WENO Methods

	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]
	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]
	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]
	Chi-Wang Shu Chi-Wang Shu (Chinese: 舒其望, born 1 January 1957) is the Theodore B. Stowell University Professor of Applied Mathematics at Brown University. He is known for his research in the fields of computational fluid dynamics, numerical solutions of conservation laws and Hamilton–Jacobi type equations. Shu has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge. Career He received his B.S. in Mathematics from the University of Science and Technology of China, Hefei, in 1982 and his Ph.D. in Mathematics from the University of California at Los Angeles in 1986. His Ph.D. thesis advisor was Stanley Osher. He started his academic career in 1987 as an assistant professor in the Division of Applied Mathematics at Brown University. He was an associate professor from 1992 to 1996 and became full professor in 1996. Honors and awards * He is the 2021 recipient of the John von Neumann Lecture Prize, the highest honor and flagship lecture of Society for Indu ... [...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]
	Numerical Differential Equations 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 ... {{disambig ... [...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]
	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]
	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]
	Chi-Wang Shu Chi-Wang Shu (Chinese: 舒其望, born 1 January 1957) is the Theodore B. Stowell University Professor of Applied Mathematics at Brown University. He is known for his research in the fields of computational fluid dynamics, numerical solutions of conservation laws and Hamilton–Jacobi type equations. Shu has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge. Career He received his B.S. in Mathematics from the University of Science and Technology of China, Hefei, in 1982 and his Ph.D. in Mathematics from the University of California at Los Angeles in 1986. His Ph.D. thesis advisor was Stanley Osher. He started his academic career in 1987 as an assistant professor in the Division of Applied Mathematics at Brown University. He was an associate professor from 1992 to 1996 and became full professor in 1996. Honors and awards * He is the 2021 recipient of the John von Neumann Lecture Prize, the highest honor and flagship lecture of Society for Indu ... [...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]
	Numerical Differential Equations 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 ... {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]