|
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] |
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] |
Computational Fluid Dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid ( liquids and gases) with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests. CFD is applied to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...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] |
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] |
Monotone Scheme
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonicity (mechanism design), a property of a social choice function. *Monotonicity criterion, a property of a voting system. *Resource monotonicity, a property of resource allocation rules and bargaining systems. In mathematics *Monotone class theorem, in measure theory *Monotone convergence theorem, in mathematics *Monotone polygon, a property of a geometric object *Monotonic function, a property of a mathematical function *Monotonicity of entailment, a property of some logical systems *Monotonically increasing, a property of number sequence Other uses *Monotone (software), an open source revision control system *Monotonic orthography, simplified spelling of modern Greek *The Monotones The Monotones were a six-member American doo-wop vocal ... [...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] |
Moscow State University
M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious university in the country. The university includes 15 research institutes, 43 faculties, more than 300 departments, and six branches (including five foreign ones in the Commonwealth of Independent States countries). Alumni of the university include past leaders of the Soviet Union and other governments. As of 2019, 13 List of Nobel laureates, Nobel laureates, six Fields Medal winners, and one Turing Award winner had been affiliated with the university. The university was ranked 18th by ''The Three University Missions Ranking'' in 2022, and 76th by the ''QS World University Rankings'' in 2022, #293 in the world by the global ''Times Higher World University Rankings'', and #326 by ''U.S. News & World Report'' in 2022. It was the highest-ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Courant–Friedrichs–Lewy Condition
In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain time in many explicit time-marching computer simulations, otherwise the simulation produces incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper. Heuristic description The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Volume Method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh. Finite volume methods can be compared and contrasted with the finite difference methods, which approximate derivatives using nodal values, or finite element methods, which create local approximations of a soluti ... [...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] |
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] |
.PNG "Click for more on -> Theorem")
