MỘT THUẬT TOÁN CẢI TIẾN VÀ CHƯƠNG TRÌNH THỰC THI CỦA THUẬT TOÁN CHO PHƯƠNG PHÁP RUNG-KUTTA THÍCH NGHI

These help to reduce a half of the number of function evaluations comparing with an adaptive Runge-Kutta method using a pair of arbitrary Runge-Kutta methods of order

A MODIFIED ALGORITHM AND ITS IMPLEMENTATION

FOR AN ADAPTIVE RUNGE-KUTTA METHOD

Tran Thi Hue *

University of Technology - TNU

ABSTRACT

Adaptive Runge-Kutta methods are used popularly in finding an approximation for the solution of

an initial value problem (IVP) because of their low computational cost and efficiency This article aims to present how to derive an adaptive Runge-Kutta method, and to present a modification for its algorithm in general, and also to explain how the algorithm works The main contribution of the article is to introduce a new pattern to adjust the step-size in a more flexible way of increasing the efficiency This pattern is created when we use a scale for a step size which is also determined basing on the estimation of the local truncation error Finally, the article emphasizes the aforementioned contents for a particular method, namely Runge-Kutta-Fehlberg, and presents its implementation as well

Keywords: initial value problem, Runge-Kutta, Runge-Kutta-Fehlberg, adaptive Runge-Kutta

method, error control.

The IVP 𝑦′ = 𝑓(𝑥, 𝑦), 𝑦(𝑥0) = 𝑦0, where

𝑦 = 𝑦(𝑥), 𝑥 ∈ ℝ, has been studied for long

time ago Many approaches have succeeded

in finding important discoveries of the

problem Numerical approach is one of the

most succeeding ones which with useful helps

of supercomputer nowadays seems to be

favorite to scientists who work in the fields of

applied mathematics Adaptive Runge-Kutta

methods take advantages from class of

Runge-Kutta methods in eliminating the need

advantages from class of adaptive methods in

reducing the errors of approximation and save

dramatically the computational cost as well

Therefore, adaptive Runge-Kutta methods are

really powerful tools for studying the solution

of an IVP

The main idea of the method is that: First, we

use a Runge-Kutta method of order 𝑝 (with 𝑠

steps) to approximate the solution at 𝑥0+ ℎ,

𝑤1∗= 𝑤0∗+ ℎ ∑𝑠 𝑏𝑖∗𝐹𝑖

* Tel: 0984 632890, Email: cuonghue1980@gmail.com

where ℎ > 0 is a step-size, and 𝑤0∗= 𝑦0,

𝐹1= 𝑓(𝑥0, 𝑤0∗), 𝐹2= 𝑓(𝑥0+ 𝑐2ℎ, 𝑤0∗+

ℎ𝑎21𝐹1),

𝐹3= 𝑓(𝑥0+ 𝑐3ℎ, 𝑤0∗+ ℎ(𝑎31𝐹1+ 𝑎32𝐹2)),

…

𝐹𝑠= 𝑓(𝑥0+ 𝑐𝑠ℎ, 𝑤0∗+ ℎ ∑𝑠−1𝑎𝑠,𝑘𝐹𝑘

𝑘=1 ) Then, we use another Runge-Kutta method of order 𝑝 + 1 approximating the solution,

𝑤1= 𝑤0+ ℎ ∑𝑠 𝑏𝑖𝐹𝑖

to estimate the error produced by (1.1), where

𝑤0= 𝑦0 and 𝑤0∗ in all formulae of

𝐹1, 𝐹2, … , 𝐹𝑠 is replaced by 𝑤0 From this, we can learn whether the local truncation error revealed by (1.1) is confidently accepted and can decide what to do in the next step (to approximate the solution at 𝑥0+ 2ℎ) The next performance needs an adjustment in step-size ℎ to meet the requirement of the error Here, in (1.1) and (1.2), parameters

𝑏𝑖∗, 𝑏𝑖, 𝑐𝑖 and 𝑎𝑖,𝑘 (𝑖 = 1,2, , 𝑠, 𝑘 = 1,2, … , 𝑖 − 1) are derived from Runge-Kutta method by identifying them with coefficients

of Taylor’s expansion up to the required order

𝑝 or 𝑝 + 1, respectively The next section presents how to find them from a particular adaptive Runge-Kutta method

In (1.1) and (1.2), 𝑠 coefficients 𝐹1, 𝐹2, … , 𝐹𝑠,

called stage derivatives, are the same These

help to reduce a half of the number of

function evaluations comparing with an

adaptive Runge-Kutta method using a pair of

arbitrary Runge-Kutta methods of order 𝑝 and

𝑝 + 1

Table 1 Butcher tableau for parameters in (1.1)

and (1.2).

