Preface

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This section is devoted to fourth order Runge--Kutta algorithm for solving first order differential equations subject to the prescribed initial condition.

Runge-Kutta of order 4

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The fourth-order formula, known as the Runge--Kutta formula, has been used extensively to obtain approximate solutions of differential equations of first, second, and higher orders. The original idea for such formulas seems to be due to C. Runge. This idea was used more effectively for first-order equations by W. Kutta and for second-order equations by E.J. Nyström. The extension to order n was made by R. Zurmuhl.

Each Runge--Kutta method is derived from an appropriate Taylor method in such a way that the final global error is of order O(h^m), so it is of order m. These methods can be constructed for any order m. All Runge--Kutta algorithms of the same order m are equivalent from numerical analysis point of view as having the same accuracy. However, there method may provide slightly different numerical answers due to round-off errors because they use different number of arithmetic operations. A trade-off to perform several slope evaluations at each step and obtain a good accuracy makes the Runge--Kutta of order 4 the most popular.

As usual, we consider an initial value problem \( y' = f(x,y) , \quad y(x_0 )= y_0 \) and seek approximate values y_k to the actual solution \( y= \phi (x) \) at mesh points \( x_n = x_0 + n\,h , \quad n=0,1,2,\ldots . \) The fourth order Runge--Kutta method is based on computing y_n+1 as follows

where coefficients are calculated as By matching coefficients with those of the Taylor method of order 4 so that the local truncation error is of order \( O( h^5 ) , \) the founders of the method were able to obtain the following system of equations: The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary. We are going to present some most useful choices for these coefficients.

 

So far the most often used is the classical fourth-order Runge-Kutta formula, which has a certain sleekness of organization about it:

where The fourth-order Runge-Kutta method requires four evaluations of the right-hand side per step h. This will be superior to the midpoint method if at least twice as large a step is possible. Generally speaking, high order does not always mean high accuracy. Predictor-corrector methods can be very much more efficient for problems where very high accuracy is a requirement.

The Runge-Kutta method iterates the x-values by simply adding a fixed step-size of h at each iteration. The y-iteration formula is far more interesting. It is a weighted average of four coefficients. Doing this,s we see that k₁ and k₄ are given a weight of 1/6 in the weighted average, whereas k₂ and k₃ are weighted 1/3, or twice as heavily as k₁ and k₄. (As usual with a weighted average, the sum of the weights 1/6, 1/3, 1/3 and 1/6 is 1.) So what are these k_i values that are being used in the weighted average?

• k₁ is simply Euler's prediction.
• k₂ is evaluated halfway across the prediction interval and gives us an estimate of the slope of the solution curve at this halfway point.
• k₃ has a formula which is quite similar to that of k₂ except that where k₁ used to be, there is now a k₂. Essentially, the f-value here is yet another estimate of the slope of the solution at the "midpoint" of the prediction interval. This time, however, the y-value of the midpoint is not based on Euler's prediction, but on the y-jump predicted already with k₂.
• k₄ evaluates f at \( x_n + h , \) which is at the extreme right of the prediction interval. The y-value coupled with this \( y_n + k_3 , \) is an estimate of the y-value at the right end of the interval, based on the y-jump just predicted by k₃.

Here is a pseudocode to implement the classical Runge--Kutta method of order 4:

procedure RK4(f,t,x,h,n)
integer j,n;            real k₁, k₂, k₃, k₄, h, t, t₀, x
external function f
output 0, t, x
t₀ ← t
for j = 1 to n
   k₁ ← h∗f(t,x)
   k₂ ← h∗f(t+h/2,x+k₁/2)
   k₂ ← h∗f(t+h/2,x+k₂/2)
   k₄ ← h∗f(t+h,x+k₃)
   x ← x + h∗(k₁ + 2k₂ + 2k₃ + k₄)/6
   t ← t₀ + j∗h
   output j, t, x
end for
end procedure RK4

Here is one step iteration code: This subroutine oneRK4 can be used to collect data into list and the plot, as the following example illustrates. Example: Demonstration of oneRK4 Example: Standard Riccati equation Example: differential equation with rational slope Example: Standard Riccati equation One step Runge--Kutta method of order 4 (RK4) can be coded as follows Next, we make preparation for output (called "preprocessing" in computer science) in the different forms. So we make the reference (or association) to possible outputs:

