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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 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] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a 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 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 e ... [...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 vector space, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Diminishing
In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten. Model equation In systems described by partial differential equations, such as the following hyperbolic advection equation, :\frac + a\frac = 0, the total variation (TV) is given by :TV(u(\cdot,t)) = \int \left, \frac \ \mathrmx , and the total variation for the discrete case is, :TV(u^n) = TV(u(\cdot,t^n)) = \sum_j \left, u_^n - u_j^n \ . where u_^n=u(x_,t^n). A numerical method is said to be total variation diminishing (TVD) if, :TV \left( u^\right) \leq TV \left( u^\right) . Characteristics A numerical scheme is said to be monotonicity preserving if the following properties are maintained: *If u^ is monotonically increasing (or decreasing) in space, then so is u^. proved the follow ... [...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 are ... [...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 wo ... [...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] |
Holistic Discretisation
Holism () is the idea that various systems (e.g. physical, biological, social) should be viewed as wholes, not merely as a collection of parts. The term "holism" was coined by Jan Smuts in his 1926 book ''Holism and Evolution''."holism, n." OED Online, Oxford University Press, September 2019, www.oed.com/view/Entry/87726. Accessed 23 October 2019. While his ideas had racist connotations, the modern use of the word generally refers to treating a person as an integrated whole, rather than as a collection of separate systems. For example, well-being may be regarded as not merely physical health, but also psychological and spiritual well-being. Meaning The exact meaning of "holism" depends on context. Jan Smuts originally used "holism" to refer to the tendency in nature to produce wholes from the ordered grouping of unit structures. However, in common usage, "holism" usually refers to the idea that a whole is greater than the sum of its parts.J. C. Poynton (1987) SMUTS'S HOLISM AND EVOL ... [...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: :''Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate.'' Professor Sergei K. 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 fo ... [...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 honor of Saint Sergius, or in Russia, of 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). It is not common in English, although the Anglo-French name Sergeant 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 originarie''. It has been speculated to derive from a more ancient Etruscan name but the etymology of the nomen Sergius is problematic. Chase hesitantly suggests a connection with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Diminishing
In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten. Model equation In systems described by partial differential equations, such as the following hyperbolic advection equation, :\frac + a\frac = 0, the total variation (TV) is given by :TV(u(\cdot,t)) = \int \left, \frac \ \mathrmx , and the total variation for the discrete case is, :TV(u^n) = TV(u(\cdot,t^n)) = \sum_j \left, u_^n - u_j^n \ . where u_^n=u(x_,t^n). A numerical method is said to be total variation diminishing (TVD) if, :TV \left( u^\right) \leq TV \left( u^\right) . Characteristics A numerical scheme is said to be monotonicity preserving if the following properties are maintained: *If u^ is monotonically increasing (or decreasing) in space, then so is u^. proved the follow ... [...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] |