0

𝑐𝑠 𝑎𝑠1 𝑎𝑠2 … … 𝑎𝑠,𝑠−1

Here, we assume that 𝑐𝑖 = ∑𝑖−1 𝑎𝑖,𝑘

𝑖 = 1,2, … , 𝑠

ERROR CONTROL

Firstly, for a traditional adaptive Runge-Kutta

algorithm, we need to estimate the local

truncation error in order to decide an

adjustment for the step-size To do so, we

attempt to bound the error by a given 𝜀 > 0

Local truncation error of (1.1) at 𝑥0+ ℎ is

𝜏1∗(ℎ) =𝑦(𝑥0+ ℎ) − 𝑤1∗

That of (1.2) is

𝜏1(ℎ) =𝑦(𝑥0+ ℎ) − 𝑤1

So, 𝜏1∗(ℎ) = (𝑦(𝑥0+ ℎ) − 𝑤1∗)/ℎ

ℎ[(𝑦(𝑥0+ ℎ) − 𝑤1) + (𝑤1− 𝑤1∗)]

= 𝜏1(ℎ) +1

ℎ(𝑤1− 𝑤1∗) (1.3) Then, it is simple to estimate the error

produced from (1.1) since 𝜏1(ℎ) has a higher

order than 𝜏1∗(ℎ) That estimate is

𝜏1∗(ℎ) ≈ (𝑤1− 𝑤1∗)/ℎ

This is enough for a traditional adaptive

Runge-Kutta method That is, from this

estimate, we require that

𝑅 ≔|𝑤1− 𝑤1∗|

So, if this condition is not met, we reduce the step-size ℎ to a smaller one, for instance, ℎ/2, then recalculate all approximation at this step with this new step-size Conversely, if this condition is met, we accept the calculation for all approximations at the step 𝑥0+ ℎ and move to the next step, 𝑥0+ 2ℎ This pattern

of adjustment seems to be awkward as it ignores the estimate of the local truncation error (1.3)

To make an important modification for the algorithm of a traditional adaptive Runge-Kutta method, we will use again (1.3) to adapt the initial step-size ℎ to a new one

Concretely, we introduce a scale q for the step-size h such that this scale will be

determined basing on the calculated estimate

of the local truncation error To see that, firstly, let 𝐾 be a constant such that 𝜏1∗(ℎ) ≈

𝐾ℎ𝑝 Adjusting the step-size to 𝑞ℎ produces a local truncation error from (1.1), which is

𝜏1∗(𝑞ℎ) ≈ 𝐾(𝑞ℎ)𝑝≈𝑞

𝑝

ℎ (𝑤1− 𝑤1∗)

To control the error, we require that

|𝜏1∗(ℎ)| ≤ 𝜀 or 𝑞𝑝(𝑤1− 𝑤1∗)/ℎ ≤ 𝜀 So,

|𝑤1− 𝑤1∗|)

1/𝑝

From this, the initial step-size ℎ at the first

step is used to find the first values of 𝑤1 and

𝑤1∗, which leads to the determination of 𝑞 for that step, then calculations were repeated This procedure requires twice the number of function evaluations per step as without the error control Here, we choose 𝑞 in a flexible way which makes the increased function-evaluation cost whorthwhile The scale 𝑞 is chosen conservatively satisfying the above bound Concretely, we take

1− 𝑤1∗|)

1/𝑝

In fact, by the penalty of evaluating the value

of function at each step point if the step are

repeated, this method of choosing 𝑞 seems to

be meaningful Moreover, we always consider

that 𝑞 ≥ 0.1 These modifications help us to

save a lot of time plugging in very small

intervals Besides, we eliminate a large

variation by requiring that 𝑞 ≤ 4 (see the

pseudo-code) This upper bound for 𝑞 can

help us to reduce the omission of sensitive

intervals The scale 𝑞 is used for the

following useful purposes:

If 𝑅 > 𝜀, we reject ℎ and adjust the step-size

to 0.1ℎ (if 𝑞 ≤ 0.1), or to 4𝑞 (if 𝑞 ≥ 4), or to

𝑞ℎ (if otherwise) Then we recalculate all

approximations at this step

If 𝑅 ≤ 𝜀, we keep computed values for this

step, but change the step-size to 0.1ℎ (if

𝑞 ≤ 0.1), or to 4𝑞 (if 𝑞 ≥ 4), or to 𝑞ℎ (if

otherwise), in the next step, to approximate

the solution at 𝑥0+ 2ℎ

For the latter, when 𝑅 ≤ 𝜖, we accept 𝑤1 as

the approximation of 𝑦(𝑥0+ ℎ)

