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2021;25(5):623-632
ISSN 2500-1590 (online)
METALLURGY AND MATERIALS SCIENCE
Original article
https://doi.org/10.21285/1814-3520-2021 -5-623-632
Dynamical control on the Adomian decomposition method for solving shallow water wave equation
Laleh Noeiaghdam10, Samad Noeiaghdam2, Denis N. Sidorov3
1 Amirkabir University of Technology, Tehran, Iran
2,3 Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, Irkutsk, Russia
South Ural State University, Chelyabinsk, Russia
3 L. A. Melentiev Institute of Energy Systems of Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia 1 l_noeiaghdam@yahoo.com
2snoei@istu.edu; noiagdams@susu.ru, https://orcid.org/0000-0002-2307-0891 3 dsidorov@isem.irk.ru, https://orcid.org/0000-0002-3131-1325
Abstract: The aim of this study is to apply a novel technique to control the accuracy and error of the Adomian decomposition method (ADM) for solving nonlinear shallow water wave equation. The ADM is among semi-analytical and powerful methods for solving many mathematical and engineering problems. We apply the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method which is based on stochastic arithmetic (SA). Also instead of applying mathematical packages we use the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. In this library we will write all codes using C++ programming codes. Applying the method we can find the optimal numerical results, error and step of the ADM and they are the main novelties of this research. The numerical results show the accuracy and efficiency of the novel scheme.
Keywords: shallow water wave problem, Adomian decomposition method, CESTAC method, CADNA library For citation: Noeiaghdam L., Noeiaghdam S., Sidorov D. N. Dynamical control on the Adomian decomposition method for solving shallow water wave equation. iPolytech Journal. 2021;25(5):623-632. https://doi.org/10.21285/1814-3520-2021-5-623-632.
МЕТАЛЛУРГИЯ И МАТЕРИАЛОВЕДЕНИЕ
Научная статья УДК 519.6
Численная валидация метода разложения Адомиана для решения нелинейного уравнения мелкой воды
Лале Нойягдам10, Самад Нойягдам2, Денис Николаевич Сидоров3
1 Технологический университет Амиркабира, г. Тегеран, Иран
2,3 Лаборатория промышленной математики, Байкальский институт БРИКС, Иркутский национальный исследовательский технический университет, г. Иркутск, Россия
2 Южно-Уральский государственный университет, г. Челябинск, Россия
3 Институт систем энергетики им. Л. А. Мелентьева СО РАН, г. Иркутск, Россия 1 l_noeiaghdam@yahoo.com
2snoei@istu.edu; noiagdams@susu.ru, https://orcid.org/0000-0002-2307-0891 3 dsidorov@isет.'^к. т, https://orcid. org/0000-0002-3131-1325
Аннотация. Целью данной работы является применение новой методики для проведения эффективного контроля точности и анализа погрешности применения метода разложения Адомиана для решения нелинейного волнового уравнения мелкой воды, возникающего при выполнении ряда важных задач в различных областях машиностроения и теории материалов. Рассмотрены три важных случая: набегание жидкости на полуплоские берега, на берега с умеренным уклоном и на берега с крутым уклоном. В исследованиях применялся метод раз-
© Noeiaghdam L., Noeiaghdam S., Sidorov D., 2021 https://vestirgtu.elpub.ru -
ложения Адомиана, являющийся полуаналитическим эффективным методом, обладающим большей гибкостью, чем прямое разложение в ряд Тейлора. В данном исследовании мы применяем метод CESTAC (Contrôle et Estimation STochastique des Arrondis de Calculs), основанный на стохастической арифметике. Также вместо применения стандартных математических пакетов в работе эффективно использована библиотека контроля точности и отладки для численных приложений CADNA (Control of Accuracy and Debugging for Numerical Applications). Программная реализация подхода с использованием библиотеки CADNA выполнена на C++ под операционную систему LINUX. Вместо использования традиционной абсолютной ошибки, которая основана на точном решении и небольшом пороговом значении, используем новое правило останова, которое основано на двух последовательных приближениях. Основная теорема метода CESTAC показывает, что количество общих значащих цифр двух последовательных приближений практически равно количеству общих значащих цифр точного и приближенного решений. Применение представленной в работе методики позволило получить оптимальные численные результаты: найти погрешность и оптимальный шаг метода разложения Адомиана, чего не позволяли делать классические подходы. В этом заключается основная новизна работы. Приведены результаты тестирования разработанной численной модели для решения уравнения мелкой воды. Таким образом, при проведении численных расчетов с использованием предлагаемого метода разложения Адомиана продемонстрирована высокая точность и эффективность разработанного подхода для решения нелинейного волнового уравнения мелкой воды.
