Parallel Solver for Oscillatory Stiff Systems of ODEs
OLASUNMBO OLAOLUWA AGBOOLA, JIMEVWO GODWIN OGHONYON,
TEMITOPE ABODUNRIN
Department of Mathematics, Covenant University
Km 10. Idiroko, Canaan Land, Ota, Ogun State,
NIGERIA
Abstract – The aim of this study will be to design Parallel solver (PS) for oscillatory stiff systems of ordinary
differential equations (ODEs). PS will be constructed via a type of specially transformed exponentially fitted
multinomial approximant in accordance with the behaviour of the solution. The method of interpolation and
collocation will be utilized. The principal local truncation errors of PS will be used to derive a suitable step size
and decide the error tolerance criteria for establishing the convergence of PS. Some examples of stiff ODEs
will be examined and compared with existing methods to show case the efficiency and accuracy of the scheme.
Parallel solver will be seen as a unique model for solving stiff ODEs without dependent on absolute stability as
required by backward differentiation formula.
Keywords- parallel solver; exponentially fitted method; stiff ODEs; error tolerance criteria; suitable variable
step size
Received: July 13, 2021. Revised: June 15, 2022. Accepted: July 18, 2022. Published: August 30, 2022.
1 Introduction
The model of a large number of technological and
applied science problems result to systems of
ODEs. The computational solution of such
problems via numerical integration,
interpolation/collocation method and many more
demands time, large space, softcode especially
when the ODEs systems is large or the rating of the
right hand side function is very costly. Thus, there
is a need for effective parallel solver method to
provide a quicker solution of such stiff systems.
The numerical solution of ODEs by Parallelism can
be split up into three families, viz the parallelism
throughout the system, parallelism throughout the
method and parallelism throughout the time. In this
study, we will solely look at parallelism throughout
the method. It can be declared clearly that
parallelism throughout the method for the solution
of ODEs takes it foundation in a family of
proficiencies referred to as block methods [6, 27-
28].
Stiff derivative equations are described as those
whose precise solution possess a condition of the
class where is named as a large prescribe
constant. This is commonly one component of the
solution, named the transient solution; the more
essential part of the solution is named the steady-
state solution. A transient component of a stiff
equation possesses magnitude , the
differential will not decompose as rapidly. Stiff
systems frequently constitute more than one
element or depend on two or more elements
connected together for effective functioning. For
instances; electric circuit, mechanical and chemical
flux systems, and traffic electronic network. Such
systems need two or more dependant variable
quantity for modelling the behaviour or function of
the systems. This can be reported in terminal figure
of a set of coupled first order differential equations.
We seek for a solution with the example of a pair of
coupled stiff systems with two or more variable
quantities
(1)
Equation (1) can be composed as a matrix equation
,
(2)
The set of differential equations can be extracted as
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. (3)
In equation (3), , and are the
driving mathematical functions [9, 17].
Definition 1: The initial value problem (1) is
ordered to be stiff if it meets
, ; i.e., whensoever
(i) , and
(ii) the stiffness ratio .
In addition, it should be mentioned that, this is
a quite a general resolution with respect to
mathematics. Stiffness takes place whensoever the
step length is restrained by stability, rather than
order onditions [10].
Definition 2: The initial value problem (1)-(3)
is stiff oscillatory or having periodic vibrations
whensoever the eigenvalues
of the Jacobian
have the
succeeding attributes:
,
or whensoever the stiffness ratio meets
and
For at least single pair of [10].
Theorem 1: Let be continuous and
periodic. Then for each , there exists a
trigonometric polynomial
such
that for all . Tantamountly, as
for
any such , there must exist a successive
polynomial
such that in a uniform manner on [7].
The parallel solver of (1) can be constituted
as the computational scheme in form of explicit and
implicit methods.
(4)
where
,
( for
are matrices.
The parallel solver is called an explicit
scheme if and only if the constant matrix is a
zero matrix or otherwise referred as an implicit
scheme [27].
Theorem 2: The A-stable multi-step
scheme
(i) must exist as an implicit, and
(ii) the almost precise A-stable multi-step
scheme is
of order p = 2 and error coefficient
.
