DOI : 10.17577/IJERTV15IS070066
- Open Access
- Authors : Vagif R. Ibrahimov, Nigar Mammadzada, Nasiba Mustafayeva, Vagif R. Ibrahimov, Nigar Mammadzada, Nasiba Mustafayeva
- Paper ID : IJERTV15IS070066
- Volume & Issue : Volume 15, Issue 07 , July – 2026
- Published (First Online): 09-07-2026
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
On Some Comparison of Explicit Runge-Kutta and Multistep Methods
Vagif R. Ibrahimov, Nigar Mammadzada, and Nasiba Mustafayeva Vagif R. Ibrahimov
Chief of the Department of Computational Mathematics, Baku State University, Baku, Azerbaijan Azerbaijan Republic Ministry of Science and Education, Institute of Mathematics and Mechanics, Baku, Azerbaijan
Nigar Mammadzada
Department of Cybersecurity, Azerbaijan Technical University, Baku, Azerbaijan
PhD Candidate, Faculty of Mechanics and Mathematics, Baku State University, Baku, Azerbaijan
Nasiba Mustafayeva
Institute of Archaeology and Anthropology of ANAS, Chief Specialist, Baku, Azerbaijan
Abstract – As is known, one of the popular methods for investigation of the numerical solution of ordinary differential equations is the RungeKutta method. At present, some modifications and variants of RungeKutta methods have been developed to solve various ordinary differential equations. The RungeKutta method is a one-step method. Therefore, it is easy to apply for the solution of many problems. As is known, every numerical method has its advantages and disadvantages. One of the main disadvantages of the RungeKutta methods is the repeated calculation of the function located on the right- hand side of the differential equation under consideration. Let us note that another popular numerical method for solving initial value problems for ordinary differential equations is represented by multistep methods with constant coefficients. As was noted above, these methods also have their own advantages and disadvantages. One
Let us consider the following problem:
= (, ), (0) = 0, 0 , (1)
which is called the initial value problem for ordinary differential equations (ODEs).
To find the numerical solution of problem (1), the segment is divided into (N) equal parts with the constant step size (h>0), and the mesh points are defined as
= 0 + ( = 0,1,2, . ). The exact and approximate values of the solution at the points () are denoted by (y ()) and (), respectively.
Let us denote the value ( , ) or , the value of the function
of the main advantages of multistep methods is the use of information
at the point (
obtained at previous mesh points. Some authors suggested using forward jumping or advanced methods. It should be noted that advanced methods can be constructed based on multistep methods. In many cases, advanced methods are more exact than multistep methods. For the sake of objectivity, let us note that advanced methods are more exact than both RungeKutta and multistep methods. However, some difficulties arise in their implementation when solving some classes of problems. Here, we investigate two known methods (RungeKutta and Multistep Methods), and less known methos so called as the advanced method. The results obtained are illustrated with the help of specific
examples.
), and suppose that problem (1) has the unique solution, which has continuous derivatives (p+1) inclusively. The function (f(x,y)) is defined in some closed set in which it has continuous partial derivatives up to some order inclusively. As is known, some scientists consider Eulers method to be a special case of the RungeKutta method, which is not correct. However, some connection can be established between these methods.
Let us consider the following method:
= + (( , ) + ( , + ( , ))) (2)
Keywords – RungeKutta methods, Ordinary Differential Equations, Multistep Methods with Constant Coefficients, Initial
+1
2
+1
Value Problems, Stability and Degree, Advanced Methods.
This method can be obtained from the following method:
(
) + (
,
)) (3)
-
INTRODUCTION
+1 = + 2 ( ,
+1
+1
As is known, one of the first numerical methods was constructed by Euler and later developed by many scientists. One of the popular methods was constructed by Runge and by Kutta at the end of the 19th century and the beginning of the 20th century. It should be noted that these methods are some families. These methods were adopted by Professor Butcher and his followers. Butcher constructed a new family of implicit nonlinear methods (see for example [1][13]).
