minerals in geologic formations in response to injection of industrial wastes, steam, or carbon dioxide; and the generation of acidic waters and leaching of metals from mine wastes. They are often relied upon to predict the migration of contaminant plumes; the mobility of radionuclides in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard S
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick", "Dickon", " Dickie", "Rich", "Rick", "Rico", "Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (other) * Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strang Splitting
Strang splitting is a numerical method for solving differential equations that are decomposable into a sum of differential operators. It is named after Gilbert Strang. It is used to speed up calculation for problems involving operators on very different time scales, for example, chemical reactions in fluid dynamics, and to solve multidimensional partial differential equations by reducing them to a sum of one-dimensional problems. Fractional step methods As a precursor to Strang splitting, consider a differential equation of the form : \frac = L_1 () + L_2 () where L_1, L_2 are differential operators. If L_1 and L_2 were constant coefficient matrices, then the exact solution to the associated initial value problem would be : y(t) = e^ y_0. If L_1 and L_2 commute, then by the exponential laws this is equivalent to : y(t) = e^ e^ y_0. If they do not, then by the Baker–Campbell–Hausdorff formula it is still possible to replace the exponential of the sum by a product 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] |
Outlines Of Mathematics And Logic
Outline or outlining may refer to: * Outline (list), a document summary, in hierarchical list format * Code folding, a method of hiding or collapsing code or text to see content in outline form * Outline drawing, a sketch depicting the outer edges of a person or object, without interior details or shading * Outline (note-taking software), a note-taking application * Outline typeface, in typography * Outline VPN, a free and open-source Shadowsocks deployment tool * Outline, the representation of a word in shorthand * Step outline, or just outline, the first summary of a story for a film script Media * Outline (novel), ''Outline'' (novel), a 2014 novel by Rachel Cusk * Outlines (collection), ''Outlines'' (collection), a 1939 collection of poems by surrealist poet Jean Venturini * The Outline (website), a news company * Outlines Festival, an annual one-day music festival held in Sheffield, United Kingdom * Outline Records, record label founded by Jane Ira Bloom * The Outline (band), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