Dahlquist suggested several methods for
outwitting the above theorem 2. Among them are
the exponentially fitting and extrapolation
processes. This study will explore the combination
of expanded exponentially fitted and extrapolation
processes to bring about parallel solver for stiff
ODEs. Nevertheless, the exponentially fitted
method agrees with behaviour of the stiff solution
and the components of the extrapolation processes
will be to implement the suited variable step size
and error tolerance criteria to enhance the
convergence of every loop. See [10, 14-15].
In literature, bookmen suggested the hybrid
multi-step and hybrid implicit Runge-Kutta to solve
(1). These methods comes with the vantage of not
demanding for initiating values and own great
stability regions. Majority of the backward
differentiation formula and block hybrid backward
differentiation formula has auto-initiating values
with good region of absolute constancy. Other
methods like block solver, parallel block backward
differentiation formula and variable step block
backward differentiation formula possesses some
vantages like auto-initiating values, parallel
execution, varying step size algorithm, step size
modification with great regions of absolute
stability. The major difficulties and challenges of
these methods is evident on the inability to use
exponentially fitted method in line with behaviour
of the system. Backward differential formulas are
over dependent on region of absolute stability
without finding a suitable step size to ensure
convergence. Again, the idea of auto initiating is
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gear towards time saving but there are single
methods of order four that consumes time with
better initiating results. On the other hand, parallel
solver will be regarded as an alternative method to
outwit the Dahlquist theorem and backward
differentiation formula by introducing
exponentially fitted method to approximate in
accordance with the behaviour of the stiff solution.
Again, parallel solver will introduce the
extrapolation processes to overcome the great
region of absolute stability by bringing about the
suitable variable step size and error tolerance
criteria which possesses the capability to change
the step size, modify the order, vary the step and
decide convergence. Parallel solver is a tedious
computation and timing consuming procedure with
a unique capacity to utilize the principal local
truncations to find a suited variable step size and
derive the error tolerance criteria [1-4, 10-11, 15-
29].
The motivation of this research emanates
from the fact that backward differentiation
formulas are considered the ideal solver for stiff
ODEs. This is due to the strong region of absolute
stability. Parallel solver is of Adams family which
is specially designed to bypass this condition by
introducing the exponentially fitted method with
the combination of variable step, variable order and
variable step size together with error tolerance
criteria. The contribution of this research study
will be using the exponentially fitted method in line
with behaviour of the exponential solution to build
the model. Again, the idea of variable step, variable
order with variable step is introduced to overcome
the barriers.
2 Developing a Parallel Solver
The parallel solver of the explicit and implicit
block methods will be developed as a combination
of variable step and variable order techniques. This
technique utilizes the for
the explicit block method with as the point of
interpolation and , as the points of
collocation. On the other hand,
is employed for the implicit
block method with as the point of
interpolation as well as , as the
points of collocation. Furthermore, this technique
will employ the expanded exponentially fitted as
the multinomial approximant defined as
. (5)
The expansion of (5) will give birth to the
expanded exponentially fitted method as
, (6)
where and for k=4 are constant
quantity to be specified in a peculiar way.
Accepting that (6) gratifies the Weierstrass
approximation theorem and correspond precisely to
the solution at some selected points of interval
to generate the approximation as
, . (7)
Interpolating and collocating (7) to gratifies (1) at
the level where
will generate the approximation
of the succeeding approximation as
(8)
Bringing together the approximations of (7) and (8)
will lead to fivefold and fourfold systems of
equation for both explicit and implicit block
methods as
,
,
,
,
. (9)
,
,
,
. (10)
Equations (9) and (10) are the unknown physical
quantities for developing the explicit and implicit
block methods to be determined. The unknown
physical quantities of equations (9) and (10) will be
substituted into equation (6) to get the continuous
explicit and implicit block methods. This
continuous explicit and implicit block method will
be evaluated at some selected points to achieve the
parallel solver for the explicit and implicit block
methods as
(11)
(12)
where w is the frequency, and
are fixed constants [7, 13-14, 20-26].
2.1 Developing the Error Tolerance Criteria
for Implementing Parallel Solver
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To launch this process, the collection of the
principal local truncation errors of
explicit block method with order method and
for the implicit block
method is utilized. Parallel solver will be used to
execute the approximation of the error tolerance
criteria of the k-step explicit block and implicit
block methods in the absence of estimating higher
differential coefficients of . Accepting that
, where represents the order
of the explicit and implicit block methods. Right
away, for a method of order , the analysis of
explicit with order of the explicit block
method will yield the principal local truncation
errors as
(13)
Similarly, inquiring into the implicit
block method with will generate the
principal local truncation errors as
(14)
and
are in
existence as distinguish entity of step size
and
will work as the accurate solution to
differential coefficient gratifying the initial
presumption .
