A Variable Step Reduction Block Solver for Stiff ODEs
JIMEVWO GODWIN OGHONYON
Department of Mathematics, Covenant University
Km 10. Idiroko, Canaan Land, Ota, Ogun State,
NIGERIA
https://covenantuniversity.edu.ng/
MATTHEW REMILEKUN ODEKUNLE
Department of Mathematics, Modibbo Adama University of Technology
P. M. B. 2076, Yola, Adamawa State,
NIGERIA
https://mautech.edu.ng/new/index.php/en/
MATTHEW ETINOSA EGHAREVBA
Department of Sociology, Covenant University
Km 10. Idiroko, Canaan Land, Ota, Ogun State,
NIGERIA
https://covenantuniversity.edu.ng/
TEMITOPE ABODUNRIN
Department of Physics, Covenant University
Km 10. Idiroko, Canaan Land, Ota, Ogun State,
NIGERIA
https://covenantuniversity.edu.ng/
Abstract: - This research study is aimed at developing variable step reduction block solver (VSRBS) for stiff
ODEs. This step reduction block solver will embrace the technic of variable step-variable order to determine suited
variable step size. The trigonometrically fitted method will represent the basis function approximation to be utilized
together with the method of interpolation and collocation to derive (VSRBS). VSRBS comes with advantages to
overcome the barrier of stability requirement pose by definition 4. Some selected modelled examples of stiff ODEs
will solved and compared with existing methods to establish the efficiency and accuracy.
Key-words- Trigonometrically Fitted Method; Stiff Odes; Variable Step Reduction; Block Solver; Tolerance
Level.
Received: July 18, 2021. Revised: April 17, 2022. Accepted: May 21, 2022. Published: June 8, 2022.
1 Introduction
Every technique for estimating the analytical solution
to initial-value problems possesses error terms that
require a higher differential of the analytical solution
of the equation. Suppose the differential will be fairly
bounded, then this technique will possess a
predictable error bound that will be utilized to
estimate the accuracy of the approximation. Still,
assume the differential increases as the steps
increase; the error will be maintained in relative
control and provided that the solution also increase in
magnitude. Problems often spring up, still, if the
magnitude of the differential increases but the
analytical solution does not. In this position, the error
will increase so high that it controls the
computations. Initial-value problems for which this is
probably to appear are referred to as stiff equations
and are often common, especially in the areas of
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oscillations, chemical responses and electric circuits
[6].
Stiff differential equations will be qualified as
those whose analytic solution holds a term of the
class , where is a high positive constant
coefficient. Thus, it forms part of the analytic
solution called the transient result. Mostly, the
essential part of the analytic solution is called the
stiff-state result. The transient part of a stiff equation
will quickly decay to nothing as increases, because
the differential of this term holds magnitude
, the differential will not decay as rapidly. As
a matter of fact, because the differential in the error
term is calculated not at , simply at a number
between nothing and , the differential terms will
increase as increases and very quickly surely. By
good fortune, stiff equations in general can be
anticipated from the real life problem from which the
equation is derived and, with caution, the error can be
maintained below control. See [6] for more info
Definition 1: Consider the form of stiff
equations of
, (1)
where and is an rectangular array
of rows and columns with eigenvalues
(presume distinct) and matching
eigenvectors . In general, the
solution to (1) assumes the class
(2)
where the is any constant coefficients and is
a particular integral. See [12-13] for more
The above introduction will be supported
with the following definitions.
Definition 2: The initial-value problem (1)
is said to be stiff whenever every of its eigenvalues
possesses negative real form and the stiffness ratio is
large. The stiffness ratio and
. See [8, 12-13] for more info.
Definition 3: The initial-value problem (1)
is said to be stiff vibrating whenever the eigenvalues
of the Jacobian
satisfies the following conditions:
,
or whenever the stiffness ratio meets
and (3)
for at least a single pair of . See [8] for
more info.
Definition 4: Stiffness appears whenever
stability demands, rather than those of accuracy
restraint the step length. See [8, 12-13] for more info.
Theorem 1: Suppose is
continuously periodic. Then for whatever
there will be a trigonometric polynomial
such that for entirely
. Equally essential, whenever any such
there must subsist a sequentially polynomial of
in a sequential order on . See [4] for more items.
