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Upwind Scheme
In computational physics, the term advection scheme refers to a class of numerical discretization methods for solving hyperbolic partial differential equations. In the so-called upwind schemes ''typically'', the so-called upstream variables are used to calculate the derivatives in a flow field. That is, derivatives are estimated using a set of data points biased to be more "upwind" of the query point, with respect to the direction of the flow. Historically, the origin of upwind methods can be traced back to the work of Richard Courant, Courant, Isaacson, and Rees who proposed the CIR method. Model equation To illustrate the method, consider the following one-dimensional linear advection equation : \frac + a \frac = 0 which describes a wave propagating along the x-axis with a velocity a. This equation is also a mathematical model for one-dimensional linear advection. Consider a typical grid point i in the domain. In a one-dimensional domain, there are only two directions associated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computational Physics
Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline (or offshoot) of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics — an area of study which supplements both theory and experiment. Overview In physics, different theories based on mathematical models provide very precise predictions on how systems behave. Unfortunately, it is often the case that solving the mathematical model for a particular system in order to produce a useful prediction is not feasible. This can occur, for instance, when the solution does not have a closed-form expression, or is too complicated. In such cases, numerical approximations are required. Computational physics is the subject that deals with these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, Academic journal, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, Technology, technical, and engineering subject areas, abandoning its literary interests. Wiley's son Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Godunov's Scheme
In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei 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 derivatives in time and space, respectively. If we integrate the hyperbolic problem over a control volume _, x_ we obtain a method of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Upwind Differencing Scheme For Convection
The upwind differencing scheme is a method used in numerical methods in computational fluid dynamics for convection–diffusion problems. This scheme is specific for Peclet number greater than 2 or less than −2 Description By taking into account the direction of the Fluid dynamics, flow, the upwind differencing scheme overcomes that inability of the central differencing scheme. This scheme is developed for strong convective flows with suppressed diffusion effects. Also known as ‘Donor Cell’ Differencing Scheme, the convected value of property \phi at the cell face is adopted from the upstream node. It can be described by Steady convection-diffusion partial Differential Equation: \frac(\rho\phi)+\nabla \cdot (\rho \mathbf \phi)\,= \nabla \cdot (\Gamma \nabla \phi) + S_ Continuity equation: \left(\rho u A \right)_ - \left(\rho u A \right)_w = 0 \, where \rho is density, \Gamma is the diffusion coefficient, \mathbf is the velocity vector, \phi is the property to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finite Difference Method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial domain and time domain (if applicable) are Discretization, discretized, or broken into a finite number of intervals, and the values of the solution at the end points of the intervals are approximated by solving algebraic equations containing finite differences and values from nearby points. Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be Nonlinear partial differential equation, nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently, and this, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Numerical Diffusion
Numerical diffusion is a difficulty with computer simulations of continua (such as fluids) wherein the simulated medium exhibits a higher diffusivity than the true medium. This phenomenon can be particularly egregious when the system should not be diffusive at all, for example an ideal fluid acquiring some spurious viscosity in a numerical model. Explanation In Eulerian simulations, time and space are divided into a discrete grid and the continuous differential equations of motion (such as the Navier–Stokes equation) are discretized into finite-difference equations. The discrete equations are in general more diffusive than the original differential equations, so that the simulated system behaves differently than the intended physical system. The amount and character of the difference depends on the system being simulated and the type of discretization that is used. Most fluid dynamics or magnetohydrodynamic simulations seek to reduce numerical diffusion to the minim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
First Order Upwind Scheme Stability
First most commonly refers to: * First, the ordinal form of the number 1 First or 1st may also refer to: Acronyms * Faint Images of the Radio Sky at Twenty-Centimeters, an astronomical survey carried out by the Very Large Array * Far Infrared and Sub-millimetre Telescope, of the Herschel Space Observatory * For Inspiration and Recognition of Science and Technology, an international youth organization * Forum of Incident Response and Security Teams, a global forum Arts and entertainment Albums * ''1st'' (album), by Streets, 1983 * ''1ST'' (SixTones album), 2021 * ''First'' (David Gates album), 1973 * ''First'', by Denise Ho, 2001 * ''First'' (O'Bryan album), 2007 * ''First'' (Raymond Lam album), 2011 Extended plays * ''1st'', by The Rasmus, 1995 * ''First'' (Baroness EP), 2004 * ''First'' (Ferlyn G EP), 2015 Songs * "First" (Lindsay Lohan song), 2005 * "First" (Cold War Kids song), 2014 * "First", by Lauren Daigle from the album '' How Can It Be'', 2015 * "First", by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Courant–Friedrichs–Lewy Condition
In mathematics, the convergence condition by Courant–Friedrichs–Lewy (CFL) 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 upper bound, given a fixed spatial increment, in many explicit time-marching computer simulations; otherwise, the simulation produces incorrect or unstable 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification). Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, ''discretization'' may also refer to modification of variable or category ''granularity'', as when multiple discrete variables are aggregated or multiple discrete categories fused. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context: one important context is numerical linear algebra, and another is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Taylor & Francis
Taylor & Francis Group is an international company originating in the United Kingdom that publishes books and academic journals. Its parts include Taylor & Francis, CRC Press, Routledge, F1000 (publisher), F1000 Research and Dovepress. It is a division of Informa, a United Kingdom-based publisher and conference company. Overview Founding The company was founded in 1852 when William Francis (chemist), William Francis joined Richard Taylor (editor), Richard Taylor in his publishing business. Taylor had founded his company in 1798. Their subjects covered agriculture, chemistry, education, engineering, geography, law, mathematics, medicine, and social sciences. Publications included the ''Philosophical Magazine''. Francis's son, Richard Taunton Francis (1883–1930), was sole partner in the firm from 1917 to 1930. Acquisitions and mergers In 1965, Taylor & Francis launched Wykeham Publications and began book publishing. T&F acquired Hemisphere Publishing in 1988, and the compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
