General One-Step Procedures

The Euler methods discussed above are the simplest methods of the great class of one-step procedures. Here, approximations xn₊1 will only be calculated at tn₊1= +tn hn solely from the approximation xn at the points tn and the increments hn.

One-step procedures can generally be written on the form

( )

1 , ,

n n n n n n

x ₊ =x + Φh x t h (7.28)

with the so-called process regulation Φ. Example 7.3

For the explicit Euler method it is true:

(

x t hn^{, ,}n n

)

f x t

( )

^,

Φ = .

To measure the quality of one-step procedures, we utilize the following terms:

Initial value problem

is the local error which emerges in this step of the method defined in (7.28).

This procedure is consistent, if it yields

( )

^{h O h}

( )

the global error is true for

Expressed in words this means that the consistency order of the global error is always one order lower than the local error.

7.1.6.1 Classical Runge-Kutta Methods

A basic disadvantage of Euler’s method is the low accuracy achieved. This demands very short integration increments h and leads to high computing times and an accumulation of round-off errors during the calculation.

Very early endeavors have been made to increase the degree of accuracy. An option is to calculate the right-hand side of differential equations at additional interpolation points.

We consider again the differential equation (7.1). If xn is given, we can integrate (7.1) in the interval

[

t t_n, _n+1

]

in order to calculate the function value xn₊1 where tn₊1 = +tn h:

We obtain the Runga-Kutta-method if we approximately integrate the right-hand side of (7.29)

Runga-Kutta Method of Second Order

The following statement results from the application of the Trapezoidal Rule in the approximation integration of the integral in (7.29)

( ) ( ) ( )

explicit Euler method. Thus, we obtain the following formula

( ) ( ( ) )

1 , ,

n n 2 n n n n n

x ₊ =x +hf x t + f x +hf x t  (7.31)

which will usually be formulated in the following calculation formula:

The Runga-Kutta method at hand is also referred to as a predictor-corrector method on the basis of the Euler method. In this context, the explicit Euler method plays the role of the predictor whereas the Trapezoidal Rule inherits the role of the corrector.

To determine the accuracy, with which this method discretizes the differential equation, we develop x in the surrounding of tn with the Taylor series at the same time, the partial derivatives of f will be shortened by:

, ,

For means of comparison, we develop (7.31) in an appropriate Taylor series

[ ] ( )

By comparing (7.33) with (7.34), it becomes clear that the error which occurs with every integration section is proportional to h³.

Hereafter, the remaining Runga-Kutta methods will only be specified by the error order and the calculation formula without derivation.

Runge-Kutta methods of third order Calculation formula:

( )

Runge-Kutta Method of Fourth Order

This method is based on Simpson’s 1 / 8-Rule and yields

( )

This version of the Runge-Kutta method is also referred to as classical Runge-Kutta method:

Stability Observation

In order to analyze the Runga-Kutta method, we consider again the test equation with a real α at first

By applying the Runge-Kutta method of fourth order in equation (7.38), we obtain the following statement

( )

²

( )

³

( )

⁴

1 1 1

1 1 2 6 24

n n

x h h h h x

γ

α α α α

+ = −((((((((((((((+ − +  (7.39)

It is obvious that the factor γ just contains the first five summands of the expansion power series where e^−αh. A comparison with the real value of the exponential function reveals that the deviation from the true value increases as α >0 and h grows and that instability is observable when α^h< −^2.785, Fig. 7.8. This is due to the numerical solution which increases with each integration section

(

^{γ >1}

)

while the true solution decreases

(

^{eαh <1}

)

^.

We receive similar results when we make the same analysis with further Runge-Kutta methods.

Similar to Euler method, we can more generally consider the case that α is a complex number. Here, we also analyze of the behavior of oscillating solutions. In this case, we do not obtain an interval of the real axis as a stable or instable area but a region in the complex plane, Fig. 7.10

Fig. 7.8: Instability area in the Runga-Kutta method of 4^th order.

7.1.6.2 General Runga-Kutta Methods

The calculation formula of an ODE discussed in section 7.1.6.1

( , )

x= f x t (7.40)

can be generally represented for a method with m function evaluations (m-stepped method)

= +

∑

can be interpreted as an approximation of the solution with one integration section in the following points in time: t_n +ζ_jh

Fig. 7.9: Transition points in the Runge-Kutta method.

We choose the coefficients to approximate xn as much as possible. A clear representation of the coefficients β ζ γ_i, _{i ij} 1≤i j, ≤m is frequently based on the following schema (Butcher

0 for .

ij j i

γ = ≥ (7.45)

the variables xi can be directly calculated from already known quantities, i.e. we refer to an explicit method (e.g. the classical Runge-Kutta methods already discussed in section 7.1.6.1).

Example 7.4: Butcher Schema for Classical Runge-Kutta Methods Euler’s method:

0 0 1

Runge-Kutta method of 4^th order:

1 1

• From section 7.1.6.1 we infer that that there is at least one Runge-Kutta method of p=m order where m≤4. The question emerges whether there also exists a method where p>m and whether there is always, at least, one method where p=m. One must negate this in both cases.

• As a matter of fact, we can prove the following connection (Table 7.1):

Table 7.1: Accumulation of the amount of transition values and achieved order.

Number of interim values 1 2 3 4 5 6 7 8 9 10

Achievable order 1 2 3 4 4 5 6 6 7 7

i.e. the method of 4^th order represents a kind of optimum in respect to this observation.

7.1.6.3 Stability Area of Runge-Kutta Methods of Order 1≤p≤4

In this case we can prove that the area of absolute stability of a method of order p results from

2

This particularly means that all methods of order p have the same stability area.

The stability border results from the complex solution z of the statement

( )

ⁱ

R z =e^θ

with arbitrary θ of the interval

(

^{0, 2π}

)

.

The stability areas of the Runge-Kutta method up to p=4 are represented in Fig. 7.10

Fig. 7.10: Stability area of the Runge-Kutta method of order 1-4.

In document List of Figures (Pldal 115-123)