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Lax–Wendroff Theorem
In computational mathematics, the Lax–Wendroff theorem, named after Peter Lax and Burton Wendroff, states that if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution. See also * Lax–Wendroff method * Godunov's scheme In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite-vol ... References * Randall J. LeVeque, Numerical methods for conservation laws, Birkhäuser, 1992 Numerical differential equations Computational fluid dynamics Theorems in analysis {{Mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Mathematics
Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006. Retrieved April 2007. A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing conjectures (particularly in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants. Areas of computational mathematics Computational mathematics emerged as a distinct part of applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Lax
Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields. In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the "Lax conjecture" grew as mathematicians working in several different areas recognized the importance of its implications in their field, until it was finally proven to be true in 2003. Life and education Lax was born in Budapest, Hungary to a Jewish family. Lax began displaying an interest in mathematics at age twelve, and soon his parents hired Rózsa Péter as a tutor for him. His parents Klara Kornfield and Henry Lax were both physici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burton Wendroff
Burton Wendroff (born March 10, 1930) is an American applied mathematician known for his contributions to the development of numerical methods for the solution of hyperbolic partial differential equations. The Lax–Wendroff method for the solution of hyperbolic PDE is named for Wendroff (as well as for Peter Lax). Wendroff is an adjunct professor at the Department of Mathematics and Statistics, University of New Mexico. He is also a retired fellow and associate at the Los Alamos National Laboratory. Wendroff is a primary author of the chess program Lachex. Together with co-author Tony Warnock, Lachex competed at two World Computer Chess Championships at Cologne (1986) and Madrid (1992). Career and research Wendroff received his B.A. degree in mathematics and physics from the New York University in 1951 and M.S. degree in mathematics from the Massachusetts Institute of Technology in 1952. After his M.S., Burt joined Los Alamos National Laboratory as a staff member. While at Los ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conservation Law (physics)
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all. A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume. From Noether's theorem, each conservation law is ... [...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] |
Hyperbolic Partial Differential Equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is : \frac = c^2 \frac The equation has the property that, if ''u'' and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), then there exists a solution for all time ''t''. The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Rela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions. Avoiding the language of distributions, one starts with a differential equation and rewrites it in such a way that no derivatives of the solution of the equation show up (the new form is called the weak formulation, and the solutions to it are called weak solutions). Somewhat surprisingly, a differential equation may have solutions which are not differentiable; and the weak formulation allows one to find such solutions. Weak solutions are important because many differential equations encountered in modelling real-world phenomena do not admit of suffici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax–Wendroff Method
The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...s. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines the governing equation is evaluated at the current time. Definition Suppose one has an equation of the following form: \frac + \frac = 0 where and are independent variables, and the initial state, is given. Linear case In the linear case, where , and is a constant, u_i^ = u_i^n - \frac A\left u_^ - u_^ \right+ \frac A^2\left u_^ -2 u_^ + u_^ \right Here n refers to the t dimension and i refers to the x di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Godunov's Scheme
In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite-volume method which solves exact, or approximate Riemann problems at each inter-cell boundary. In its basic form, Godunov's method is first order accurate in both space and time, yet can be used as a base scheme for developing higher-order methods. Basic scheme Following the classical finite-volume method framework, we seek to track a finite set of discrete unknowns, : Q^_i = \frac \int_ ^ q(t^n, x)\, dx where the x_ = x_ + \left( i - 1/2 \right) \Delta x and t^n = n \Delta t form a discrete set of points for the hyperbolic problem: : q_t + ( f( q ) )_x = 0, where the indices t and x indicate the derivations in time and space, respectively. If we integrate the hyperbolic problem over a control volume _, x_ we obtain a me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randall J
Randall may refer to the following: Places United States *Randall, California, former name of White Hall, California, an unincorporated community * Randall, Indiana, a former town *Randall, Iowa, a city *Randall, Kansas, a city *Randall, Minnesota, a city * Randall, West Virginia, an unincorporated community *Randall, Wisconsin, a town *Randall, Burnett County, Wisconsin, an unincorporated community *Randall County, Texas * Randall Creek, in Nebraska and South Dakota *Randall's Island, part of New York City *Camp Randall, Madison, Wisconsin, a former army camp, on the National Register of Historic Places *Fort Randall, South Dakota, a former military base, on the National Register of Historic Places Elsewhere *Mount Randall, Victoria Land, Antarctica *Randall Rocks, Graham Land, Antarctica *Randall, a community in the town of New Tecumseth, Ontario, Canada Businesses *Randall Amplifiers, a manufacturer of guitar amplifiers *Randall House Publications, American publisher *Randall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Differential Equations
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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] |
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] |