Ключевые слова: волновая задача на мелководье, метод разложения Адомиана, метод CESTAC, библиотека CADNA
Для цитирования: Ноэягдам Л., Ноэягдам С., Сидоров Д. Н. Численная валидация метода разложения Адомиана для решения нелинейного уравнения мелкой воды // iPolytech Journal. 2021. Т. 25. № 5. С. 623-632. https://doi.org/10.21285/1814-3520-2021-5-623-632.
INTRODUCTION
Finding numerical and accurate methods for solving wave equation is one of important topics in applied mathematics [1-26]. The problem has special and important applications to simulate Tsunami phenomenon. Tsunamis can be categorized as long waves. In general form we apply the solitary waves or combinations of negative and positive solitary-like waves to simulate this phenomenon. Also many other applications of the shallow water wave problems can be found in the field of metallurgy and materials science. The horizontal centrifugal casting using shallow water model has been discussed by Bohacek et al. in [4]. Also Bresch et al. in [5] have focused on some compressible fluid models specially the shallow water system. In [13] Nobel and Vila have discussed the thin power-law film flow
down an inclined plane: consistent shallow-water models.
In this study we discuss the following problem which is the specific case of the run-up of two dimensional long waves incident upon a uniform sloping beach connected to an open ocean with a uniform depth (fig. 1).
We consider the following nonlinear shallow-water equations:
r)t + (u(h + ri))x = 0; ut + uux + g^x = 0, ( )
where wave domain is displayed by the depth averaged velocity is showed by u, the variable depth is presented by h, the acceleration of gravity is ehibited by g. Also the initial conditions are given
Fig. 1. Definition Sketch for solitary wave run-up Рис. 1. Эскиз определения для набега уединенной волны
2021 ;25(5):623-632
ISSN 2500-1590 (online)
r](x, 0) = Hsech2
3 H
x;
(2)
u
(x, o ) = 2Jgd,
where the initial wave height and stationary elevation are displayed by H and d respectively.
Noeiaghdam et al. in [15] applied the ho-motopy analysis method for solving this problem. Approximate traveling wave solution has been discussed by some authors in [1, 12]. A new analytical solution for nonlinear shallow water-wave equation can be found in [3]. In [2] the stability analysis of two dimensional extended shallow water wave equation has been illustrated. Also in [22] the space-time fractional shallow water wave equation has been studied.
Adomian decomposition method is among semi-analytical and powerful methods for solving many kinds of linear and non-linear problems. Solving Volterra integral equation with discontinuous kernel [18], first kind integral equations with hyper-singular kernels [20], ordinary differential equations [10] and fuzzy Convection-Diffusion equation [11] are only some of applications of the ADM.
In the mentioned papers and many other studies the computations have been obtained using floating point arithmetic (FPA). In FPA in order to show the accuracy we should apply the traditional conditions which are based on the exact solution and as follows
IQ-Qjl^e.orlQj-Qj^Ke.
(3)
In real life problems that we do not have exact solution it will be impossible to use the conditions. Also the optimal value of e is unknown for us. Thus choosing small or large values for e either we will not be able to find the accurate results or we will have many iterations without improving the accuracy.
In this study instead of applying FPA, we use SA and a novel condition based on successive iterations and as
\Qj-Qj-1\ = @.0.
(4)
@ . 0 shows the informatical zero which can be produced only in SA when we apply the CESTAC method and the CADNA library. It shows that number of common significant digits (NCSDs) of two successive iterations are almost zero. The CESTAC method has been presented by Vignes and Laporte [21]. Also a French research team has been improved the method and also the library [6-9]. In [16] the CESTAC method has been applied to control the accuracy of a nonlinear fractional order model of COVID-19. A dynamical control on the reverse osmosis system can be found in [14, 19]. Also in [17] the CESTAC method has been used to validate the results of numerical methods for solving integral equations.