As is known, one of the main disadvantages of the classical RungeKutta methods is the repeated calculation of the value
Note that the accuracy of these methods is the same. But one of them is explicit, and the other is implicit. Here, considering the investigation of explicit methods, we study the RungeKutta methods. By simple comparison, we receive that method (2) is from the class of RungeKutta methods. It is not difficult to construct methods which differ from methods (2) or (3). For this, let us consider the following midpoint method in the class of RungeKutta methods.
This method is presented as follows:
+1 = + ( 1, 1). (4)
of the function which is on the right-hand side of the differential equation.
+
2
+
2
By this way, here two methods from the class of RungeKutta methods are constructed.
As is known, one of the popular methods is the multistep
+1 = + (1 + 42 + 3)/6 , (7) where,
= ( , ),
method with constant coefficients. One of the well-known representatives of these methods is the trapezoidal method, which can be received as a partial case from the multistep
1
2
= (
+ 2 ,
+ 1), 2
method as the special case. Note that, in the application of the trapezoidal method, there arise some difficulties with the calculation of the value . For this aim, one can use the following method (2), which is not identical with method (3). Note that the trapezoidal method has accuracy which is equal to for the local truncation error.
And now let us consider the following method:
3 = ( + , + 22 1).
The method presented here has accuracy = 3. Here, is the order of the presented method. It is not difficult to demonstrate a constructed more exact method with accuracy = 3. For this, let us consider the following:
1 = (),
2 1 3 2 2
+1
=
+ (5
+ 8+1
+2
)/12. (5)
2 = ( + 211 + 21
+ 2 22 ( + )),
2 1 3 2
This method differs from known methods in that here the value
, which needs to be determined, is used. Along with this, such a property is often called forward jumping or advanced method.
3 = ( + 311 + 322 + 31 + 2 32 (
+ 2)).
Method (5) is the first method constructed with this property. There are some designs for using above present method.
Most scientists conducting research in the field of ecology work in the field of biological sciences. Usually, the concepts of ecosystem and biosystem are studied within biological sciences. Recently, representatives of other sciences have often encountered ecological problems. Mathematical methods are used to investigate ecological problems. Similar studies
By using the above-presented equations, one can con\struct the RungeKutta method with the degree = 3. In our case, one can construct the following method:
+1 = + (1 + 2 + 43)/6, (8) here ,
concerning Volterra-type equations, finite-difference methods,
Simpson-type modifications, and related numerical models have also been carried out y many authors. Taking this into
2 = ( + 1 +
3
2
2 + 5
1
1
3(2 + 2)) , 10
1 1
account, this paper studies some well-known numerical
methods and considers some of their properties.
3 = ( +
8 1 +
8 2 +
2 40
3(2
80
-
COMPARISON OF SOME NUMERICAL METHODS FOR SOLVING THE INITIAL VALUE PROBLEM FOR
ODES
In the introduction, some information is given about the problems which are studied here. For this aim, some classes of methods are suggested, and comparisons are made of the studied problems.
As is known, here, considering the comparison of some
+ 2)).
Note that here, three methods are constructed which have the degree = 3. All the methods have the same degree = 3. But methods (8) and (7) are independent from in other words, the function ()is independent from the argument .
And now let us consider the following RungeKutta method with the degree
= 4, which can be represented in the following form:
(1 + 22 + 23 + 4)
families, let us consider the methods and for this study, the following general RungeKutta method is used:
+1 = + , (6)
+1 = +
Here,
. (9)
6
where,
=1
1 = (, ), 2 = ( + 2 , +
2
1), 2
1 = (, ),
2 = ( + 2, + (211)),
3 = ( + 2, +
2 ), 4 = (+1, + 3).
3 = ( + 3, + (311 + 322)),
..
Let us consider the case when the following function ()can
be presented as follows:
= ( + , + (11 + 22 + + ,11)).
= ( ),
= ( + ),
1
2 2
Let us consider the following RungeKutta method of third order:
= ( + ) , = ( + ).
3 4
2
From here, we receive the following:
+1 = + (() + 4(+1/2) + (+1))/3, (10)
which is identical with the Simpson method.
