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Adams–Moulton Methods
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 analysis, 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 firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoidal Rule (differential Equations)
In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method and a linear multistep method. Method Suppose that we want to solve the differential equation y' = f(t,y). The trapezoidal rule is given by the formula y_ = y_n + \tfrac 1 2 h \Big( f(t_n,y_n) + f(t_,y_) \Big), where h = t_ - t_n is the step size. This is an implicit method: the value y_ appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear. One possible method for solving this equation is Newton's method. We can use the Euler method to get a fairly good estimate for the solution, which can be used as the initial guess of Newton's method. Cutting short, using only the guess from Eulers method is ... [...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 b ... [...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 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] |
Lax Equivalence Theorem
In numerical analysis, the Lax equivalence theorem is a fundamental theorem in the analysis of linear finite difference methods for the numerical solution of linear partial differential equations. It states that for a linear consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and only if it is stable. The importance of the theorem is that while the convergence of the solution of the linear finite difference method to the solution of the linear partial differential equation is what is desired, it is ordinarily difficult to establish because the numerical method is defined by a recurrence relation while the differential equation involves a differentiable function. However, consistency—the requirement that the linear finite difference method approximates the correct linear partial differential equation—is straightforward to verify, and stability is typically much easier to show than convergence (and would be needed in an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germund Dahlquist
Germund Dahlquist (16 January 1925 – 8 February 2005) was a Sweden, 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 Licentiate (degree), 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 ... [...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 Sweden, Swedish mathematician known primarily for his early contributions to the theory of numerical analysis as applied to differential equations. Dahlquist began to study mathematics ..., 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 gives ... [...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] |
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] |
Numerical Ordinary Differential Equations
Numerical methods for ordinary differential equations are methods used to find Numerical analysis, 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 firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