Authors have suggested that computing stiff and
extremely vibrating initial-value problems generally
demands the acceptance of several computational
methods. Among them includes; [1] integrated the
stiff ODEs using block backward differentiation
formulas of order six. The method is derived via the
expansion of linear multistep method and executed
with fixed step size. MATLAB solver ode15s is used
to achieve the computational result with better
accuracy. Suitable step and convergence of the result
is not established. [[3] formulated the new numerical
method for solving stiff initial value problems. The
derivation of the method is done with linear operator
and implemented with fixed step size. Stiff problems
solved have vibrating and oscillating solutions.
Solving the method is done with fixed step size to
determine the maximum errors. [9] proposed the
diagonally implicit block backward differentiation
formula ( which relies on the best
choice of the parameter that has optimum stability
attributes resulting to more precise results. Although,
( is self-initiating but utilize the
uniform step size to carry out the implementation
process. Convergence of the result is done with
uniform step size. [10] suggested the derivation of
diagonally implicit block backward differentiation
formulas for solving stiff Initial value problems. The
derivation and implementation are carried out using
Lagrange polynomial and fixed step size. [10]
consider the linear and nonlinear stiff problems
whose analytical solutions are exponential in nature
with fixed step size. [11] implemented the BBDF
for solving stiff ordinary differential equations with
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oscillating solutions. This method is derived using
Lagrange polynomial and solves stiff problems with
oscillating solutions and uses fixed step size to
implement the method. [14] developed an accurate
block solver for stiff initial value problems. Linear
difference operator is used to derive the method.
Variable step size and tolerance contributes
immensely to the successful implementation of the
method. Problems solved have oscillating and
vibrating solutions. [24] proposed the numerical
algorithm for solving stiff ordinary differential
equations. An efficient scheme in selecting the step
size and order has been introduced and implemented
throughout the numerical calculation. Test problem
considered has analytical solution with exponentially
and trigonometrically fitted in nature. Lagrange
polynomial is used as basis function approximation.
Variable step size and tolerance level were both
utilized to establish the convergence of the result.
[25] implemented fully implicit block method with
four-point for computing ODEs. The derivation of
the method is done with Lagrange polynomial and
problems solved have oscillating/vibrating solutions.
Variable step size implementation and tolerance level
were used as well. The BBDF has strong nature of
region of absolute stability that is considered as a
better method to proffer solutions to stiff problems.
This study will suggest variable step reduction block
solver for stiff ODEs for the purpose of introducing
variable step-variable order and finding suitable
variable step size to provide a better solution to
trigonometrically exact solution. Trigonometrically
fitted methods used as basis function approximation
agrees with the oscillatory or vibration solutions to
ensure stability of the results achieved. VSRBS have
the capacity to determine for every loop a suited
variable step to overcome the stability demand of
definition 4. Again, VSRBS is introduced to ensure
better efficiency and accuracy. See [12-13].
The motivation of this study originates from [12-
13] to yield better efficiency and precision via the
introduction of variable step-variable order-variable
step size. Secondly, VSRBS is proposed to bypass
the obstacle pose by backward differentiation
formula to adopt region of absolute stability as the
criteria for better result. Thus, this study will
implement variable step-variable order and finding a
suitable variable step size. Again, trigonometrically
fitted method will be utilized as the basis function
approximation to suit the trigonometrically exact
solution or oscillating and vibrating solutions whose
solutions are trigonometrically in nature. This idea of
using trigonometrically fitted method supersedes the
use of Lagrange polynomial and other basis function
utilized by other researchers. See [1, 11, 14, 25] for
details.
2 Formulation of Variable Step
Reduction Block Solver
The variable step reduction block solver formulation
will employ the concept of variable step and variable
order strategy. This involves the combination of
block predictor method and block corrector method.
The block predictor corrector method takes as
point of interpolation and as
points of collocation while the block corrector
method use and as points of
collocation. The block predictor method has
and for the
block corrector method. The step reduction block
solver utilizes the trigonometrically fitted method as
the basis function approximation in accordance with
the oscillating and vibrating solutions. The
trigonometrically fitted method for the 3-step block
predictor method of order 4 is defined as
(4)
Similarly, the trigonometrically fitted method for the
2-step block corrector method of order 3 is defined as
(5)
Interpolating and collocating (4) and (5) using the
selected points of block predictor method and block
corrected method will yield the expression as
(6)
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7)
Equations (6) and (7) represent the Mathematica
matrix format written under Mathematica Kernel 9.