Here, some methods with the degree = 3 and with the degree
= 4 are constructed.
One popular technique that uses this inequality for error control is the RungeKuttaFehlberg method. This technique uses a RungeKutta method with local truncation error of order five:
For the methods (7) and (8), , in our case, = 1 4 . From method (9), it is received that method (9) has the degree = 4. As is known, the following method has the degree = 5, and the amount of is equal to 6().
By this way, we prove the following theorem:
Theorem. Let be the degree of accuracy of an explicit RungeKutta type method and let s denote the number of quantities . For p=1,2,3,4, it is possible to take s=p. However, for p>4, the number of quantities becomes greater than p.
Indeed, each quantity corresponds to one evaluation of the
16 6656
28561 9
function f (x, y) at a given step. Thus, the number of quantities
+1 = + 135 1 + 12825 3 + 56430 4 50 5
2
+ 55 6,
to estimate the local error in a RungeKutta method of order four, given by
is the number of stages of the method. For the first four degrees, this number coincides with the degree of accuracy. Beginning from the fifth degree, this relation changes. In the RungeKuttaFehlberg method, fourth and fifth-degree approximations are obtained by using six evaluations of f. Therefore, for p=5, one has 1, 2, , 6.
For higher degrees, the number of stages increases further. For
25 1408
2197 1
single explicit RungeKutta methods, known constructions
+1 = + 216 1 + 2565 3 + 4104 4 5 5,
where,
1 = (, ),
give s=7 for p=6, s=9 for p=7, and s=11 for p=8. In embedded pairs, this number may be larger; for example, a 5,6 pair usually use eight stages. For p=9, known constructions use more than nine stages; in particular, Khashin constructed a ninth-order method with thirteen stages (Khashin, 2009), while Butchers lower bound gives s 12 (Verner, 2013).
Thus, after p=4, increasing the degree of accuracy requires
1 increasing the number of quantities . This explains why high-
2 = ( + 4 , + 4 1),
degree explicit RungeKutta methods are more accurate, but also more expensive computationally. Recent RungeKutta
3 = ( +
3
8 , +
3
32 1 +
9
32 2),
pairs of orders 8,9 also confirm the practical use of such high- order explicit pairs in high-precision computations (Kovalnogov et al., 2024).
12
1932
7200
7296
4 = ( +
13 , + 2197 1 2197 2 + 2197 3),
-
MULTISTEP METHOD
As is known, one of the popular methods is the multistep
439
3680
845
method with constant coefficients. This method has been
5 = ( + , + 216 1 82 + 513 3 4104 4).
studied by many famous scientists. Thus, the method is usually presented as follows (see for example [14] [43]):
8 3544
1859
6 = ( + 2 , 27 1 + 22 2565 3 + 4104 4
+ = +, = 0,1,2, , . (11)
11
40 5) .
=0
=0
An advantage to this method is that only six evaluations of are required per step. Arbitrary RungeKutta methods of orders four and five used together require at least four evaluations of for the fourth-order method and an additional six for the fifth- order method, for a total of at least ten functional evaluations. And now, let us consider the comparison of some methods which are recommended here to solve some ordinary differential equations. For this, let us consider the following orders of exactness, which are applied for comparison of methods of Runge-Kutta type. Note that the first method has the degree = 3, and the other methods also have the degree =
3. The next method which is used here has a degree = 4. Thus, we find that when moving from one method to another, accuracy increases depending on the quantity.
It is known that numerical methods are usually based on the
concepts of stability and degree, which can be formulated in the following form:
DEFINITION 1. Method (11) is called stable if the roots of the following polynomial
() = + 11 + + 1 + 0
are located in the unit circle, on the boundary of which there are no multiple roots.
DEFINITION 2. The integer value is called as the degree of method (11), if the following asymptotic equality holds:
(( + ) ( + )) = (+1), 0. (12)
=0
Method (4) is stable and has the degree as By Dahlquists rule, we obtain that if method (11) is stable, then
+2 = + 3 ((, ) + 4(+1, +1) + (+2, +2)). (19)
2 [
] + 2.
