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runge-kutta methods

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Runge–Kutta methods

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In numerical analysis, the Runge–Kutta methods (pronounced /??u?ge?kuta/) are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta.

See the article on numerical ordinary differential equations for more background and other methods. See also List of Runge-Kutta methods.

Contents

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The common fourth-order Runge–Kutta method

One member of the family of Runge–Kutta methods is so commonly used that it is often referred to as "RK4" or simply as "the Runge–Kutta method".

Let an initial value problem be specified as follows.

Then, the RK4 method for this problem is given by the following equations:

where yn + 1 is the RK4 approximation of y(tn + 1), and


Thus, the next value (yn + 1) is determined by the present value (yn) plus the product of the size of the interval (h) and an estimated slope. The slope is a weighted average of slopes:

  • k1 is the slope at the beginning of the interval;
  • k2 is the slope at the midpoint of the interval, using slope k1 to determine the value of y at the point tn + h / 2 using Euler's method;
  • k3 is again the slope at the midpoint, but now using the slope k2 to determine the y-value;
  • k4 is the slope at the end of the interval, with its y-value determined using k3.

In averaging the four slopes, greater weight is given to the slopes at the midpoint:

The RK4 method is a fourth-order method, meaning that the error per step is on the order of h5, while the total accumulated error has order h4.

Note that the above formulae are valid for both scalar- and vector-valued functions (i.e., y can be a vector and f an operator). For example one can integrate Schrödinger's equation using the Hamiltonian operator as function f.

Explicit Runge–Kutta methods

The family of explicit Runge–Kutta methods is a generalization of the RK4 method mentioned above. It is given by

where

(Note: the above equations have different but equivalent definitions in different texts).

To specify a particular method, one needs to provide the integer s (the number of stages), and the coefficients aij (for 1 ≤ j < is), bi (for i = 1, 2, ..., s) and ci (for i = 2, 3, ..., s). These data are usually arranged in a mnemonic device, known as a Butcher tableau (after John C. Butcher):

  0
  c2 a21
  c3 a31 a32
   
  cs as1 as2 as,s − 1  
    b1 b2 bs − 1 bs

The Runge–Kutta method is consistent if

There are also accompanying requirements if we require the method to have a certain order p, meaning that the truncation error is O(hp+1). These can be derived from the definition of the truncation error itself. For example, a 2-stage method has order 2 if b1 + b2 = 1, b2c2 = 1/2, and b2a21 = 1/2.

Examples

The RK4 method falls in this framework. Its tableau is:

  0
  1/2 1/2
  1/2 0 1/2
  1 0 0 1  
    1/6 1/3 1/3 1/6

However, the simplest Runge–Kutta method is the (forward) Euler method, given by the formula yn + 1 = yn + hf(tn,yn). This is the only consistent explicit Runge–Kutta method with one stage. The corresponding tableau is:

  0  
    1

An example of a second-order method with two stages is provided by the midpoint method

The corresponding tableau is:

  0
  1/2 1/2  
    0 1

Note that this 'midpoint' method is not the optimal RK2 method. An alternative is provided by Heun's method, where the 1/2's in the tableau above are replaced by 1's and the b's row is [1/2, 1/2]. If one wants to minimize the truncation error, the method below should be used (Atkinson p. 423). Other important methods are Fehlberg, Cash-Karp and Dormand-Prince. Also, read the article on Adaptive Stepsize.

Usage

The following is an example usage of a two-stage explicit Runge–Kutta method:

  0
  2/3 2/3  
    1/4 3/4

to solve the initial-value problem

with step size h=0.025.