• list
• line
• plot
• table

There is a special Mathematica command <| followed by |> . What is inside these two marks is the definition of references. Example: Oscillating slope function

 

Kutta's algorithm (1901) of order four:

where If one wants to see the whole list of intermediate values calculated by this algorithm, then replace the last line with the following: Example: Kutta method solve the Riccati equation

We demonstrate other implementations of Kutta's algorithm using Mathematica. For simplicity, we reconsider the IVP: \( y' = x^2 - y^2 , \quad y(1) =1 .\)

Now we apply the same code with modified last line

Another version:

 

Gill's method:

where

Another version:

Then we exercute the code by revisiting our previous example: \( y' = x^2 - y^2 , \quad y(1) =1 .\)

 

Example: linear differential equation

 

Here is the fifth order Runge--Kutta method proposed by Butcher:

where We impliment the above algorithm using Mathematica as follows.

Another version of Butcher's method:

Example: RK5 solves the linear differential equation

 

One way to guarantee accuracy in the solution of an initial value problem is t solve the problem twice using step sizes h and h/2 and compare answers at the mesh points corresponding to the largest step size. However, this requires a significant amount of computation for the smaller step length and must be repeated if it is determined that the agreement is not good enough.

The Runge--Kutta--Fehlberg method (denoted RKF45) or Fehlberg method was developed by the German mathematician Erwin Fehlberg (1911--1990) in 1969 NASA report. The novelty of Fehlberg's method is that it is an embedded method from the Runge-Kutta family, and it has a procedure to determine if the proper step size h is being used. At each step, two different approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step length is reduced. If the answers agree to more significant digits than required, the step size is increased. RKF45 method is a method of order \( O( h^4 ) \) with an error estimator of order \( O( h^5 ) . \)

Initially, Erwin Fehlberg introduced a so-called six stage Runge-Kutta method that requires six function evaluations per step. These function values can be combined with one set of coefficients to produce a fifth-order accurate approximation and with another set of coefficients to produce an independent fourth-order accurate approximation. Comparing these two approximations provides an error estimate and resulting step size control. Notice that it takes six stages to get fifth order. It is not possible to get fifth order with only five function evaluations per step. The new ode45 introduced in the late 1990s is based on an algorithm of Dormand and Prince (DOPRI). It uses six stages, employs the FSAL strategy, provides fourth and fifth order formulas, has local extrapolation and a companion interpolant. The new ode45 is so effective that the interpolant is called up to produce more finely spaced output at no additional cost measured in function evaluations. The codes for the two routines ode23 and ode45 are very similar.

Butcher tableau for Fehlberg's 4(5) method

Here is a pseudocode for the Runge--Kutta--Fehlberg method RK45:

procedure RK45(f,t,x,h,n,ε)
real ε, k₁, k₂, k₃, k₄, k₅, k₆, h, t, t₀, x, x₄
external function f
real c₂₀ ← 0.25, c₂₁ ← 0.25
real c₃₀ ← 0.375, c₃₁ ← 0.09375, c₃₂ ← 0.28125
real c₄₀ ← 12./13., c₄₁ ← 1932./2197.
real c₄₂ ← -7200./2197., c₄₃ ← 7296./2197.
real c₅₁ ← 439./216., c₅₂ ← -8.
real c₅₃ ← 3680./513., c₅₄ ← -845./4104.
real c₆₀ ← 0.5, c₆₁ ← -8./27., c₆₂ ← 2.
real c₆₃ ← -3544./2565., c₆₄ ← 1859./4104.
real c₆₅ ← -0.275
real a₁ ← 25./216., a₂ ← 0., a₃ ← 1408./2565.
real a₄ ← 2197./4104., a₅ ← -0.2
real b₁ ← 16./135., b₂ ← 0., b₃ ← 6656./12825.
real b₄ ← 28561./56430., b₅ ← -0.18
real b₆ ← 2./55.,
k₁ ← h∗f(t,x)
k₂ ← h∗ f(t+h∗c₂₀, x+c₂₁∗k₁)
k₃ ← h∗ f(t+h∗c₃₀, x+c₃₁∗k₁ + c₃₂∗k₂)
k₄ ← h∗ f(t+h∗c₄₀, x+c₄₁∗k₁ + c₄₂∗k₂ + c₄₃∗k₃)
k₅ ← h∗ f(t+h, x+c₅₁∗k₁ + c₅₂∗k₂ + c₅₃∗k₃ + c₅₄∗k₄)
k₆ ← h∗ f(t+h∗c₆₀, x+c₆₁∗k₁ + c₆₂∗k₂ + c₆₃∗k₃ + c₆₄∗k₄ + c₆₅∗k₅)
x₄ ← x + a₁k₁ + a₃k₃ + a₅k₅
x₄ ← x + b₁k₁ + b₃k₃ + b₄k₄ + b₅k₅ + b₆k₆
t ← t + h
ε ← |x - x₄|
end procedure RK45