RUNGE-KUTTA-FELHBERG METHOD

abovementioned modification for a specified

combines a Runge-Kutta method of order 4

and one of order 5 This method was

proposed by Erwin Felhberg [2] The

advantage of this method comparing with

other adaptive Runge-Kutta method can be

explained basing on the relationship between

the number of stages 𝑠 and the order 𝑝 of a

method As J Butcher proved in [1] that if

𝑝 > 5 then the difference between 𝑠 and 𝑝

becomes large and large Therefore, the

orders around 𝑝 = 4 is expected to maximize

the exactness of the approximation, and

minimize the number of calculations

For Runge-Kutta-Felhberg method, we take

𝑠 = 5 for the method of order 𝑝 = 4, and

𝑠 = 6 for the other one of order 𝑝 + 1 = 5

To derive parameters in (1.1) and (1.2), first,

we use the Taylor expansion up to order 𝑝 for the solution 𝑦(𝑥) at 𝑥0+ ℎ at 𝑥0 to get 𝑦(𝑥0+ ℎ) = 𝑦0+ ∑𝑝𝑘=1ℎ𝑘!𝑘𝑑𝑑𝑥𝑘𝑦𝑘(𝑥0) (1.5) Using the notation of ‘rooted tree’, introduced

in [1], we can rewrite (1.7) in the form

𝑦(𝑥0+ ℎ)

= 𝑦0+ ∑ ℎ𝑟(𝑡)

𝜎(𝑡)𝛾(𝑡)

𝑡∈𝑇𝑝

𝐹(𝑡)(𝑦0) (1.6)

where 𝑇𝑝 is the set of all rooted trees 𝑡 of order 𝑟(𝑡) ≤ 𝑝

Next, we use the Taylor series for s stage derivative 𝐹1, 𝐹2, … , 𝐹𝑠 and sum up them in (1.1), and (1.2) as well, to get the approximation of 𝑦(𝑥0+ ℎ),

𝑦(𝑥0+ ℎ)

= 𝑦0+ ∑ℎ𝑟(𝑡)

𝜎(𝑡)Φ(𝑡)

𝑡∈𝑇𝑝

𝐹(𝑡)(𝑦0), (1.7)

with Φ(t) is called elementary weights of a rooted tree 𝑡 When matching terms of the same order of ℎ in (1.8) and (1.9), we imply the order condition [1], which states that

Φ(𝑡) = 1/𝛾(𝑡) , ∀𝑡 ∈ 𝑇𝑝 (1.8)

Since (1.10), with 𝑐1= 0, we find parameters

of Runge-Kutta-Felhberg method from Table 2

Table 2 Value of functions 𝛷(𝑡), and 𝛾(𝑡) on rooted trees of order 𝑟(𝑡) ≤ 5, that is 𝑡 ∈ 𝑇5.

Roote

d tree

𝒕 ∈ 𝑻𝟓

Tree order

𝑖=2 𝑐𝑖 2

𝑖=2 𝑐𝑖2 3

2≤𝑗<𝑖 2≤𝑖≤6 𝑎 𝑖𝑗 𝑐 𝑗 6

𝑖=2 𝑐𝑖3 4

2≤𝑗<𝑖 2≤𝑖≤6 𝑐 𝑖 𝑎 𝑖𝑗 𝑐 𝑗 8

4 ∑2≤𝑗<𝑖𝑏̂𝑖

2≤𝑖≤6 𝑎𝑖𝑗𝑐𝑗2 12

4 ∑ 𝑏̂ 𝑖

2≤𝑘<𝑗<𝑖 2≤𝑖≤6 𝑎 𝑖𝑗 𝑎 𝑗𝑘 𝑐 𝑘 24

6 𝑖=2 𝑐 𝑖4 5

5 ∑2≤𝑗<𝑖𝑏̂𝑖

2≤𝑖≤6 𝑐𝑖2 𝑎𝑖𝑗𝑐𝑗 10

2≤𝑗<𝑖 2≤𝑖≤6 𝑐 𝑖 𝑎 𝑖𝑗 𝑐 𝑗2 15

2≤𝑗<𝑖 2≤𝑖≤6 𝑎 𝑖𝑗 𝑐𝑗3 20

𝑖=1 (∑ 𝑎𝑖𝑗𝑐𝑗 2≤𝑗<𝑖 )