2
Similar studies have been carried out by many authors. To solve certain problems, method (18) is preferable to method (19),
It follows that, if method (11) is stable, then there exists a stable method with degree = 4 for = 2 and = 3. Note that the values of max for = 2 and = 3 do not match.
The stable method with degree max = 4 for = 2 is the Simpson method, and the method with degree max = 4 for
= 3 can be presented as:
since method (18) is a one-step method. If the values are given, then by using method (18), one can compute the values. However, when using method (17), at each step it is necessary to compute the intermediate values and.
-
ADVANCED METHODS AND THEIR
=
3
+ (
+3
+3
+ ),
= (5). (13)
INVESTIGATION
+3
8
+3
+2
+1
And know let us consider the constrution the multistep methods of advanced type. Advanced methods have been
Some authors call this method Simpsons rule. As was noted, the region of stability for the Simpson method consists of one point, which is called the origin of the coordinate system.
By using the predictor-corrector method, one can expand the region of stability. For example, let us consider the following predictor-corrector method:
investigated by the well-known scientist Cowell in the 19th century, or as the forward-jumping method.
Cowell, by using the so-named method, calculated the orbit of Halleys comet.
Let us note that the advanced methods, or forward-jumping methods, have been investigated by the well-known scientist
4
= + (2 + 2 ). (14)
Cowell in the 19th century. Advanced methods, in one version,
+1 3 3
1
2
can be written as:
=
+ ( +4 + ). (15)
+ = +, n = 0,1, , ; > 0. (20)
+1
1
3 +1
1
=0
=0
Note that the local truncation errors for these methods can be presented as follows:
(13) = 28 5(5) + (6), (14) = 1 5(5) + (6). (16)
In formal form, one can say that method (10) can be received from (20) as a partial case. If in method (20) we put = 0, in this case we receive the known multistep method. For the case of objectivity, let us note that the main properties of these
90
1
90
1
methods differ in that the condition < , which is satisfied because > 0.
Simpson method let us write in the following form:
( )
-
The coefficients ( = 0,1, , ),( = 0,1, , )
are real numbers, and 0.
+1 = + 6 ( ,
+ 4(+1/2, +1/2) +
(+1, +1)) (17)
here we use the hybrid point.
For obtaining more accurate results, one can use the following
-
The polynomials
() = , () =
predictor-corrector method:
=0
=0
have no common factor different from a constant.
+1/2 = + 2 (, ),
+1/2 = + 2 ((, ) + (+1/2, +1/2))
Note that method (17) is obtained by changing the step size by method. From the above description, it is clear that the predictor-corrector approach can be applied to using method
(17). In this case, we recieved the following:
+1 = + 6 ((, ) + 4(+1/2, +1/2)
+ (+1, +1)). (18)
For the computation of the value +1, one may use either method (13) or method (14). Method (13) is implicit, whereas method (14) is explicit. In method (18), if we replace with 2, then one can be presented:
-
The polynomials ()and ()satisfy the conditions:
(1) = 0, (1) = (1) 0, 1.
Usually, the condition is called the necessary condition for the convergence of method (20). Numerical methods of type (20) have been constructed by some well-known scientists such as Laplace, Steklov, etc. The advanced methods constructed by these scientists may or may not obey the Dahlquist law.
The following method has been constructed in the related literature:
11 8
+2 = 19 + 19 +1
(10 + 57+1 + 24+2 +3)
+ (21)
57
It is proved that method (21) is stable and has order. It follows
that if method (20) is stable, then it is more accurate than method (2).
However, method (20) has some disadvantages. For example, in the application of this method to solve a problem, it is necessary to calculate the value +( ),which participates in method (20). In our case, it arises the calculation of the value +3 before calculation +2.
Let us consider the following theorem.
Theorem. Suppose that method (20) is stable and has degree . Then, in the class of methods (20), there are stable methods with degree + + 1, for the case, 3 + 1.
It should be noted that the properties of the resulting method depend on the properties of the method used for calculating the value +( ).
For the illustration,
the bilateral method, the signs of the main terms of the local truncation errors should be different.