Continuing to build the presumption that
for small measures of h,
(15)
and the efficiency of the error control procedure
banks on this presumption (15).
On deduction of (14) from (13) and disregarding
terms of degree
as well as presume (15)
will result to the error tolerance criteria for the
principal local truncation errors as
(16)
.
Stating the arguments that
,
and
are referred to as the
predicting and correcting approximant of the
principal local truncation errors for the explicit
block and implicit block methods.
,
and
are distinctly addressed as the
principal local truncation errors and are the
boundaries of the error tolerance criteria for
implementing parallel algorithm
In addition, the approximants of the
principal local truncation errors (16) will be utilize
to resolve whether to allow the effect of the loop
with the current step size or repeat the loop with a
reduce step size. This procedure is verified based
on the trial run carried out by (16) [5, 8-9, 15-16,
20-26].
2.2 Step Size Adjustment and Error Control
Procedures for Parallel Solver
The global error of (16) can be approximated by
(17)
,
where constitute the solution to the first
derivative equation gratifying the initial condition
.
Imagine if we immediately reconstruct the
situation with a new step size producing new
approximants
,
and
. To check and control the global
error to inside , we select such that
(18)
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Employing (14), we get
(19)
Therefore, we require to select with
,
(20)
.
Accordingly, this will require the change in step
size from , where q gratifies
,
,
(21)
.
Thus, a number of estimate suppositions
have been established in this development, hence in
practical applications the new step size is selected
in a conservative manner. A step size change for
parallel algorithm is more pricey and tedious in
terms of functional valuations than for a multi-step
method [9].
According to [15-16], the broad computing
experience that has been compiled throughout the
years suggest that the primal to greater efficiency
and error control in explicit block and implicit
block methods is the capability to change
automatically not just the step size, simply also the
order (and step number of the methods utilized).
3 Numerical Examples of Stiff ODEs
The numerical examples of stiff ODEs consider for
this research study are those with stiff oscillatory
and vibrating solutions.
System Problem 1
The first example is a virtually sinusoidal
problem defined in the interval .
with analytical solution
Author: [4, 20].
System Problem 2
,
,
,
with analytical solution:
For System 2, .
Author: [12, 19].
System Problem 3
,
,
,
with analytical solution:
.
Author: [19]
4 Results and Discussion
The numerical examples of system1, system 2 and
system 3 are all oscillatory stiff systems of ordinary
differential equations which must gratifies the
condition of definition 1 and 2 with respect to the
oscillating behaviour or periodic vibrations. Most
block backward differentiation formula derivations
are carried out employing the multinomial
approximant, Lagrange multinomial, backward
difference multinomial and Taylor series expansion
of the linear operator. The PS in accordance with
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definition 1 and 2 is formulated using exponentially
fitted which is one of the principal justification to
outwit the Dahlquist theorem and backward
differentiation formula. These aspects yield a very
good precision and efficiency with the introduction
of the suitable variable step size and error tolerance
criteria. The strength of the parallel solver lies in
the ability to find a suitable step size and generate
the error tolerance criteria to foster the convergence
of the loop. The PS performs better compare to
due to the task involved
in designing a suitable variable step size for each
loop to ensure the convergence at every error
tolerance criteria. Again, PS implements an
expanded exponentially fitted multinomial
approximant based on the oscillatory behaviour of
the solution as seen in the numerical examples.
This strategy will ensure faster convergence of the
loop with very good precision and efficiency. On
the other hand, is auto-initiating with
good region of absolute stability and fixed large
step length to ensure the implementation.
Therefore, shows well stable properties
with precision and efficiency while the
are
all centred on trimming the entire number of paces,
computing time utilized and possessing good
stability region [4, 12, 19-20].
Table 1. Numerical Results for System Problem 1
MU
MAXE
ETC
Table 2. Numerical Results for System Problem 2
MU
MAXE
ETC
3NBBDF
3NBBDF
Table 3. Numerical Results for System Problem3
MU
MAXE
ETC
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4.1 Nomenclature
The nomenclatures utilized in the tables represent
the following meaning.
: parallel solver of solution
: parallel solver of solution
: method used
: maximum errors
: error tolerance criteria
: hybrid block second derivative
backward differentiation formula [4].