The tableau above yields the equivalent corresponding equations below defining the method:

k1 = yn
t0 = 1    
  y0 = 1
t1 = 1.025
  k1 = y0 = 1 f(t0,k1) = 2.557407725 k2 = y0 + 2 / 3hf(t0,k1) = 1.042623462
  y1 = y0 + h(1 / 4 * f(t0,k1) + 3 / 4 * f(t0 + 2 / 3h,k2)) = 1.066869388
t2 = 1.05
  k1 = y1 = 1.066869388 f(t1,k1) = 2.813524695 k2 = y1 + 2 / 3hf(t1,k1) = 1.113761467
  y2 = y1 + h(1 / 4 * f(t1,k1) + 3 / 4 * f(t1 + 2 / 3h,k2)) = 1.141332181
t3 = 1.075
  k1 = y2 = 1.141332181 f(t2,k1) = 3.183536647 k2 = y2 + 2 / 3hf(t2,k1) = 1.194391125
  y3 = y2 + h(1 / 4 * f(t2,k1) + 3 / 4 * f(t2 + 2 / 3h,k2)) = 1.227417567
t4 = 1.1
  k1 = y3 = 1.227417567 f(t3,k1) = 3.796866512 k2 = y3 + 2 / 3hf(t3,k1) = 1.290698676
  y4 = y3 + h(1 / 4 * f(t3,k1) + 3 / 4 * f(t3 + 2 / 3h,k2)) = 1.335079087

The numerical solutions correspond to the underlined values. Note that f(ti,k1) has been calculated to avoid recalculation in the yis.

Adaptive Runge-Kutta methods

The adaptive methods are designed to produce an estimate of the local truncation error of a single Runge-Kutta step. This is done by having two methods in the tableau, one with order p and one with order p − 1.

The lower-order step is given by

where the ki are the same as for the higher order method. Then the error is

which is O(hp). The Butcher Tableau for this kind of method is extended to give the values of :

  0
  c2 a21
  c3 a31 a32
   
  cs as1 as2 as,s − 1  
    b1 b2 bs − 1 bs
   

The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4. Its extended Butcher Tableau is:

  0
  1/4 1/4
  3/8 3/32 9/32
  12/13 1932/2197 −7200/2197 7296/2197
  1 439/216 −8 3680/513 -845/4104
  1/2 −8/27 2 −3544/2565 1859/4104 −11/40  
    16/135 0 6656/12825 28561/56430 −9/50 2/55
    25/216 0 1408/2565 2197/4104 −1/5 0

However, the simplest adaptive Runge-Kutta method involves combining the Heun method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is:

  0
  1 1  
    1/2 1/2
    1 0

The error estimate is used to control the stepsize.

Other adaptive Runge–Kutta methods are the Bogacki–Shampine method (orders 3 and 2), the Cash–Karp method and the Dormand–Prince method (both with orders 5 and 4).

Implicit Runge-Kutta methods

The implicit methods are more general than the explicit ones. The distinction shows up in the Butcher Tableau: for an implicit method, the coefficient matrix aij is not necessarily lower triangular:

The approximate solution to the initial value problem reflects the greater number of coefficients:

Due to the fullness of the matrix aij, the evaluation of each ki is now considerably involved and dependent on the specific function f(t,y). Despite the difficulties, implicit methods are of great importance due to their high (possibly unconditional) stability, which is especially important in the solution of partial differential equations. The simplest example of an implicit Runge-Kutta method is the backward Euler method:

The Butcher Tableau for this is simply:

It can be difficult to make sense of even this simple implicit method, as seen from the expression for k1:

In this case, the awkward expression above can be simplified by noting that

so that

from which

follows. Though simpler then the "raw" representation before manipulation, this is an implicit relation so that the actual solution is problem dependent. Multistep implicit methods have been used with success by some researchers. The combination of stability, higher order accuracy with fewer steps, and stepping that depends only on the previous value makes them attractive; however the complicated problem-specific implementation and the fact that ki must often be approximated iteratively means that they are not common.

[edit] References

  • J. C. Butcher, Numerical methods for ordinary differential equations, ISBN 0471967580
  • George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Chapter 6.)
  • Ernst Hairer, Syvert Paul Nørsett, and Gerhard Wanner. Solving ordinary differential equations I: Nonstiff problems, second edition. Berlin: Springer Verlag, 1993. ISBN 3-540-56670-8.
  • William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Sections 16.1 and 16.2.)
  • Kaw, Autar; Kalu, Egwu (2008), Numerical Methods with Applications (1st ed.), www.autarkaw.com .
  • Kendall E. Atkinson. An Introduction to Numerical Analysis. John Wiley & Sons - 1989
  • F. Cellier, E. Kofman. Continuous System Simulation. Springer Verlag, 2006. ISBN 0-387-26102-8.


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