Solving (6) and (7) using Mathematica Kernel 9 and
substituting into (4) and (5) will bring forth the
continuous scheme of both the block predictor
method and block corrector method as
. (8)
. (9)
Evaluating equations (9) and (10) at
will yield block predictor
method and block corrector method of:
, (10)
, (11)
is the recognized frequency,
is called
fixed constant coefficients. Equations (10) and (11)
are called the Variable step reduction block solver
(VSRBS) implemented via variable step and variable
order strategy. See [12-13, 15-23] for info.
2.1 Deriving the Tolerance Level of Variable
Step Reduction Block Solver
To obtain the derivation of the tolerance lever of
variable step reduction block solver, block
predictor method of order 4 and of block
corrector method of order 3 with different order and
step is been considered. A collection of [2, 5-7, 12-
13, 19-23] suggest the possibility of finding estimate
of the principal local truncation error of the block
predictor method-block corrector pair without
solving higher differential constant coefficients,
. Making the presumption that whenever
,
and represents the order of block predictor
method and block corrector method. Right way,
inquiry of block predictor method of fourth
order to yield the principal local truncation errors as
stated by
(15)
.
In likewise manner, investigating the breakdown of
block corrector method will give rise to
principal local truncation errors as defined by
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(16)
,
and
exists as a
separate device with step size h. represents the
analytical result of the differential constant
coefficient which agrees with the assumptions
. See [2, 5-7, 12-13, 19-23].
Moving ahead, the assumptions is fixed for lesser
valuates of h to get
. (17)
Step reduction block solver banks on the assumption
(17) to be implemented.
Furthermore, solving (15) and (16) and truncating
terms of degree and will achieve
the step reduction block solver estimates
(18)
.
Stating the assertions that
,
and
are known as the block predicting
and block correcting estimate achieved by the block
solver order.
,
and
will
distinctly be called principal local truncation errors.
and will be tolerance level of the block
solver.
Again, the results of (18) is employ to take
decision to accept or reject the computed results of
the loop or redo the loop with a lesser suitable
variable step size. Accepting the computed results is
certainly based on successful loop as specified by
(18). See [2, 5-7, 12-13, 19-23] for more details.
2.2 Variable Step Size Variation Strategy for
VSRBS
This study uses the principal local truncation errors
of order four (4) of the 3-step block predictor method
and order three (3) of the 2-step block corrector
method to introduce the idea. The block predictor
method of order four (4) and the block corrector
method of order three (3) will be employed to find
the suitable vary step of the step reduction block
solver. The principal local truncation errors of the
block predictor method-block corrector method will
be computed by this mathematical expression
(19)
,
where define the exact result to the first order stiff
problem satisfying the initial condition
.
Presume rebuilding the process of utilizing
anew suitable variable step size to generate a
anew estimates of
,
and
. To ascertain and ensure the principal
local truncation errors in , choosing such that
(20)
Employing the principal local truncation errors of the
block predictor method and block corrector method
together with (20) will achieve the result as
(21)
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.
Thus, requesting for the new selection of by
resolving (21) will further gives
,
(22)
.
Hence, requesting for change in step size from
, where will be adjusted as
,
, (23)
.
In addition, the successful execution of VSRBS
relies on (22) and (23). This demand of employing
the 3-step block predictor method of order four and
2-step block corrector method of order three
combines with (22) or (23) is iteratively solved to
agree with the tolerance level. Again, this iteration
process is implemented repeatedly until the newly
selected suitable step size satisfies the tolerance level.
Whenever the newly selected variable step size is
successful achieved then it eventually becomes the
suitable vary step size to get the desired results with
better accuracy and efficiency. Vary step size
strategies involve varying the step size during the
iteration process until the tolerance levels are
achieved. A step size changes for SRBS is very
costly to implement with high demand to achieve and
as such, mathematical software package is used to
easy the execution. See [2, 5-7, 12-13, 19-23] for
details.
3 Examples of Stiff Problems
Stiff problems solved involve trigonometrically
solution with oscillating and vibrating behaviour.
Three stiff problems were considered and solved
using the VSRBS. These stiff problems were
extracted from [9, 11, 25]
Stiff Problem 1
A two-torso orbit mildly stiff problem
,
.
Exact result:
.
Source. See [25] more info.
Stiff Problem 2
.
Exact result: ,
Source: see [9] for details.
Stiff Problem 3
,
, .
Exact result:
.
