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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 Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ... 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 pu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of 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 as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Accuracy
In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider u, the exact solution to a differential equation in an appropriate normed space (V,, , \ , , ). Consider a numerical approximation u_h, where h is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method. The numerical solution u_h is said to be nth-order accurate if the error E(h):= , , u-u_h, , is proportional to the step-size h to the nth power: : E(h) = , , u-u_h, , \leq Ch^n where the constant C is independent of h and usually depends on the solution u. Using the big O notation an nth-order accurate numerical method is notated as : , , u-u_h, , = O(h^n) This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Godunov's Theorem
In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high-resolution schemes for the numerical solution of partial differential equations. The theorem states that: Professor Sergei Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name. The theorem We generally follow Wesseling (2001). Aside Assume a continuum problem described by a PDE is to be computed using a numerical scheme based upon a uniform computational grid and a one-step, constant step ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MUSCL Scheme
In the study of partial differential equations, the MUSCL scheme is a finite volume method that can provide highly accurate numerical solutions for a given system, even in cases where the solutions exhibit shocks, discontinuities, or large gradients. MUSCL stands for ''Monotonic Upstream-centered Scheme for Conservation Laws'' (van Leer, 1979), and the term was introduced in a seminal paper by Bram van Leer (van Leer, 1979). In this paper he constructed the first ''high-order'', '' total variation diminishing'' (TVD) scheme where he obtained second order spatial accuracy. The idea is to replace the piecewise constant approximation of Godunov's scheme by reconstructed states, derived from cell-averaged states obtained from the previous time-step. For each cell, slope limited, reconstructed left and right states are obtained and used to calculate fluxes at the cell boundaries (edges). These fluxes can, in turn, be used as input to a ''Riemann solver'', following which the solutions a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flux Limiter
Flux limiters are used in high resolution schemes – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations (PDEs). They are used in high resolution schemes, such as the MUSCL scheme, to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution scheme, make the solutions total variation diminishing (TVD). Note that flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have the effect of limiting the solution gradient near shocks or discontinuities. In general, the term flux limiter is used when the limiter acts on system ''fluxes'', and slope limiter is used when the limiter acts on system ''states'' (like pressure, velocity etc.). How they ... [...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-Shan Jiang 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 Order, ORDER ... * ENO methods References Further reading * * {{Numerical PDE Numerical differential equations Computational fluid dynamics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holistic Discretisation
Holism is the interdisciplinary idea that systems possess properties as wholes apart from the properties of their component parts.Julian Tudor Hart (2010''The Political Economy of Health Care''pp.106, 258 The aphorism "The whole is greater than the sum of its parts", typically attributed to Aristotle, is often given as a summary of this proposal. The concept of holism can inform the methodology for a broad array of scientific fields and lifestyle practices. When applications of holism are said to reveal properties of a whole system beyond those of its parts, these qualities are referred to as emergent properties of that system. Holism in all contexts is often placed in opposition to reductionism, a dominant notion in the philosophy of science that systems containing parts contain no unique properties beyond those parts. Proponents of holism consider the search for emergent properties within systems to be demonstrative of their perspective. Background The term "holism" was coined b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei K
Sergius is a male given name of Ancient Roman origin after the name of the Latin ''gens'' Sergia or Sergii of regal and republican ages. It is a common Christian name, in honour of Saint Sergius, or in Kyivan Rus', of Sergius of the Holy Caves (Saint Sergius the Obedient of the Kiev Caves), one of saint Fathers of Kyiv, Saint Sergius of Radonezh, and has been the name of four popes. It has given rise to numerous variants, present today mainly in the Romance (Serge, Sergio, Sergi) and Slavic languages (Serhii, Sergey, Serguei, Srđan). It is not common in English, although the Anglo-French name Sargent is possibly related to it. Etymology The name originates from the Roman ''nomen'' (patrician family name) ''Sergius'', after the name of the Roman ''gens'' of Latin origins Sergia or Sergii from Alba Longa, Old Latium, counted by Theodor Mommsen as one of the oldest Roman families, one of the original 100 ''gentes originaria''. It has been speculated to derive from a more ancien ... [...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] |