In this paper, we apply the famous Adomian decomposition method for solving problem (1). Also using the CESTAC method and the CADNA library we try to control the accuracy of the results. The main novelty of this study is to find the optimal approximations, optimal error and optimal step of the method.
ADOMIAN DECOMPOSITION METHOD
Consider the following problem:
Lu + R(u) + F(u) = g{t),
(5)
where is the operator of the highest-ordered derivatives with respect to and is the remainder of the linear operator. Also we show the nonlinear term with F(u). We can write
Lu = g(t) - R(u) - F(u).
(6)
Now we can define the following inverse operator:
% 1 = Í (■) d t■
(7)
Applying the inverse operator for both sides of (7) we get
u = f0 + L~t 1 [g (t)-R(u)-F (u)] , (8)
where is the solution of homogeneous equation Lu = 0 involving the constants of integration. The integration constants involved in the
solution of the mentioned homogeneous equation are to be determined by the initial or bound- where ary condition according as the problem is initial-value problem or boundary-value problem.
According to the ADM we can assume that the solution can be obtained in the form of the following series solution:
= =0,n = 0,1,2,... (11)
n\ dln
A=0
MAIN IDEA
Applying the following operators:
и (x,t) = ZZ о^п (x,t), (9)
and also for the nonlinear term F(u) we have
F (и) = Щ= о А п, (10)
(Ltr](x, t) + Lx[u(pc, t)h(x)] + Lx[u(x, t)r\{x, t)] = 0, I Ltu(x, t) + u(x, t)Lxu(x, t) + gLxrj(x, t) = 0.
L =- L = —
Lt = dt,Lx = dx '
(12)
and also the inverse operator L^ = J^ (.)dt on (1) we get
(13)
Now we can apply the inverse operator L~t 1 for both sides of (13) as
fn(x, t) = T](x, 0) - L^Lxlu^x, t)h(x)] - L^Lx[u(x, t)rj(x, t)],
(u(x, t) = u(x, 0) - L^lu^x, t)Lxu(x, t)] - gL'[1Lx[r](x, t)].
According to the traditional ADM for nonline- B0 = u0xu0, ar terms ur} and uux the following terms can be applied
ur} = Zn=0 An,uux = Xn=0 Bn, (15) B2 = u0xu2 + u1 xu1 + u2xu0,
where
i40 = U0T]0, Ai = u„ih + Utfo,
A2 = u0r]2 + u1t]1 + U2t]0,
= W<)»?3 + «1^2 + U2?ll + U2tl0,
B3 — u0xu3 + ulxu2 + u2xul + U3XU0'
(16)
Thus substituting
Г (x,t) = Г n (x,t), и (x, fy = £n - о un (x, fy, in (14) we get
and
,^n=0 Vnty^) = V&,0) LtlLx[Tm=0 un(x,t)h(x)] LtlLx[Tm=0 Ап],
hn=0 Un(x, t) = u(x, 0) - L-t^o Bn] - gL?Lx\Zn=0 Vn(x, t)].
(14)
(17)
(18)
(19)
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Following Adomian approach we obtain the following recursive relation:
rj0(x,t) = r](x, 0), u0(x, t) = u(x, 0),
(20)
i(x, t)
= -L^L^u^x.^hipc)] - L^LxiAk], uk+l(.x> t)
= -L?[Bk] - 9Lt%[rik(x> t)]-
The -th order approximate solution can be obtained using the following relations
fm (X, t) — Ym=0 rjn(x, t),
Um (x, 0 — ^n=0 Un (x, 0 ■
(21)
CESTAC method-CADNA library. Assume that some representable values are produced by computer and they are collected in set . Then W e A can be produced for w e R with J mantissa bits of the binary FPA in the following form
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where sign of w showed by x, missing segment of the mantissa presented by 2~JZ> and the binary exponent of the result characterized by . Moreover, in single and double precisions Jl = 2 4,5 3 respectively [6, 21, 23].