2

20

2≤𝑘<𝑗<𝑖 2≤𝑖≤6 𝑐 𝑖 𝑎 𝑖𝑗 𝑎 𝑗𝑘 𝑐 𝑘 30

2≤𝑘<𝑗<𝑖 2≤𝑖≤6 𝑎 𝑖𝑗 𝑐 𝑗 𝑎 𝑗𝑘 𝑐 𝑘 40

5 ∑2≤𝑘<𝑗<𝑖𝑏̂𝑖

2≤𝑖≤6 𝑎𝑖𝑗𝑎𝑗𝑘𝑐𝑘 60

5 ∑2≤𝑙<𝑘<𝑗<𝑖𝑏̂𝑖

2≤𝑖≤6 𝑎𝑖𝑗𝑎𝑗𝑘𝑎𝑘𝑙𝑐𝑙

120

We find parameters 𝑏𝑖, 𝑏𝑖∗, 𝑐𝑖, and 𝑎𝑖𝑗 (∀𝑖, 𝑗 =

1,2, … ,6, 𝑖 > 𝑗), basing on 17 equations

obtained from Table 2 with the use of (1.10),

where 𝑏̂𝑖’s are replaced by 𝑏𝑖’s and 𝑏𝑖∗’s,

respectively One of solutions of this system

is given in Table 3

Note that from Table 3, 𝑏6∗= 0, so the

method (1.1) has 𝑠 = 5 steps, and order

𝑝 = 4 (since each rooted tree 𝑡 of order

𝑟(𝑡) = 5 has Φ(𝑡) = 0)

aforementioned modification is performed by

simply taking 𝑞 somewhat consecutively,

2|𝑤 1 − 𝑤1∗ |)

1/4

2|𝑤 1 − 𝑤1∗ |)

1/4

The estimate for the local truncation error

𝜏1∗(ℎ) is denoted by

𝑅 =|𝑤1 −𝑤1∗|

55 |

Table 3 Parameters of Runge-Kutta-Felhberg

method

0

1

4 1

4

3

8 323 9

32

12

13 1932

2197 −7200

2197 7296

2197

216 −8 3680

513 −8454104

1

2 −827 2 −3544

2565 18594104 −1140

16

135 0 6656

12825 2856156430 −950 552

25

216 0 1408

2565 21974104 −15 0 IMPLEMENTATION OF RUNGE-KUTTA-FELHBERG METHOD

The following pseudocode describes the algorithm for the Runge-Kutta-Felhberg method to approximate the solution of the IVP:

𝑦′= 𝑓(𝑥, 𝑦), 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑦(𝑎) = 𝛼, such that the local truncation error bounded

by a given tolerance 𝜀 > 0

INPUT function 𝑓, endpoints 𝑎 and 𝑏, initial

value 𝛼, tolerance 𝜀, maximum step-size ℎ𝑚𝑎𝑥, minimum step-size ℎ𝑚𝑖𝑛

OUTPUT mesh point 𝑥, current step-size ℎ, approximation 𝑤 of 𝑦(𝑥), or message that ℎ𝑚𝑖𝑛 is exceeded (the procedure fails)

Step 1 (Initiate the procedure)

𝑥 ≔ 𝑎; 𝑤 ≔ 𝛼, ℎ = ℎ𝑚𝑎𝑥, 𝐹𝐿𝐴𝐺 ≔ 1;

OUTPUT (𝑥, 𝑤)

Step 2 While (𝐹𝐿𝐴𝐺 = 1) do steps 3-9 Step 3 𝐹 1 ≔ ℎ𝑓(𝑥, 𝑤);

𝐹2≔ ℎ𝑓 (𝑥 +14ℎ, 𝑤 +14𝐹1) ;

𝐹3≔ ℎ𝑓 (𝑥 +3ℎ8 , 𝑤 +3𝐹1

32 ) ;

𝐹4≔ ℎ𝑓 (𝑥 +12ℎ13 , 𝑤 +1932𝐹1

2197 ) ;

𝐹 5 ≔ ℎ𝑓 (𝑥 + ℎ, 𝑤 +439𝐹1

216 − 8𝐹 2 +3680𝐹3

513 −845𝐹4

4104 ) ;

𝐹6≔ ℎ𝑓 (𝑥 +ℎ2, 𝑤 −8𝐹1

27 + 2𝐹2−3544𝐹3

2565 +1859𝐹4

4104 −11𝐹5

40 ) ;

Step 4 𝑅: =1ℎ|𝐹1

55 |

Step 5 If 𝑅 < 𝜀 then do

𝑥 ≔ 𝑥 + ℎ; (Adopt the approximation.)