: BDF method [12].
: BBDF method [12].
: BBDF method [12].
: fifth order new BBDF
method [19].
: fifth order Block Backward
Differentiation Formulas [20].
5 Conclusion
Parallel solver for oscillatory stiff systems of ODEs
has been suggested. Parallel solver is a fusion of
the explicit and implicit blocks method developed
via interpolation and collocation methods with the
help of the exponentially fitted method as the
polynomial. The exponentially fitted method and
the components of the extrapolation processes such
as variable step, variable order and suitable variable
step size were used to outwit the Dahlquist obstacle
and backward differentiation formulas. This
combination is geared to foster error control with
an improve accuracy, greater efficiency and
maximize errors. Three problems were examined
under the following error tolerance criteria;
and compared with PS. The
convergence of PS is made possible with the help
of deciding a suitable step size to meet the error
tolerance criteria which in turn lead to achieving
lesser maximum error. The following methods of
has good stability properties which is
implemented using large step size compare to
that requires the determination of a
suitable variable step size and error tolerance
criteria during implementation. Table 1, Table 2
and Table 3 presents the end result of the PS
compare with other subsisting methods of
.Thus, PS which involves tedious
computing strategies of implementing variable step,
variable order and finding suitable variable step
size has the advantage of high precision, high
efficiency with more preferred maximum errors
compare to subsisting methods of
.
Further Study:
The further study will be to implement parallel
solver in higher order of ODEs.
Acknowledgments:
The authors would like to thank Covenant
University for providing all round support
throughout this research study.
References:
[1] P. Agarwal, I. H. Ibrahim, A new type of
hybrid multistep multiderivative formula
for solving stiff IVPs, Advances in
Difference Equations, Vol. 286, 2019, pp.
1-14.
[2] O. A. Akinfenwa, S. Jator, N. Yoa. An
eight order backward differentiation
formula with continuous coefficients for
stiff ordinary differential equations,
International Journal of Mathematical and
Computational Sciences, Vol. 5, No. 2,
2011, pp. 160-165.
[3] O. A. Akinfenwa, S. A. Okunuga, B. I.
Akinnukawe, Multiderivative hybrid
implicit Runge-Kutta method for solving
stiff system of a first order differential
equation, Far East Journal of
Mathematical Sciences (FJMS), Vol. 106,
No. 2, 2018, pp. 543-562.
[4] O. A. Akinfenwa, R. I. Abdulganiy, B. I.
Akinnukawe, S. A. Okunuga, Seventh
order hybrid block method for solution of
first order stiff systems of initial value
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problems, Journal of the Egyptian
Mathematical Society, Vol. 28, 2020, pp. 1-
11.
[5] U. M. Ascher, L. R. Petzoid, Computer
Methods for Ordinary Differential
Equations and Differential-Algebraic
Equations, SIAM, USA, 1998.
[6] I. G. Birta, O. Abou-Rabia, Parallel Block
Predictor-Corrector Methods for ODEs,
IEEE Transactions on Computers , Vol. 36,
No. 3, 1987, pp. 299-311.
[7] M. Bond, Convolutions the Weierstrass
approximation theorem, 2009.
[8] J. R. Dormand, Numerical methods for
differential equations, CRC Press, New
York, 1996.
[9] J. D. Faires, R. L. Burden, Numerical
Analysis, Initial-value problems for ODEs,
Brooks Cole, Dublin City University, 2012.
[10] S. O. Fatunla, Numerical methods for
initial value problems in ordinary
differential equations, Academic Press,
Inc., New York, 1988.
[11] H. M. Ijam, Z. B. Ibrahim, Z. A. Majid, N.
Senu, Stability analysis of a diagonally
implicit scheme of block backward
differentiation formula for stiff
pharmacokinetics models, Advances in
Difference Equations, Vol. 2020, 2020, pp.
1-22.
[12] Z. B. Ibrahim, K. I. Othman, S.
Mohammed, Implicit -point block
backward differentiation formula for
solving first-order stiff ODEs, Applied
Mathematics and Computation, Vol. 186,
No 1, 2007, pp. 558-565.
[13] S. N. Jator, On a class of hybrid methods
for , International Journal of
Pure and Applied Mathematics, Vol. 59,
No. 4, 2010, pp. 381-395.