Source. See [11] for info.
4 Results and Discussion
This study considers specifically stiff problems
whose exact result is trigonometrically in nature with
oscillating and vibrating solutions. Again, this study
will solve stiff problems with tolerance level and
without tolerance level. The following tolerance level
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of
were used during the implementation process. The
results of the three stiff problems were displayed in
Table 1, Table 2 and Table 3. Headings in the tables
are defined in the nomenclature. The VSRBS is
coded under Mathematica Kernel 9 and implemented
as well. The process of implementation can be
viewed from sections 2.1 and 2.2. VSRBS is very
tedious computational procedure that involves the
combination of equations (11) and (12) together with
equation (18) written in Mathematica language. See
[9, 11, 25] for more info.
Table 1. Result of Stiff Problem I
MTHDEMPLOYED
MAXERROR
TOL
4PIFI
VSRBS
4PIFI
VSRBS
4PIFI
VSRBS
4PIFI
VSRBS
4PIFI
VSRBS
Table 2. Result of Stiff Problem 2
MTHDEMPLOYED
MAXERROR
TOL
-DIBBDF(0.50)
-DIBBDF(0.95)
VSRBS
-DIBBDF(-0.75)
-DIBBDF(-60)
VSRBS
-DIBBDF(0.95)
VSRBS
-DIBBDF(-0.75)
-DIBBDF(-0.60)
-DIBBDF(0.50)
VSRBS
-DIBBDF(0.95)
VSRBS
-DIBBDF(-0.75)
-DIBBDF(-0.60)
-DIBBDF(0.50)
VSRBS
Table 3. Result of Stiff Problem 3
MTHDEM
PLOYED
MAXERROR
TOL
BBDF
VSRBS
BBDF
VSRBS
BBDF
VSRBS
BBDF
VSRBS
4.1 Nomenclature
The following nomenclature will be used to show the
results in Tables 1, 2 and 3.
VSRBS : variable step reduction block solver
TOL : the tolerance level employed
Maxerror : the magnitude of the maximum errors of
VSRBS.
Mthdused: method employed.
4PIFI: implementation of the four-point one-block
fully implicit method using variable step size. See
[25] for more info.
-DIBBDF (: –Diagonally implicit block
backward differentiation formula ( value). See [9]
for more info.
BBDF-: block backward differentiation -
formulas. See [11] for more details.
5 Conclusion
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A variable step reduction block solver for stiff ODEs
has been suggested. This VSRBS emanates from the
3-step block predictor method of order four and 2-
step block corrector method of order three. The
VSRBS has the capacity to vary the step-vary the
order and implement a suitable vary step size with
the support of the tolerance level. The derivation of
the VSRBS is done via a special trigonometrically
fitted method used as the basis function
approximation for the purpose of approximating the
trigonometrically exact solution. The (VSRBS)
evaluated three stiff problems and compare the
results with existing methods. The performance of
VSRBS competes favourably with [25] in terms of
the maximum errors as a result of finding a suitable
variable step size for VSRBS to satisfy the tolerance
level. [9, 11, 25] belongs to the backward
differentiation formula (family) which has been
strictly designed to solve stiff problems with strong
region of absolute stability compare to VSRBS of
Adams family which is projected for non-stiff
problems. VSRBS involves tedious computation of
using a specially designed block predictor and block
corrector method to find a suitable variable step size
to satisfy the tolerance criteria. The VSRBS performs
better than [9, 11] due to the execution of (4) and (5)
as the basis function approximation compare to
others using Lagrange polynomial and Newton
iteration as basis function approximation. Also, the
successful implementation is attributed to
implementing variable step-variable order-finding a
suitable variable step size at every loop process.
Thus, the VSRBS is efficient and accurate for stiff
ordinary differential equations. Further studies will
be to design a block solver with the capacity to
handle exponentially exact solution.
Acknowledgements:
The authors would like to thank and appreciate
Covenant University for providing sponsorship
throughout the study period of time. Many thanks to
the anonymous reviewers for their immense
contribution.
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Conflicts of Interest:
The authors state that there are no conflicts of interest
regarding the publication of this article and among
the authors.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Jimevwo Godwin Oghonyon found the idea, method
and implemented the code using Mathematica.
-Matthew Remilekun Odekunle carried out the proof
reading and supervision.
-Matthew Etinosa Egharevba assisted in the
supervision.
-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.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC by 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_
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