𝑤 ≔ 𝑤 +25𝐹1

216 +

1408𝐹3

2565 +

2197𝐹4

4104 −

𝐹5

5 ;

OUTPUT (𝑥, 𝑤, ℎ); end do; End If

Step 6 𝑞 ≔ 0.84(𝜀/𝑅)1/4

Step 7 If 𝑞 ≤ 0.1 then ℎ ≔ 0.1ℎ

elseif 𝑞 ≥ 4 then ℎ ≔ 4ℎ

else ℎ ≔ 𝑞ℎ

End If

Step 8 If ℎ > ℎ𝑚𝑎𝑥 then ℎ ≔ ℎ𝑚𝑎𝑥;

End If

Step 9 If 𝑥 ≥ 𝑏 then 𝐹𝐿𝐴𝐺 ≔ 0

elseif 𝑥 + ℎ > 𝑏 then ℎ ≔ 𝑏 − 𝑥

elseif ℎ < ℎ𝑚𝑖𝑛 then 𝐹𝐿𝐴𝐺 ≔ 0;

OUTPUT(“minimum h exceeded”)

(Procedure fails!)

STOP

EXAMPLE

We illustrate the result harvested from a

program written by basing on the algorithm to

approximate the solution of the IVP:

𝑦′= 𝑦 − 𝑥2+ 1, 0 ≤ 𝑥 ≤ 1.5, 𝑦(0) = −1

We take the tolerance 𝜀 = 10−6, and the

0.25, ℎ𝑚𝑖𝑛 = 0.01, respectively

The approximations are represented in Table

4 It is easy to see that the exact solution is

𝑦(𝑥) = (𝑥 + 1)2− 2𝑒𝑥

So, we can determine the absolute error of the

approximation at each mesh point 𝑥𝑖 This

table also reveals the value of 𝑦(𝑥) (in

column 2) and the absolute error (in column

5) at each mesh point Column 4 and 5

indicate the approximations obtained by the

Runge-Kutta method of order 4 and 5,

respectively Column 6 presents the estimate

for the local truncation error 𝜏𝑖+1∗ (ℎ)

produced by the Runge-Kutta method of order

4 at the operating mesh point

COMPAIRISON WITH THE ALGORITHM

RUNGE-KUTTA METHOD

The modification we made here is worthwhile

in the sense of increasing efficiency It reduces dramatically the number of function-evaluations, especially in the case of a very small tolerance, compare to a traditional one This point is easily illustrated with a traditional algorithm in which we adjust the

step-size h by reducing it to a half each time

the condition (1.4) is not met However, in this strategy, we do not intervene to an increasing adjustment of the step-size at each time the condition (1.4) is met Table 5 represents the comparison of results obtained from such a traditional algorithm and from the modified algorithm It shows us that the flexibility of verifying the step-size gains the upper hand in diminishing the number of intermediate stages The IVP illustrated here is

𝑦′ = 𝑦 − 𝑥2+ 1, 0 ≤ 𝑥 ≤ 1.5, 𝑦(0) = 0.5, with the tolerance 𝜀 > 0, and the maximum,

0.01, respectively

In Table 5, the first and the second column indicate the number of intermediate stages performed for the traditional algorithm and the modified algorithm, correspondingly The third column presents the respective tolerance

𝜀 Definitely, the exactness of the results from the modified algorithm is lower than that from the traditional one, however, it is still in the accepted limit In fact, the requirement for the accuracy does not request a further exactness than what it really needs Therefore, the higher exactness of the results from the

Moreover, if we compare the rate of the increasing number of immediate stages and the decreasing error produced by the traditional algorithm with that produced by the modified algorithm, we recognize easily that the compensation which the traditional has to pay is too expensive

