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Linear Multistep Method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of ''linear'' multistep methods, a linear combination of the previous points and derivative values is used. Definitions Numerical methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Ordinary Differential Equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. The problem A first-order differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest Ray Moulton
Forest Ray Moulton (April 29, 1872 – December 7, 1952) was an American astronomer. Biography He was born in Le Roy, Michigan, and was educated at Albion College. After graduating in 1894 (Bachelor of Arts, A.B.), he performed his graduate studies at the University of Chicago and gained a Doctor of Philosophy, Ph.D. in 1899. At the University of Chicago he was associate in astronomy (1898–1900), instructor (1900–03), assistant professor (1903–08), associate professor (1908–12), and professor after 1912. He is noted for being a proponent, along with Thomas Chrowder Chamberlin, Thomas Chamberlin, of the Chamberlin–Moulton planetesimal hypothesis that the planets coalesced from smaller bodies they termed planetesimals. Their hypothesis called for the close passage of another star to trigger this condensation, a concept that has since fallen out of favor. In the first decades of the twentieth century, some additional small satellites were discovered to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Energy Gain
In computer simulations of mechanical systems, energy drift is the gradual change in the total energy of a closed system over time. According to the laws of mechanics, the energy should be a constant of motion and should not change. However, in simulations the energy might fluctuate on a short time scale and increase or decrease on a very long time scale due to numerical integration artifacts that arise with the use of a finite time step Δ''t''. This is somewhat similar to the flying ice cube problem, whereby numerical errors in handling equipartition of energy can change vibrational energy into translational energy. More specifically, the energy tends to increase exponentially; its increase can be understood intuitively because each step introduces a small perturbation δv to the true velocity vtrue, which (if uncorrelated with v, which will be true for simple integration methods) results in a second-order increase in the energy :E = \sum m \mathbf^ = \sum m \mathbf_\mathrm^ + \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiff Equation
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution. When integrating a differential equation numerically, one would expect the requisite step size to be relatively small in a region where the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with slope nearly zero. For some problems this is not the case. In order for a numerical method to give a reliable solution to the differential system sometimes the step size is required to be at an unacceptably small level in a region where the solution curve is very smooth. The phenomenon is known as ''stiffness''. In some cases there may be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Truncation Error
Truncation errors in numerical integration are of two kinds: * ''local truncation errors'' – the error caused by one iteration, and * ''global truncation errors'' – the cumulative error caused by many iterations. Definitions Suppose we have a continuous differential equation : y' = f(t,y), \qquad y(t_0) = y_0, \qquad t \geq t_0 and we wish to compute an approximation y_n of the true solution y(t_n) at discrete time steps t_1,t_2,\ldots,t_N . For simplicity, assume the time steps are equally spaced: : h = t_n - t_, \qquad n=1,2,\ldots,N. Suppose we compute the sequence y_n with a one-step method of the form : y_n = y_ + h A(t_, y_, h, f). The function A is called the ''increment function'', and can be interpreted as an estimate of the slope \frac . Local truncation error The local truncation error \tau_n is the error that our increment function, A , causes during a single iteration, assuming perfect knowledge of the true solution at the previous ite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference Method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is 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, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently which, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis. Today, FDM are one of the most common approaches to the numerical solution of PDE, along with finite element metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax Equivalence Theorem
Los Angeles International Airport , commonly referred to as LAX (with each letter pronounced individually), is the primary international airport serving Los Angeles, California and its surrounding metropolitan area. LAX is located in the Westchester neighborhood of Los Angeles, southwest of Downtown Los Angeles, with the commercial and residential areas of Westchester to the north, the city of El Segundo to the south and the city of Inglewood to the east. LAX is the closest airport to the Westside and the South Bay. The airport is operated by Los Angeles World Airports (LAWA), a branch of the Los Angeles city government, that also operates Van Nuys Airport for general aviation. The airport covers of land and has four parallel runways. In 2019, LAX handled 88,068,013 passengers, making it the world's third-busiest and the United States' second-busiest airport following Hartsfield–Jackson Atlanta International Airport. As the largest and busiest international airp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germund Dahlquist
Germund Dahlquist (16 January 1925 – 8 February 2005) was a Swedish mathematician known primarily for his early contributions to the theory of numerical analysis as applied to differential equations. Dahlquist began to study mathematics at Stockholm University in 1942 at the age of 17, where he cites the Danish mathematician Harald Bohr (who was living in exile after the occupation of Denmark during World War II) as a profound influence. He received the degree of licentiat from Stockholm University in 1949, before taking a break from his studies to work at the Swedish Board of Computer Machinery (Matematikmaskinnämnden), working on (among other things) the early computer BESK, Sweden's first. During this time, he also worked with Carl-Gustaf Rossby on early numerical weather forecasts. Dahlquist returned to Stockholm University to complete his Ph.D., ''Stability and Error Bounds in the Numerical Solution of Ordinary Differential Equations'', which he defended in 1958, with Fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Stability
Zero-stability, also known as D-stability in honor of Germund Dahlquist Germund Dahlquist (16 January 1925 – 8 February 2005) was a Swedish mathematician known primarily for his early contributions to the theory of numerical analysis as applied to differential equations. Dahlquist began to study mathematics at Sto ..., refers to the stability of a numerical scheme applied to the simple initial value problem y'(x) = 0. A linear multistep method is ''zero-stable'' if all roots of the characteristic equation that arises on applying the method to y'(x) = 0 have magnitude less than or equal to unity, and that all roots with unit magnitude are simple. This is called the ''root condition'' and means that the parasitic solutions of the recurrence relation will not grow exponentially. Example The following third-order method has the highest order possible for any explicit two-step method for solving y'(x) = f(x): y_ + 4 y_ - 5y_n = h(4f_ + 2 f_n). If f(x)=0 identically, this gi ... [...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] |
Local Truncation Error
Truncation errors in numerical integration are of two kinds: * ''local truncation errors'' – the error caused by one iteration, and * ''global truncation errors'' – the cumulative error caused by many iterations. Definitions Suppose we have a continuous differential equation : y' = f(t,y), \qquad y(t_0) = y_0, \qquad t \geq t_0 and we wish to compute an approximation y_n of the true solution y(t_n) at discrete time steps t_1,t_2,\ldots,t_N . For simplicity, assume the time steps are equally spaced: : h = t_n - t_, \qquad n=1,2,\ldots,N. Suppose we compute the sequence y_n with a one-step method of the form : y_n = y_ + h A(t_, y_, h, f). The function A is called the ''increment function'', and can be interpreted as an estimate of the slope \frac . Local truncation error The local truncation error \tau_n is the error that our increment function, A , causes during a single iteration, assuming perfect knowledge of the true solution at the previous ite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