We now describe a simple adaptive procedure in the RK45 procedure, the fourth- and fifth-order approximations for x(t+h), say x₄ and x₅, are computed from six function evaluations, and the error estimate ε = |x₄ - x₅| is known. From user specified bounds on the allowable error estimate, the step size h is doubled or halved as needed to keep ε within these bounds. A range for the allowable step size h is also specified by the user. Clearly, the user must set the bounds carefully so that the adaptive procedure does not get caught in a loop, trying repeatedly to halve and double the step size from the same point to meet error bounds that are too restrictive for the given differential equation. This adaptive process includes the following steps.

1. Given a step size h and an initial value x(t), the RK45 routine computes the value x(t+h) and an error estimate ε.
2. If ε_min < ε < ε_max, then the step size h is not changed and the the next step size is taken by repeating step I with initial value x(t+h).
3. If ε < ε_min, then h is replaced by 2h, provided that |2h| < h_max.
4. If ε > ε_max, then h is replaced by h/2, provided that |h/2| > h_min.
5. If h_min ≤ |h| ≤ h_max, then the step is repeated by returning to step I with x(t) and the new h value.

The corrsponding algorithm is presented below under the name of RK45adaptive. In the parameter list of the pseudocode, f is the function f(t,x) for the differential equation \( x' = f(t,x) , \) variables t and x contain the initial values, h is the initial step size, t₀ is the initial value for independed variable t, t_b is the final value fort, itmax is the maximum number of steps to be taken in going from a = t₀ to b = t_b, ε_min and ε_max are lower and upper bounds on the allowable error estimate ε, h_min and h_max are bounds on the step size h, and iflag is an error flag that returns one of the following values.

iflag	meaning
0	Successful march from t0 to tb
1	Maximum number of iterations reached

procedure RK45adaptive(f, t, x, h, t_b, itmax, ε_min, ε_max, h_min, h_max, iflag)
integer iflag, itmax, n;    external function f
real ε, ε_min, ε_max, d, h, h_min, h_max, t, t_b, x, x_save, t_save
real  δ ← ½×10^-5
output   0, h, t, x
iflag ← 1
k ← 0
while   k < itmax
     k ← k+1
     if |h| < h_min then   h ← sign(h)h_min
     if |h| > h_max then   h ← sign(h)h_max
     d ← |t_b - t|
     if d ≤ |h| then
          iflag ← 0
          if d ≤ δ ⋅ max{ |t_b|, |t|}   then   exit loop
          h ← sign(h)d
     end if
     x_save ← x
     t_save ← t
     call RK45(f, t, x, h, ε)
     output n, h, t, x, ε
     if iflag == 0  then   exit loop
     if ε < ε_min   then   h ← 2h
     if ε > ε_max   then
          h ← h/2
          x ← x_save
          t ← t_save
          k ← k-1
     end if
end while
end procedure RK45adaptive

The optimal step size h can be determined by multilying the scalar s times the current step size h. The scalar s is

where z_k+1 is approximation based on fifth order RK method and y_k+1 is approximation based on fourth order RK method.

Other Runge--Kutta methods could be found on
https://en.wikipedia.org/wiki/List_of_Runge-Kutta_methods

Some other popular 4/5 versions of Runge--Kutta methods:

• The Dormand--Prince 4/5 pair (Dormand, J. R.; Prince, P. J. (1980), "A family of embedded Runge--Kutta formulae", Journal of Computational and Applied Mathematics, 6 (1): 19–26, doi:10.1016/0771-050X(80)90013-3;
  Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press, pp. 82--84, ISBN 0-8493-9433-3.).
• The Cash--Karp pair. It was proposed by Professor Jeff R. Cash from Imperial College London and Alan H. Karp from IBM Scientific Center. (J. R. Cash, A. H. Karp. "A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides", ACM Transactions on Mathematical Software 16: 201--222, 1990. doi:10.1145/79505.79507).
• The Tsitorous 4/5