Assume ( is the casual variable that uniformly distributed on [-1,1]. After making perturbation on final mantissa bit of w we will have (i) and (a) as mean and standard deviation for results of W which they have important role in accuracy of W. Repeating this process / times for Wbi = 1,...,J we will have quasi Gaussian distribution for results. It means that fi for these data equals to the exact w. It is clear that we should find i and a based on W0i = 1,...,J. For more consideration, the following algorithm is presented where is the value of distribution as the confidence interval is with freedom degree [7-9]. We should note that in the CESTAC method instead of applying usual applications such as Mathematica and Maple we use the CADNA library [23].
Definition 1. [15, 16] Number of common significant digits for two real numbers can be defined as
(22)
= logic
Xi
X1-X2
(23)
Theorem 1. Applying the ADM for solving the shallow water wave problem (1), the NCSDs of the exact and approximate solutions are almost equal to the NCSDs of two successive approximations as
CrimVm+l ~
c « c
umium+1 U,Um'
NUMERICAL ILLUSTRATIONS
Assuming H = 2 and d = 20, we discuss the results in three forms 1 - Semi-flat shores (h(x) = 0 . 2x - 20 ), 2 - Moderate-slope shores (h(x) = x - 1 0 0), 3 - Sharp-slope shores
(h(x) = 5 x - 5 0 0). We apply the ADM for solving problem (1) and we use the CESTAC method and the CADNA library to validate the results. Fig. 2, 3 and 4 show the approximate solutions for the mentioned shores for x = t = 0. 1. According to results of tab. 1 and 2, optimal approximation and optimal error of method for solving shallow water wave equation are obtained. Thus the optimal approximation of is , the optimal step is
and the optimal error . Also,
for the approximate solution of is
1.40115 3 672165483 , the optimal step is and the optimal error .
Ноэягдам Л., Ноэягдам С., Сидоров Д. Н. Численная валидация метода разложения Адомиана для решения.
Fig. 2. Approximate solution for semi-flat shores mode: а - n(x,t); b - u(x,t) Рис. 2. Приближенное решение для режима пологих берегов: а - n(x,t); b - u(x,t)
Fig. 3. Approximate solution for moderate-slope shores mode: а - n(x,t); b - u(x,t) Рис. 3. Приближенное решение для режима берегов с умеренным уклоном: а - n(x,t); b - u(x,t)
Fig. 4. Approximate solution for sharp-slope shores mode: а - n(x,t); b - u(x,t) Рис. 4. Приближенное решение для режима берегов с резким уклоном: а - n(x,t); b - u(x,t)
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Table 1. Applying the CESTAC method for numerical results of u(x, t)
Таблица 1. Применение метода CESTAC для численных результатов u(x,t)
Table 2. Applying the CESTAC method for numerical results of ц{х, t)
Таблица 2. Применение метода CESTAC для численных результатов n(x,t)
m Щи 1 um - um+il
1 1.9999962462046976 1.999996246
2 1.9999489773636574 4.72688 x 10"5
3 2.GGG32222992G2164 3.7325x 10"4
4 2.GGG5G4545245128 1.8231 x 10"4
5 2.GGG5981472797174 9.36x 10"5
6 2.GGG64342675G37G8 4.5x 10"5
7 2.GGG6415328545538 @.G.
m Пт \Пт - Пт+ll
1 1.400697371G2946G1 1.4007
2 1.4007378405649158 4.04695x 10"5
3 1.400961G914243738 2.2325x 10"4
4 1.401G7545341977G2 1.143x 10"4
5 1.401131280443496 5.58x 10"5
6 1.401159855883387 2.8x 10"5
7 1.401153672165483 @.G.
CONCLUSIONS
The ADM is a good and accurate method for solving linear and non-linear problems. Applying the method we found the approximate solution of the non-linear shallow water wave problem. Combining the ADM with the CESTAC method
we tried to find the optimal approximation, step and error of the method. We applied the CADNA library to find the results. Plotting the graphs of approximate solution for semi-flat, moderateslope and sharp-slope shores we showed the efficiency of the method.
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Список источников
1. Abdelsalam U. M. Traveling wave solutions for shallow water equations // Journal of Ocean Engineering and Science. 2017. Vol. 2. Iss. 1. P. 28-33. https://doi.org/10.1016/j.joes.2017.02.002.