SUMMARY

The adaptive Runge-Kutta methods have the

advantages in finding a highly exact

approximation for the solution of an IVP It is

a favor in solving a non-stiff differential

equation numerically The paper presented an

improvement of the algorithm for any

improvement can be easily achieved by

introducing a scale for the adjusted step-size

in a flexible way which makes the twice

traditional adaptive Runge-Kutta method

worthwhile Nowadays, with the help of

supercomputers, it is quite simple to translate

this algorithm into one that is easy to execute

by a process of high performance computing

This reduces dramatically time for calculating

and increasing so much the exactness of the approximations

REFERENCES

1 John C Butcher (2008), Numerical Methods for Ordinary Differential Equations, 2nd Edition, John Wiley & Sons, Ltd

2 Richard L Burden, J Douglas Faires (2010),

Numerical Analysis, 9th Edition, Brooks/Cole

3 John C Butcher (1963), “Coefficients for the

study of Runge-Kutta integration processes” Journal of Australian Mathematical Society, 3(2),

pp 185-201

4 Michiel Hazewinkel, (2001), “Runge-Kutta method”, https://www.encyclopediaofmath.org/ind

Mathematics, Springer Science+Business Media B.V / Kluwer Academic Publishers

Table 4 Approximations for Example

Absolute Error

|𝒚(𝒙𝒊) − 𝒘𝒊∗| 𝒘𝒊 by RK5

Estimate 𝑹 𝒊

of 𝝉𝒊+𝟏∗ (𝒉)

0.25 -1.005550834 0.25 -1.005550873 0.000000039 -1.005550725 5.97× 10 −7 0.407146 -1.024987041 0.157145948 -1.024987153 0.000000112 -1.024986925 5.14× 10 −7

0.563048 -1.068914234 0.155901799 -1.06891447 0.000000236 -1.068914115 8.1× 10 −7

0.701105 -1.138199742 0.13805677 -1.138200102 0.00000036 -1.138199649 7.25× 10 −7

0.826775 -1.23476275 0.125670756 -1.234763232 0.000000482 -1.234762696 6.59× 10 −7

0.943955 -1.361290913 0.117179383 -1.361291519 0.000000606 -1.36129091 6.24× 10 −7

1.054692 -1.520421635 0.110736981 -1.520422373 0.000000738 -1.520421698 6.042× 10 −7

1.160198 -1.71467541 0.10550681 -1.714676288 0.000000878 -1.714675552 5.91× 10 −7

1.261289 -1.94650959 0.101090617 -1.946510613 0.000001023 -1.946509819 5.81× 10 −7

1.358555 -2.21835268 0.097265817 -2.218353859 0.000001179 -2.218353009 5.74× 10 −7

1.452434 -2.532574686 0.093879137 -2.532576029 0.000001343 -2.532575126 5.68× 10 −7

1.5 -2.71337814 0.047565983 -2.713379551 0.000001411 -2.713378645 4.32× 10 −8

Table 5 Comparison of the results from a traditional algorithm and the modified algorithm

Number of

intermediate stages

by the traditional

algorithm

Number of intermediate stages

by the modified algorithm

Tolerance

𝜺

Absolute error at the last step (at x=1.5) from the traditional procedure

Absolute error at the last step (at x=1.5) from the modified procedure

TÓM TẮT

MỘT THUẬT TOÁN CẢI TIẾN VÀ CHƯƠNG TRÌNH THỰC THI

CỦA THUẬT TOÁN CHO PHƯƠNG PHÁP RUNG-KUTTA THÍCH NGHI

Trần Thị Huê *

Trường Đại học Kỹ thuật Công nghiệp - ĐH Thái Nguyên

Các phương pháp Runge-Kutta thích nghi được sử dụng rộng rãi để tìm nghiệm xấp xỉ của bài toán phương trình vi phân với giá trị ban đầu bởi tính hiệu quả và khối lượng tính toán tương đối nhỏ Bài báo này sẽ giới thiệu cách xây dựng phương pháp Runge-Kutta trong trường hợp tổng quát, và

sự cải tiến thuật toán của phương pháp Đóng ghóp quan trọng của bài báo này là việc đưa ra một cách thức mới cho việc điều chỉnh kích thước bước linh hoạt hơn nhằm tăng hiệu quả tính toán Phần sau của bài báo nhằm giới thiệu và nhấn mạnh các nội dung cải tiến nói trên cho một phương pháp cụ thể, phương pháp Runge-Kutta-Felhberg, cũng như chương trình thực thi của thuật toán

Từ khóa: bài toán giá trị ban đầu, phương pháp Runge-Kutta, phương pháp

Runge-Kutta-Felhberg, phương pháp Runge-Kutta thích nghị, điều khiển sai số.

Ngày nhận bài: 22/8/2018; Ngày phản biện: 11/9/2018; Ngày duyệt đăng: 12/10/2018

* Tel: 0984 632890, Email: cuonghue1980@gmail.com