[14] S. N. Jator, L. Lee, Implementing a
seventh-order linear multistep
method in a predictor-corrector mode
or block mode: which is more
efficient for the general second order
initial value problem, Springer Plus,
Vol. 3, 2014, p. 1-8.
[15] J. D. Lambert, Computational methods in
ordinary differential equations, John Wiley
& Sons, Inc., New York, 1973.
[16] J. D. Lambert, Numerical methods for
ordinary differential systems, John Wiley
& Sons, Inc., New York, 1991.
[17] P. M. Shankar, Differential equations: A
problem solving approach based on
MATLAB®, CRC Press, New York, 2018.
[18] H. Musa, M. B. Suleiman, F. Ismail, N.
Senu, Z. B. Ibrahim. An accurate block
solver for stiff initial value problems,
Applied Mathematics, Vol. 2013, 2013, pp.
1-11.
[19] H. Musa., M. B. Suleiman, F. Ismail, N.
Senu, Z. A. Majid, Z. B. Ibrahim, A new
fifth order implicit block method for
solving first order stiff ordinary differential
equations, Malaysian Journal of
Mathematical Sciences, Vol. 8, No S, 2014,
pp. 45-49.
[20] A. A. M. N. Nor, Z. B. Ibrahim, K. I.
Othman, M. Suleiman, Numerical solution
of first order stiff ordinary differential
equations using fifth order block backward
differential formulas, Sains Malaysiana,
Vol. 41, No. 4, 2012, pp. 489-492.
[21] A. A. Nasarudin, Z. B. Ibrahim, H. Rosali,
On the integration of stiff ODEs using
block backward differentiation formulas of
order six, Symmetry, Vol. 12, 2020, pp. 1-
13.
[22] J. G. Oghonyon, S. A. Okunuga, N. A
Omoregbe, O. O. Agboola, A
computational approach in estimating the
amount of pond and determining the long
time behavioural representation of pond
pollution, Global Journal of Pure and
Applied Mathematics, Vol. 11, No. 5, 2015,
pp. 2773-2786.
[23] J. G. Oghonyon, J. Ehigie, S. K. Eke,
Investigating the convergence of some
selected properties on block predictor-
corrector methods and it’s applications,
Journal of Engineering and Applied
Sciences, Vol. 11, No. 11, 2017, pp. 2402-
2408.
[24] J. G. Oghonyon, O. A. Adesanya, H.
Akewe, H. I. Okagbue, Softcode of multi-
processing Milne’s device for estimating
first-order ordinary differential equations,
Asian Journal of Scientific Research, Vol.
11, No. 4, 2018, pp. 553-559.
[25] J. G. Oghonyon, O. F. Imaga, P. O.
Ogunniyi, The reversed estimation of
variable step size implementation for
solving nonstiff ordinary differential
equations, International Journal of Civil
Engineering and Technology, Vol. 9, No.
8, 2018, pp. 332-340.
WSEAS TRANSACTIONS on MATHEMATICS
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Olasunmbo Olaoluwa Agboola,
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Volume 21, 2022
[26] J. G. Oghonyon, S. A. Okunuga, H. I.
Okagbue, Expanded trigonometrically
matched block variable-step-size technics
for computing oscillating vibrations,
Lecture Notes in Engineering and
Computer Science, Vol. 2239, 2019, pp.
552-557.
[27] L. K. Yap, F. Ismail, S. Mohamed, Block
Methods for Special Second Order
Ordinary differential equations, Lap
Lambert Academic Publishing, USA, 2011.
[28] B. I. Zarina, I. O. Khairil, M. Suleiman,
Variable step block backward
differentiation formula for solving first
order stiff ODEs, Proceedings of the World
Congress on Engineering, Vol. 2, 2007, pp.
1-5.
[29] B. I. Zarina, I. O. Khairil, Parallel block
backward differentiation formulas for
solving large systems of ordinary
differential equations, International
Journal of Mathematical and
Computational Sciences, Vol. 4, No. 4,
2010, pp. 479-482.
Conflict of Interest
The authors declare that there is no conflict of
interest regarding the publication of this article and
among authors.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Olasunmbo Olaoluwa Agoola carried out the
editing and supervision.
Jimevwo Godwin Oghonyon devised the idea,
method used and implemented the code using
Mathematica.
Temitope Abodunrin provided the logistics and
technical support.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research study is funded by Covenant
University, Ota.
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