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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 and partial differential equations, as is required in computer simulations of 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 implement. Implicit methods are used because many pro ... [...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 matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and 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. 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 = C. One ADI iteration consists of the following steps:1. Solve for X^, where \left( A - \beta_ I\right) X^ = X^\left( B - \beta_ I \right) + C. 2. Solve for X^, where X^\left( B - \alpha_ I \right) = \left( A - \alpha_ I\r ... [...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 the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an 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 place of f(t_, y_). The backward Euler method is an implicit method: the new approxima ... [...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 mid-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 better 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 of the ... [...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 * 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 mathematics * Striatal-enrich ... [...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. 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 previous equation is the ''advective form'' of the Burgers' equation. The ''conservative form'' is found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Of Beam-warming
Basis 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 trading, a trading strategy consisting of the purchase of a security and the sale of a similar security ** Basis of futures, the value differential between a future and the spot price ** Basis (options), the value differential between a call option and a put option ** 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 and technology *Basis function * Basis (linear algebra) **Dual basis ** Orthonormal basis ** Schauder basis *Basis (universal algebra) * Basis of a matroid *Generating set of an ideal: ** Gröbner basis ** Hilbert's basis theorem * Generating set of a group *Base (topology) * Change of basis * Greedoid * Normal basis * Polynomial basis * Radial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoidal Rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for 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 trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by 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 ,b/math> such that a=x_0 < x_1 < \cdots < x_ < x_N = b and be the length of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 matrices commonly arise fr ... [...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] |
Relaxation (iterative Method)
In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems. Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. They are also used for the solution of linear equations for linear least-squares problems and also for systems of linear inequalities, such as those arising in linear programming. They have also been developed for solving nonlinear systems of equations. Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. Relaxation methods are used to solve the linear equations resulti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Differences
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 the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted \Delta is the operator that maps a function to the function \Delta /math> defined by :\Delta x)= f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term "fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
