Runge Kutta Method

5 Key excerpts on "Runge Kutta Method"

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  eBook - ePub

  Numerical Methods for Differential Equations

  A Computational Approach

  • J.R. Dormand(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 4 Runge–Kutta methods 4.1  Introduction

  Runge–Kutta methods have been popular with practitioners for many years. Originally developed by Runge towards the end of the nineteenth century and generalised by Kutta in the early twentieth century, these methods are very convenient to implement, when compared with the Taylor polynomial scheme, which requires the formation and evaluation of higher derivatives. This drawback of the Taylor method is particularly serious in the treatment of systems of differential equations but, with the growing popularity of computer software for symbolic manipulation, it seems likely that the series approach will receive further attention.

  Modern development of Runge–Kutta processes has occurred since 1960, mainly as a direct result of the advances due to J.C.Butcher in the development and simplification of RK error coefficients. The importance of Butcher’s work in this regard cannot be emphasized too greatly. His contributions have paved the way for the development of efficient high order processes on which modern differential equation software depends. However, the availability of powerful computers is itself a motivation for the development of new methods to yield ever greater accuracy.

  Some simple Runge-Kutta processes have been introduced in earlier chapters and, in Chapter 3 , the basis for the construction of formulae has been presented. An important observation from the last chapter is that, for a specific number of stages, the RK formula is not unique. This means that there exist free parameters which must be chosen. It is essential to adopt a systematic approach in making these choices and, in this chapter, criteria for building new formulae will be introduced and justified.

  One of the problems confronting any scientist or engineer with an initial value problem to solve is how to choose the best formula for the required task. With the available alternatives and the range of problems to be solved, this choice is not always straightforward. Although this chapter does not provide the ‘best’ RK formula, it does give details of the worst numerical process for solving differential equations!

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  Numerical Analysis: A R Mitchell 75th Birthday Volume

  A R Mitchell 75th Birthday Volume

  • D F Griffiths, Alistair Watson(Authors)
  • 1996(Publication Date)
  • World Scientific

    (Publisher)

  39 R U N G E -K U T T A M E T H O D S AS MATHEMATICAL OBJECTS J. C BUTCHER Mathematics Department, The University of Auckland Private Bag 92019, Auckland, New Zealand E-mail: [email protected] ABSTRACT Sets of Runge-Kutta methods, closely related in certain specific ways, are re-garded as comprising equivalence classes. Using these classes as mathematical objects, an algebraic system is built up that can be used to represent the es-sential numerical properties of Runge-Kutta methods and of sequences of meth-ods applied sequentially over several steps. By generalizing the formulation slightly, the algebraic system is able to represent other computationally signif-icant quantities. The principal application of this theory is in the analysis of multistage-multivalue-multiderivative methods. However, even for the narrow class of Runge-Kutta methods, it suggests a generalization of the order concept to include effective order. A new Runge-Kutta method of effective order 5 is given, as is a new method which achieves order 5 behaviour in a different way: by moving additional information from step to step. 1. Introduction Runge-Kutta methods have become very popular both as computational tech-niques and as thesis subjects. To survey all the wealth of applications and theoretical knowledge that has come into existence, even within the last 30 years, would be a formidable task and it will be avoided in this paper. Rather than present a Michelin guide to Runge-Kutta methods, we will here look at some aspects of these methods that make them worthy of study as mathematical objects. What could possibly justify doing this? The answer is that the sort of objects that will be constructed have a wider role as a means of studying more general numerical methods. Thus, while we will shun abstraction for its own sake, we will certainly not shun abstraction for the sake of insight and practical application.

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  Elementary Differential Equations with Linear Algebra

  • Albert L. Rabenstein(Author)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  430 Numerical Methods 8.3 RUNGE-KUTTA METHODS In the previous section we derived the formula h 2 + f{Xn-U W„-i)/ y (x n -i, W n -l)] from the Taylor formula approximation h 2 The underlying idea of the Runge-Kutta methods is exemplified by the approach we now consider. The first formula of this section is to be replaced by one of the form where a, fe, x' n -15 and W n _ l are to be chosen so that the right-hand members of the two equations differ only by a term proportional to h 3 . Thus both formulas have second-order accuracy. The advantage of the last formula is that only one function, /, need be evaluated, although at two points. In particular, f x and/ y are not involved. The determination of a, b, x' n _ t and w/_ x requires the use of the mean value theorem for functions of two variables. It is fairly elementary, but tedious. We content ourselves with the final result, 2 which may be stated as follows: ^ = ^-1+1^0 + ^1 (8.7) where ko = W(Xn-l> W n-ll fcl = ¥ ( * » -1 + K ™n-1 + fc o)· The method (8.7) is called a second-order Runge-Kutta method. In some instances a Runge-Kutta method and Taylor method of the same order may 2 The quantities a, b, x' n _ x , and W n _ x are not uniquely determined, and other choices are possible. 8.3 Runge-Kutta Method 431 give identical results (see Exercise 9), but this does not usually happen. Example We illustrate the use of formula (8.7) by calculating w x for a step size h = 0.01 for the initial value problem y' = -* 2 + y, y(0)= l. Since x 0 = 0 and vv 0 = y 0 = 1, we have, step by step, fc 0 = .01(-0+ 1) = 0.01, k l = .01[-(.01) 2 + 1.01] = 0.010099, w x = 1 + .5(.01 + .010099) = 1.0100495. If the calculations are continued until x = 1.0, the approximate value ob-tained for y(l) is 2.28168. The value obtained by the second-order Taylor method is 2.28176, and the exact value is 2.28172.