Note that in the construction of methods, one of the main questions is the determination of the signs for some members of the used methods. By using this, Dahlquist proved that if method (1) is stable and has the maximum degree, then the condition is satisfied. If method (22) is stable and has the maximum degree, then and
+1+ < 0,if +1 0( = 1,2, , , 1 ).
-
-
NUMERICAL RESULTS
And now, for the illustration of the above-obtained results, let us consider the following example problem:
= cos , (0) = 0, 0 1
with the exact solution:
() = sin . (24)
For solving this problem, let us apply Simpsons method and its modification, presented in the following form:
5
+ 8
+4 + ). (25)
+1
= +
+1
12
+2
. (22)
+1 = +
6 (+1
1
+
2
The local truncation error for this method is:
Modified method,
= 0.1
Simpson,
= 0.1
Modified method, = 0.05
Simpson,
= 0.05
Modified method, = 0.01
Simpson,
= 0.01
0.2
6.5 9
1.0 7
4.3 10
6.8 9
6.8 13
1.1 11
0.6
1.8 8
3.0 7
1.2 9
1.5 8
1.8 12
3.1 11
1.0
2.5 8
4.4 7
1.8 9
2.8 8
2.8 11
4.6 11
4(4)
The results of the solution are given in the following table. Table 1. Solution of problem (24) by Simpsons method and its modification.
= + (5).
24
For the calculation of the value +2, let us use the following method:
The numerical results correspond to the theoretical results.
+2
= 3+1
2
+ .
12
-
CONCLUSION
As is known, in solving many problems, there appears the need for the construction of more exact numerical methods for
By using this method in (22), we receive:
solving some applied problems. Therefore, many famous scientists began to study the construction of numerical methods
=
+ 8+1+5 (
, 3
2
+ )
for solving various problems (see for example [44][71]). The
+1
(23)
12 12
+2
+1
12
first approximate numerical method was constructed by Euler, which made it possible to solve initial value problems for
This method is not A-stable. However, it is possible to change method (23) to the following method:
= + 3+1 ,
+2 +1 2
and by using this in method (22), we receive an A-stable method. By the above, we have shown some advantages of the predictor-corrector method.
Very often, the question arises about the reliability of the obtained values by some numerical methods. For solving this problem, it is recommended to use bilateral methods. It is easy to construct that the bilateral method has some relation with the predictorcorrector methods. As is known, in predictor corrector methods, one can use methods for which the remainder terms are the same. However, in the cnstruction of
ordinary differential equations numerically. The theory of Euler was developed by Adams, Runge, Kutta, and others.
At the beginning of the 19th century, they began to construct a new numerical method, which was called the finite difference method. In the middle of the twentieth century, new constructions of multistep methods with constant coefficients were developed. New works written by some scientists appeared in the mid-twentieth century. Among them, one can note the works written by well-known scientists such as Shura- Bura, Bakhvalov, G. Dahlquist, and Butcher. Butcher developed the theory of RungeKutta methods, as a result of which new directions emerged, which are called Butcher theory.
For the sake of objectivity, let us note that some well-known scientists developed the theory of explicit RungeKutta methods and constructed methods with an order of accuracy of 14 or more. Note that here it is suggested to investigate the
following method, which has not been investigated at a high enough level. This method is called the forward jumping or advanced method. Let us note that the advanced methods are more exact than the multistep methods. But these methods have some disadvantages, which are connected with the construction of more complex methods.
+1 .
= + (5 + 8+1 +2)
12
Here, the given method is presented, by using which one can show the way for using the above-given method. We hope that this direction will find its followers.
ACKNOWLEDGMENT
The authors thank Academician T. Aliyev and Academician A. Abbasov for their valuable advice. The authors also thank the reviewers for their helpful comments. This work was supported by the Azerbaijan Science Foundation under Grant No. AEF-MGC-2025-1(54)-20/05/1-M-05.
CONFLICT OF INTEREST
The authors state express that there is no conflict-of-interest misunderstanding between them. We hereby confirm that all the methods in this manuscript are ours
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