2. Adeyemo O. D., Khalique C. M. Stability analysis, symmetry solutions and conserved currents of a two-dimensional extended shallow water wave equation of fluid mechanics // Partial Differential Equations in Applied Mathematics. 2021. Vol. 4. P. 100134. https://doi.org/10.1016/j.padiff.2021.100134.
3. Aydin B., Kânoglu U. New analytical solution for nonlinear shallow water-wave equations // Pure and Applied Geophysics. 2017. Vol. 174. P. 3209-3218. https://doi.org/10.1007/s00024-017-1508-z.
4. Bohacek J., Kharicha A., Ludwig A., Wu Menghuai. Shallow water model for horizontal centrifugal casting // Materials Science and Engineering: IOP Conference Series. 2012. Vol. 33. https://doi.org/10.1088/1757-899X/33/1/012032.
5. Bresch D., Desjardins B., Lin Chi-Kun. On some compressible fluid models: Korteweg, lubrication, and shallow water systems // Communications in Partial Differential Equations. 2003. Vol. 28. Iss. 3-4. P. 843-868. https://doi.org/10.1081/PDE-120020499.
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INFORMATION ABOUT THE AUTHORS
Laleh Noeiaghdam,
Postgraduate Student,
Department of Civil and Environmental Engineering, Amirkabir University of Technology, 350, Hafez Ave, Valiasr Square, Tehran, Iran 1591634311
Samad Noeiaghdam,
Senior Lecturer,
Industrial Mathematics Laboratory,
Baikal School of BRICS,
Irkutsk National Research Technical University,
83 Lermontov St., Irkutsk 664074, Russia;
Senior Researcher of the Department of Applied
Mathematics and Programming,
South Ural State University,
76, Lenin pr., Chelyabinsk 454080, Russia
ИНФОРМАЦИЯ ОБ АВТОРАХ
Нойягдам Лале,
аспирант,
факультет гражданского строительства и охраны окружающей среды,
Технологический университет Амиркабира, 1591634311, г. Тегеран, пр. Хафеза, 350, пл. Валиасра, Иран
Нойягдам Самад,
старший преподаватель,
Лаборатория промышленной математики,
Байкальский институт БРИКС,
Иркутский национальный исследовательский
технический университет,
664074, г. Иркутск, ул. Лермонтова, 83, Россия;
старший научный сотрудник кафедры
прикладной математики и программирования,
Южно-Уральский государственный университет,
454080, г. Челябинск, пр. Ленина, 76, Россия
Denis N. Sidorov,
Dr. Sci. (Physics & Mathematics), Professor of RAS,
Head of the Laboratory of Industrial Mathematics,
Baikal School of BRICS,
Irkutsk National Research Technical University;
83 Lermontov St., Irkutsk 664074, Russia;
Chief Researcher of the Department of Applied
Mathematics,
L. A. Melentiev Institute of Energy Systems of Siberian Branch of the Russian Academy of Sciences, 130, Lermontov St., Irkutsk 664033, Russia
Contribution of the authors
The authors contributed equally to this article.
Conflict of interests
The authors declare no conflicts of interests.
The final manuscript has been read and approved by all the co-authors.
Information about the article
The article was submitted 20.08.2021; approved after reviewing 29.09.2021; accepted for publication 22.10.2021.
Сидоров Денис Николаевич,
доктор физико-математических наук, профессор РАН, заведующий Лабораторией промышленной математики,
Байкальский институт БРИКС,
Иркутский национальный исследовательский
технический университет,
664074, г. Иркутск, ул. Лермонтова, 83, Россия;
главный научный сотрудник
Отдела прикладной математики,
Институт систем энергетики им. Л. А. Мелентьева
СО РАН,
664033, г. Иркутск, ул. Лермонтова, 130, Россия
Вклад авторов
Все авторы сделали эквивалентный вклад в подготовку публикации.
Конфликт интересов
Авторы заявляют об отсутствии конфликта интересов.
Все авторы прочитали и одобрили окончательный вариант рукописи.
Информация о статье
Статья поступила в редакцию 20.08.2021; одобрена после рецензирования 29.09.2021; принята к публикации 22.10.2021.