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  Numerical Analysis: Historical Developments in the 20th Century

  • C. Brezinski, L. Wuytack(Authors)
  • 2012(Publication Date)
  • Elsevier Science

    (Publisher)

  Fig. 2 , includes the formulations of methods with two derivative calculations per step, based on the mid-point and trapezoidal quadrature rules, respectively.

  Fig. 2 An extract from the Runge paper.

  2.3 The contributions of Heun and Kutta

  Following the important and prophetic work of Adams and of Runge, the new century began with further contributions to what is now known as the Runge–Kutta method, by Heun [40 ] and Kutta [45 ]. In particular, the famous method in Kutta’s paper is often known as the Runge–Kutta method. Heun’s contribution was to raise the order of the method from two and three, as in Runge’s paper, to four. This is an especially significant contribution because, for the first time, numerical methods for differential equations went beyond the use of what are essentially quadrature formulas. Even though second-order Runge methods can be looked at in this light, because the derivatives of the solution are computed from accurate enough approximations so as not to disturb the second-order behaviour, this is no longer true for orders greater than this. Write a three stage method in the form

  Y 1

  =

  y 0

  ,

  F 1

  = f

  x 0

  Y 1

  ,

  Y 2

  =

  y 0

  + h

  a 2

  F 1

  ,

  F 2

  = f

  x 0

  + h

  c 2

  ,

  Y 2

  ,

  Y 3

  =

  y 0

  + h

  a 31

  F 1

  +

  a 32

  F 2

  ,

  F 3

  = f

  x 0

  + h

  c 3

  ,

  Y 3

  ,

  y 1

  =

  y 0

  + h

  b 1

  F 1

  +

  b 2

  F 2

  +

  b 3

  F 3

  ,

  where a 21 , a 31 , a 32 , b 1 , b 2 , b 3 , c2 , c 3 are constants that characterize a particular method in this family. We can view computation of the stage values Y 1 , identical to the initial value for the step, Y 2 , which approximates the solution at x 0  + hc 2 and Y 3 , which approximates the solution at x 0  + hc 3 as temporary steps, whose only purpose is to permit the evaluation of F 1 , F 2 and F 3 as approximations to y ′(x 0 ), y ′(x 0  + hc 2 ) and y ′(x 0  + hc 3

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  Differential Equations

  A Dynamical Systems Approach to Theory and Practice

  • Marcelo Viana, José M. Espinar(Authors)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  By (4.11), its global error is O ( h ) . 118 4. Numerical integration 4.2. Runge–Kutta methods Runge–Kutta methods, which will be presented shortly, keep the strategy of using affine approximations. However, rather than following the inclination to be simply x ( t 0 ) , one uses an average of (estimates of) the derivatives of the solution at carefully chosen points t ∈ [ t 0 , t 1 ] . The goal is to obtain better approximations than those given by the Euler method, without raising the computation costs too much. The choice of such points and of their weights in the average is not unique, and it also depends on the desired order of approximation. We shall start by illustrating this approach with a simple example. Later we shall discuss this family of methods in general and give other representa-tive examples. To simplify our presentation, initially we shall consider only first order equations in dimension d = 1 . However, it is easy to extend these ideas to any order and any dimension, as we shall explain in Section 4.2.3. 4.2.1. Heun method. The idea is to approximate the solution x ( t ) by an affine function x 0 + m ( t − t 0 ) where, instead of m = x ( t 0 ) , we take the slope to be the average of the derivatives of x at the points t 0 and t 1 = t 0 + h : (4.15) m = 1 2 ( x ( t 0 ) + x ( t 1 ) ) = 1 2 ( F ( t 0 , x ( t 0 )) + F ( t 1 , x ( t 1 )) ) . The problem with (4.15) is that the right hand side depends on x ( t 1 ) , which is precisely what we want to calculate! However, it is possible to circumvent this difficulty in the following manner (see Figure 4.4). t x t 0 t 1 x ( t ) x = k 2 ( t − t 1 ) + ˜ x 1 x = k 2 ( t − t 0 ) + x 0 x = 1 2 ( k 1 + k 2 )( t − t 0 ) + x 0 x = k 1 ( t − t 0 ) + x 0 Figure 4.4. Geometric interpretation of the Heun method. Initially, we find some approximation ˜ x 1 for x ( t 1 ) , not necessarily a very good one. This can be done using the Euler method, for example.

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