|
Beam And Warming Scheme
In numerical mathematics, Beam and Warming scheme or Beam–Warming implicit scheme introduced in 1978 by Richard M. Beam and R. F. Warming, is a second order accurate Explicit and implicit methods, implicit scheme, mainly used for solving non-linear hyperbolic equations. It is not used much nowadays. Introduction This scheme is a spatially factored, non iterative, Alternating direction implicit method, ADI scheme and uses Backward Euler method, implicit Euler to perform the time Integration. The algorithm is in delta-form, linearized through implementation of a Taylor series, Taylor-series. Hence observed as increments of the conserved variables. In this an efficient factored algorithm is obtained by evaluating the spatial cross derivatives explicitly. This allows for direct derivation of scheme and efficient solution using this computational algorithm. The efficiency is because although it is three-time-level scheme, but requires only two time levels of data storage. This resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Explicit And Implicit Methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary differential equation, ordinary and partial differential equations, as is required in computer simulations of Process (science), physical processes. ''Explicit methods'' calculate the state of a system at a later time from the state of the system at the current time, while ''implicit methods'' find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if Y(t) is the current system state and Y(t+\Delta t) is the state at the later time (\Delta t is a small time step), then, for an explicit method : Y(t+\Delta t) = F(Y(t))\, while for an implicit method one solves an equation : G\Big(Y(t), Y(t+\Delta t)\Big)=0 \qquad (1)\, to find Y(t+\Delta t). Computation Implicit methods require an extra computation (solving the above equation), and they can be much harder to impl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Direction Implicit Method
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester equation, Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and Control theory, control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve Parabolic partial differential equation, parabolic and Elliptic partial differential equation, elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions.. It is an example of an operator splitting method. The method was developed at Humble Oil in the mid-1950s by Jim Douglas Jr, Henry Rachford, and Don Peaceman. ADI for matrix equations The method The ADI method is a two step iteration process that alternately updates the column and row spaces of an approximate solution to AX - XB = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backward Euler Method
In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for ordinary differential equations, numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an explicit and implicit methods, implicit method. The backward Euler method has error of order one in time. Description Consider the ordinary differential equation : \frac = f(t,y) with initial value y(t_0) = y_0. Here the function f and the initial data t_0 and y_0 are known; the function y depends on the real variable t and is unknown. A numerical method produces a sequence y_0, y_1, y_2, \ldots such that y_k approximates y(t_0+kh) , where h is called the step size. The backward Euler method computes the approximations using : y_ = y_k + h f(t_, y_). This differs from the (forward) Euler method in that the forward method uses f(t_k, y_k) in p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Steps In Beam And Warming
Step(s) or STEP may refer to: Common meanings * Steps, making a staircase * Walking * Dance move * Military step, or march ** Marching Arts Films and television * ''Steps'' (TV series), Hong Kong * ''Step'' (film), US, 2017 Literature * ''Steps'' (novel), by Jerzy Kosinski * Systematic Training for Effective Parenting, a book series Music * Step (music), pitch change * Steps (pop group), UK * ''Step'' (Kara album), 2011, South Korea ** "Step" (Kara song) * ''Step'' (Meg album), 2007, Japan * "Step" (Vampire Weekend song) * "Step" (ClariS song) Organizations * STEP (company), Belgium * Society of Trust and Estate Practitioners, international professional body for advisers who specialise in inheritance and succession planning * Board on Science, Technology, and Economic Policy of the U.S. National Academies * Solving the E-waste Problem, a UN organization Science, technology, and mathematics * Step (software), a physics simulator in KDE * Step function, in mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burgers' Equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. For a given field u(x,t) and diffusion coefficient (or ''kinematic viscosity'', as in the original fluid mechanical context) \nu, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system: :\frac + u \frac = \nu\frac. The term u\partial u/\partial x can also be rewritten as \partial(u^2/2)/\partial x. When the diffusion term is absent (i.e. \nu=0), Burgers' equation becomes the inviscid Burgers' equation: :\frac + u \frac = 0, which is a prototype for conservation equations that can develop discontinuities (shock waves). The reason f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conservation Form
Conservation form or ''Eulerian form'' refers to an arrangement of an equation or system of equations, usually representing a hyperbolic system, that emphasizes that a property represented is conserved, i.e. a type of continuity equation. The term is usually used in the context of continuum mechanics. General form Equations in conservation form take the form \frac + \boldsymbol \nabla \cdot \mathbf f(\xi) = 0 for any conserved quantity \xi, with a suitable function \mathbf f. An equation of this form can be transformed into an integral equation \frac d \int_V \xi ~ dV = -\oint_ \mathbf f(\xi) \cdot \boldsymbol \nu ~ dS using the divergence theorem. The integral equation states that the change rate of the integral of the quantity \xi over an arbitrary control volume V is given by the flux \mathbf f(\xi) through the boundary of the control volume, with \boldsymbol \nu being the outer surface normal through the boundary. \xi is neither produced nor consumed inside of V and is hence co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Of Beam-warming
Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis swap, an interest rate swap * Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation * Tax basis, cost of an asset Securities markets and trading strategies * Basis trading, a trading strategy consisting of the purchase of a security and the sale of a similar security Fixed income markets: * Treasury basis trade, a leveraged arbitrage strategy exploiting price differences between Treasury securities and futures contracts * Index arbitrage, a strategy that exploits price differences between a stock index and its futures contract Commodities and physical assets: * Commo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoidal Rule
In calculus, the trapezoidal rule (or trapezium rule in British English) is a technique for numerical integration, i.e., approximating the definite integral: \int_a^b f(x) \, dx. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that \int_^ f(x) \, dx \approx (b-a) \cdot \tfrac(f(a)+f(b)). The integral can be even better approximated by Partition of an interval, partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let \ be a partition of [a,b] such that a=x_0 < x_1 < \cdots < x_ < x_N = b and be the length of the -th subinterval (that is, ), then |
Tridiagonal Matrix Algorithm
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve Tridiagonal matrix, tridiagonal systems of equations. A tridiagonal system for ''n'' unknowns may be written as :a_i x_ + b_i x_i + c_i x_ = d_i, where a_1 = 0 and c_n = 0. : \begin b_1 & c_1 & & & 0 \\ a_2 & b_2 & c_2 & & \\ & a_3 & b_3 & \ddots & \\ & & \ddots & \ddots & c_ \\ 0 & & & a_n & b_n \end \begin x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end = \begin d_1 \\ d_2 \\ d_3 \\ \vdots \\ d_n \end . For such systems, the solution can be obtained in O(n) operations instead of O(n^3) required by Gaussian elimination. A first sweep eliminates the a_i's, and then an (abbreviated) backward substitution produces the solution. Examples of such matrice ... [...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 Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ... 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 and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
