|
ENO Methods
ENO (essentially non-oscillatory) methods are classes of high-resolution schemes in numerical solution of differential equations. History The first ENO scheme was developed by Harten, Engquist, Osher and Chakravarthy in 1987. In 1994, the first weighted version of ENO was developed. See also *High-resolution scheme *WENO methods In numerical solution of differential equations, WENO (weighted essentially non-oscillatory) methods are classes of high-resolution schemes. WENO are used in the numerical solution of hyperbolic partial differential equations. These methods were d ... * Shock-capturing method References Numerical differential equations Computational fluid dynamics {{fluiddynamics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
High-resolution Scheme
High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties: *Second- or higher-Order of accuracy, order spatial accuracy is obtained in smooth parts of the solution. *Solutions are free from spurious oscillations or wiggles. *High accuracy is obtained around shocks and discontinuities. *The number of mesh points containing the wave is small compared with a first-order scheme with similar accuracy. General methods are often not adequate for accurate resolution of steep gradient phenomena; they usually introduce non-physical effects such as ''smearing'' of the solution or ''spurious oscillations''. Since publication of ''Godunov's order barrier theorem'', which proved that linear methods cannot provide non-oscillatory solutions higher than first order (Godunov 1954, Godunov 1959), these difficulties have attracted much attention and a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ami Harten
Amiram Harten (1946 – 1994) was an American/Israeli applied mathematician. Harten made fundamental contribution to the development of high-resolution schemes for the solution of hyperbolic partial differential equations. Among other contributions, he developed the total variation diminishing scheme, which gives an oscillation free solution for flow with shocks. In 1980s, Harten along with Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy developed the essentially non-oscillatory (ENO) schemes. The article on ENO, titled, ''Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III'' was published in Journal of Computational Physics, in 1987 and is one of the most cited papers in the field of scientific computing. It was republished in 1997 in the same journal. Harten is listed as an ISI highly cited researcher. In 1990 Harten gave a talk on "Recent developments in shock-capturing schemes" at the International Congress of Mathematicians The Internat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Björn Engquist
Björn Engquist (also ''Bjorn Engquist''; born 2 June 1945 in Stockholm) has been a leading contributor in the areas of multiscale modeling and scientific computing, and a productive educator of applied mathematicians. Life He received his PhD in numerical analysis from University of Uppsala in 1975, and taught there during the following years while also holding a professorship at the University of California, Los Angeles. In 2001, he moved to Princeton University as the Michael Henry Stater University Professor of Mathematics and served as the director of the Program in Applied and Computational Mathematics. He has also been professor at the Royal Institute of Technology in Stockholm since 1993, and is director of the Parallel and Scientific Computing Institute. Engquist currently holds the Computational and Applied Mathematics Chair I at the Institute for Computational Engineering and Sciences at the University of Texas at Austin, after leaving Princeton in 2005. Research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley Osher
Stanley Osher (born April 24, 1942) is an American mathematician, known for his many contributions in shock capturing, level-set methods, and PDE-based methods in computer vision and image processing. Osher is a professor at the University of California, Los Angeles (UCLA), Director of Special Projects in the Institute for Pure and Applied Mathematics (IPAM) and member of the California NanoSystems Institute (CNSI) at UCLA. He has a daughter, Kathryn, and a son, Joel. Education * BS, Brooklyn College, 1962 * MS, New York University, 1964 * PhD, New York University, 1966 Research interests * Level-set methods for computing moving fronts * Approximation methods for hyperbolic conservation laws and Hamilton–Jacobi equations * Total variation (TV) and other PDE-based image processing techniques * Scientific computing * Applied partial differential equations * L1/TV-based convex optimization Osher is listed as an ISI highly cited researcher. Research contributions Osher was th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WENO Methods
In numerical solution of differential equations, WENO (weighted essentially non-oscillatory) methods are classes of high-resolution schemes. WENO are used in the numerical solution of hyperbolic partial differential equations. These methods were developed from ENO methods (essentially non-oscillatory). The first WENO scheme was developed by Liu, Osher and Chan in 1994. In 1996, Guang-Sh and Chi-Wang Shu developed a new WENO scheme called WENO-JS. Nowadays, there are many WENO methods. See also *High-resolution scheme High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties: *Second- or higher-order spatial accur ... * ENO methods References Further reading * * {{Numerical PDE Numerical differential equations Computational fluid dynamics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
High-resolution Scheme
High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties: *Second- or higher-Order of accuracy, order spatial accuracy is obtained in smooth parts of the solution. *Solutions are free from spurious oscillations or wiggles. *High accuracy is obtained around shocks and discontinuities. *The number of mesh points containing the wave is small compared with a first-order scheme with similar accuracy. General methods are often not adequate for accurate resolution of steep gradient phenomena; they usually introduce non-physical effects such as ''smearing'' of the solution or ''spurious oscillations''. Since publication of ''Godunov's order barrier theorem'', which proved that linear methods cannot provide non-oscillatory solutions higher than first order (Godunov 1954, Godunov 1959), these difficulties have attracted much attention and a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shock-capturing Method
In computational fluid dynamics, shock-capturing methods are a class of techniques for computing inviscid flows with shock waves. The computation of flow containing shock waves is an extremely difficult task because such flows result in sharp, discontinuous changes in flow variables such as pressure, temperature, density, and velocity across the shock. Method In shock-capturing methods, the governing equations of inviscid flows (i.e. Euler equations) are cast in conservation form and any shock waves or discontinuities are computed as part of the solution. Here, no special treatment is employed to take care of the shocks themselves, which is in contrast to the shock-fitting method, where shock waves are explicitly introduced in the solution using appropriate shock relations ( Rankine–Hugoniot relations). The shock waves predicted by shock-capturing methods are generally not sharp and may be smeared over several grid elements. Also, classical shock-capturing methods have the disadvan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Differential Equations
{{disambig ...
Numerical may refer to: * Number * Numerical digit * 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
